Source: ProCGroups.FoxDifferential.Completed.Continuous.Universal.AugmentationQuotient
1import ProCGroups.FoxDifferential.Completed.Continuous.Universal.NaturalTopology
2import ProCGroups.FoxDifferential.Completed.ProCIntegerCoefficients.AugmentationIdeal.Kernel
4/-!
5# Fox differential: completed — continuous — universal — augmentation quotient
7The principal declarations in this module are:
9- `zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard`
10 The algebraic product \(I(\ker \psi)I(G)\) inside the algebraic standard source augmentation
11 ideal.
12- `zcCompletedGroupAlgebraStandardAugmentationIdealProjection`
13 Projection of the algebraic standard augmentation ideal to a finite augmentation stage.
14- `zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_generator_mem`
15 A generator \((n-1)s\) lies in the algebraic product \(I(\ker \psi)I(G)\).
16- `zcCompletedGroupAlgebraStandardAugmentationIdealProjection_val`
17 The value of the augmentation-ideal projection is the value of the corresponding finite-stage
18 projection.
19-/
21namespace FoxDifferential
23noncomputable section
25open ProCGroups.ProC
27universe u
29section KernelAugmentationQuotient
31variable (C : ProCGroups.FiniteGroupClass.{u})
32variable {G H : Type u}
33variable [Group G] [TopologicalSpace G] [IsTopologicalGroup G]
34variable [Group H] [TopologicalSpace H] [IsTopologicalGroup H]
36/--
37The algebraic product \(I(\ker \psi)I(G)\) inside the algebraic standard source augmentation
38ideal.
39-/
40def zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard
41 (psi : ContinuousMonoidHom G H) :
42 Submodule (ZCCompletedGroupAlgebra C G)
43 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :=
44 Submodule.span (ZCCompletedGroupAlgebra C G)
45 (Set.range fun p :
46 ProfiniteKernelSubgroup psi × zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
47 (zcGroupLike C G p.1.1 - 1) • p.2)
49omit [IsTopologicalGroup H] in
50/-- A generator \((n-1)s\) lies in the algebraic product \(I(\ker \psi)I(G)\). -/
51theorem zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_generator_mem
52 (psi : ContinuousMonoidHom G H) (n : ProfiniteKernelSubgroup psi)
53 (s : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
54 (zcGroupLike C G n.1 - 1) • s ∈
55 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :=
56 Submodule.subset_span (Set.mem_range_self (n, s))
58/-- Projection of the algebraic standard augmentation ideal to a finite augmentation stage. -/
59def zcCompletedGroupAlgebraStandardAugmentationIdealProjection
61 (i : ZCCompletedGroupAlgebraIndex C G) :
62 zcCompletedGroupAlgebraStandardAugmentationIdeal C G →
63 zcCompletedGroupAlgebraStageAugmentationIdeal C G i := by
64 intro x
65 let xAug : ZCCompletedGroupAlgebraAugmentationIdeal C G :=
66 ⟨x, zcCompletedGroupAlgebraStandardAugmentationIdeal_le_augmentationIdeal C G x.2⟩
67 exact zcCompletedGroupAlgebraAugmentationIdealProjection C G i xAug
69/--
70The value of the augmentation-ideal projection is the value of the corresponding finite-stage
71projection.
72-/
73@[simp]
74theorem zcCompletedGroupAlgebraStandardAugmentationIdealProjection_val
76 (i : ZCCompletedGroupAlgebraIndex C G)
77 (x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
78 ((zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x :
79 ZCCompletedGroupAlgebraStage C G i)) =
80 zcCompletedGroupAlgebraProjection C G i x :=
81 rfl
83/-- The standard augmentation-ideal projection is continuous. -/
84theorem continuous_zcCompletedGroupAlgebraStandardAugmentationIdealProjection
86 (i : ZCCompletedGroupAlgebraIndex C G) :
87 Continuous (zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i) := by
88 have hval : Continuous (fun x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
89 zcCompletedGroupAlgebraProjection C G i (x : ZCCompletedGroupAlgebra C G)) :=
90 (continuous_zcCompletedGroupAlgebraProjection C G i).comp continuous_subtype_val
91 exact Continuous.subtype_mk hval
92 (fun x => (zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x).2)
94/--
95On a finite coefficient stage, the identity crossed-differential boundary is a left inverse to
96the additive lift from the finite augmentation ideal.
97-/
98theorem zcCompletedGAStageAugmentationIdeal_identityBoundary_monoidAlgebraToIdentity
99 (i : ZCCompletedGroupAlgebraIndex C G)
100 (x : zcCompletedGroupAlgebraStageAugmentationIdeal C G i) :
101 identityCrossedDifferentialBoundary
102 (monoidAlgebraToIdentityCrossedDifferentialModule
103 (S := ModNCompletedCoeff i.1.modulus)
104 (G := CompletedGroupAlgebraQuotientInClass G C i.2)
105 (x : ZCCompletedGroupAlgebraStage C G i)) =
106 (x : ZCCompletedGroupAlgebraStage C G i) := by
107 letI : Fact (0 < i.1.modulus) := ⟨i.1.positive⟩
108 have hxaug :
109 (MonoidAlgebra.lift
110 (ModNCompletedCoeff i.1.modulus)
111 (ModNCompletedCoeff i.1.modulus)
112 (CompletedGroupAlgebraQuotientInClass G C i.2)
113 (1 : CompletedGroupAlgebraQuotientInClass G C i.2 →* ModNCompletedCoeff i.1.modulus))
114 (x : ZCCompletedGroupAlgebraStage C G i) = 0 := by
115 simpa [modNCompletedGroupAlgebraStageAugmentationInClass] using
116 (mem_zcCompletedGroupAlgebraStageAugmentationIdeal_iff
117 (C := C) (H := G) (i := i) (x := (x : ZCCompletedGroupAlgebraStage C G i))).1 x.2
118 exact
119 idCrossedDiffBoundary_monoidAlgebraToModule_of_augmentation_eq_zero
120 (S := ModNCompletedCoeff i.1.modulus)
121 (G := CompletedGroupAlgebraQuotientInClass G C i.2)
122 (a := (x : ZCCompletedGroupAlgebraStage C G i))
123 hxaug
125/-- The finite-stage augmentation-ideal projection is compatible with zero. -/
126@[simp]
127theorem zcCompletedGroupAlgebraStandardAugmentationIdealProjection_zero
129 (i : ZCCompletedGroupAlgebraIndex C G) :
130 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i
131 (0 : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) = 0 := by
132 apply Subtype.ext
133 simp only [zcCompletedGroupAlgebraStandardAugmentationIdealProjection_val, ZeroMemClass.coe_zero,
134 zcCompletedGroupAlgebraProjection_zero]
136/-- The finite-stage augmentation-ideal projection is compatible with addition. -/
137@[simp]
138theorem zcCompletedGroupAlgebraStandardAugmentationIdealProjection_add
140 (i : ZCCompletedGroupAlgebraIndex C G)
141 (x y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
142 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i (x + y) =
143 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x +
144 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i y := by
145 apply Subtype.ext
146 simp only [zcCompletedGroupAlgebraStandardAugmentationIdealProjection_val, Submodule.coe_add,
147 zcCompletedGroupAlgebraProjection_add]
149/-- The finite-stage augmentation-ideal projection is compatible with scalar multiplication. -/
150@[simp]
151theorem zcCompletedGroupAlgebraStandardAugmentationIdealProjection_smul
153 (i : ZCCompletedGroupAlgebraIndex C G)
154 (a : ZCCompletedGroupAlgebra C G)
155 (x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
156 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i (a • x) =
157 zcCompletedGroupAlgebraProjection C G i a •
158 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x := by
159 apply Subtype.ext
160 change zcCompletedGroupAlgebraProjection C G i (a * (x : ZCCompletedGroupAlgebra C G)) =
161 zcCompletedGroupAlgebraProjection C G i a *
162 zcCompletedGroupAlgebraProjection C G i (x : ZCCompletedGroupAlgebra C G)
163 rw [zcCompletedGroupAlgebraProjection_mul]
165/--
166The finite-stage projection of the standard augmentation ideal, as a semilinear map over the
167completed group-algebra projection.
168-/
169def zcCompletedGroupAlgebraStandardAugmentationIdealProjectionLinear
171 (i : ZCCompletedGroupAlgebraIndex C G) :
172 zcCompletedGroupAlgebraStandardAugmentationIdeal C G →ₛₗ[
173 zcCompletedGroupAlgebraProjectionRingHom C G i]
174 zcCompletedGroupAlgebraStageAugmentationIdeal C G i where
175 toFun := zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i
176 map_add' := by
177 intro x y
178 exact zcCompletedGroupAlgebraStandardAugmentationIdealProjection_add C i x y
179 map_smul' := by
180 intro a x
181 exact zcCompletedGroupAlgebraStandardAugmentationIdealProjection_smul C i a x
183/--
184The linear finite-stage projection of the standard completed augmentation ideal has the same value
185as its underlying projection.
186-/
187@[simp]
188theorem zcCompletedGroupAlgebraStandardAugmentationIdealProjectionLinear_apply
190 (i : ZCCompletedGroupAlgebraIndex C G)
191 (x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
192 zcCompletedGroupAlgebraStandardAugmentationIdealProjectionLinear C i x =
193 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x :=
194 rfl
196/-- The finite-stage augmentation-ideal projection is compatible with subtraction. -/
197@[simp]
198theorem zcCompletedGroupAlgebraStandardAugmentationIdealProjection_sub
200 (i : ZCCompletedGroupAlgebraIndex C G)
201 (x y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
202 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i (x - y) =
203 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x -
204 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i y := by
205 simpa using
206 map_sub (zcCompletedGroupAlgebraStandardAugmentationIdealProjectionLinear C i) x y
208/-- The product of all finite-stage projections of the standard completed augmentation ideal. -/
209def zcCompletedGroupAlgebraStandardAugmentationIdealProjectionProduct
211 zcCompletedGroupAlgebraStandardAugmentationIdeal C G →
212 ∀ i : ZCCompletedGroupAlgebraIndex C G,
213 zcCompletedGroupAlgebraStageAugmentationIdeal C G i :=
214 fun x i => zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x
216/--
217Evaluating the product of standard augmentation-ideal projections at `i` returns the projection at
218`i`.
219-/
220@[simp]
221theorem zcCompletedGroupAlgebraStandardAugmentationIdealProjectionProduct_apply
223 (x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
224 (i : ZCCompletedGroupAlgebraIndex C G) :
225 zcCompletedGroupAlgebraStandardAugmentationIdealProjectionProduct C x i =
226 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x :=
227 rfl
229/-- Finite-stage projections separate points of the standard completed augmentation ideal. -/
230theorem zcCompletedGroupAlgebraStandardAugmentationIdealProjectionProduct_injective
232 Function.Injective
233 (zcCompletedGroupAlgebraStandardAugmentationIdealProjectionProduct C (G := G)) := by
234 intro x y hxy
235 apply Subtype.ext
236 apply Subtype.ext
237 funext i
238 exact congrArg Subtype.val (congrFun hxy i)
240/--
241Extensionality for standard completed augmentation ideal elements by finite-stage projections.
242-/
243theorem zcCompletedGroupAlgebraStandardAugmentationIdealProjection_ext
245 {x y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G}
246 (h : ∀ i : ZCCompletedGroupAlgebraIndex C G,
247 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x =
248 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i y) :
249 x = y :=
250 zcCompletedGroupAlgebraStandardAugmentationIdealProjectionProduct_injective C
251 (by
252 funext i
253 exact h i)
255/--
256Every finite standard augmentation stage is hit by the completed standard augmentation ideal
257projection.
258-/
259theorem zcCompletedGroupAlgebraStandardAugmentationIdealProjection_surjective
261 (i : ZCCompletedGroupAlgebraIndex C G) :
262 Function.Surjective
263 (zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i) := by
264 intro x
265 rcases zcCompletedGroupAlgebraStageAugmentationIdeal_mem_projection_standard
266 (C := C) (H := G) i x with
267 ⟨y, hy, hproj⟩
268 refine ⟨⟨y, hy⟩, ?_⟩
269 apply Subtype.ext
270 simpa [zcCompletedGroupAlgebraStandardAugmentationIdealProjection_val] using hproj
272/--
273The finite-stage product \(I(\ker)\,I(G/U)\) attached to the open-image quotient at a source
274stage.
275-/
276def zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
278 (hForm : ProCGroups.FiniteGroupClass.Formation C)
279 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
280 (hfopen : IsOpenMap psi)
281 (i : ZCCompletedGroupAlgebraIndex C G) :
282 Submodule (ZCCompletedGroupAlgebraStage C G i)
283 (zcCompletedGroupAlgebraStageAugmentationIdeal C G i) :=
284 Submodule.span (ZCCompletedGroupAlgebraStage C G i)
285 (Set.range fun p :
286 (zcCompletedGroupAlgebraOpenImageQuotientMap C hC hForm psi hpsi hfopen i).ker ×
287 zcCompletedGroupAlgebraStageAugmentationIdeal C G i =>
288 (MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
289 (CompletedGroupAlgebraQuotientInClass G C i.2) p.1.1 - 1) • p.2)
291/-- A finite-stage product generator belongs to the finite-stage product submodule. -/
292theorem zcCompletedGAOpenImageKernelAugmentationIdealMulStageStandard_generator_mem
295 (hForm : ProCGroups.FiniteGroupClass.Formation C)
296 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
297 (hfopen : IsOpenMap psi)
298 (i : ZCCompletedGroupAlgebraIndex C G)
299 (q : (zcCompletedGroupAlgebraOpenImageQuotientMap C hC hForm psi hpsi hfopen i).ker)
300 (s : zcCompletedGroupAlgebraStageAugmentationIdeal C G i) :
301 (MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
302 (CompletedGroupAlgebraQuotientInClass G C i.2) q.1 - 1) • s ∈
303 zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
304 C hC hForm psi hpsi hfopen i :=
305 Submodule.subset_span (Set.mem_range_self (q, s))
307/-- An actual kernel element determines a class in the finite-stage open-image kernel. -/
308def zcCompletedGroupAlgebraOpenImageKernelClass
310 (hForm : ProCGroups.FiniteGroupClass.Formation C)
311 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
312 (hfopen : IsOpenMap psi)
313 (i : ZCCompletedGroupAlgebraIndex C G)
314 (n : ProfiniteKernelSubgroup psi) :
315 (zcCompletedGroupAlgebraOpenImageQuotientMap C hC hForm psi hpsi hfopen i).ker := by
316 refine ⟨QuotientGroup.mk'
317 ((((OrderDual.ofDual i.2).1 : OpenNormalSubgroup G) : Subgroup G)) n.1, ?_⟩
318 rw [MonoidHom.mem_ker]
319 rw [zcCompletedGroupAlgebraOpenImageQuotientMap_mk]
320 change QuotientGroup.mk'
321 ((((OrderDual.ofDual
322 (zcCompletedGroupAlgebraOpenImageIndexInClass C hForm psi hpsi hfopen i)).1 :
323 OpenNormalSubgroup H) : Subgroup H)) (psi n.1) = 1
324 rw [show psi n.1 = 1 from n.2]
325 simp only [QuotientGroup.mk'_apply, QuotientGroup.mk_one]
327/--
328The open-image kernel class has the expected underlying subgroup, characterized by finite-stage
329Fox coordinate formulas.
330-/
331@[simp]
332theorem zcCompletedGroupAlgebraOpenImageKernelClass_val
335 (hForm : ProCGroups.FiniteGroupClass.Formation C)
336 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
337 (hfopen : IsOpenMap psi)
338 (i : ZCCompletedGroupAlgebraIndex C G)
339 (n : ProfiniteKernelSubgroup psi) :
340 (zcCompletedGroupAlgebraOpenImageKernelClass C hC hForm psi hpsi hfopen i n).1 =
341 QuotientGroup.mk'
342 ((((OrderDual.ofDual i.2).1 : OpenNormalSubgroup G) : Subgroup G)) n.1 :=
343 rfl
345/-- The finite-stage augmentation-ideal projection is compatible with scalar multiplication. -/
346@[simp 900]
347theorem zcCompletedGroupAlgebraStandardAugmentationIdealProjection_kernel_generator_smul
350 (hForm : ProCGroups.FiniteGroupClass.Formation C)
351 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
352 (hfopen : IsOpenMap psi)
353 (i : ZCCompletedGroupAlgebraIndex C G)
354 (n : ProfiniteKernelSubgroup psi)
355 (s : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
356 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i
357 ((zcGroupLike C G n.1 - 1) • s) =
358 (MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
359 (CompletedGroupAlgebraQuotientInClass G C i.2)
360 (zcCompletedGroupAlgebraOpenImageKernelClass
361 C hC hForm psi hpsi hfopen i n).1 - 1) •
362 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i s := by
363 apply Subtype.ext
364 change zcCompletedGroupAlgebraProjection C G i
365 ((zcGroupLike C G n.1 - 1) * (s : ZCCompletedGroupAlgebra C G)) =
366 (MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
367 (CompletedGroupAlgebraQuotientInClass G C i.2)
368 (QuotientGroup.mk'
369 ((((OrderDual.ofDual i.2).1 : OpenNormalSubgroup G) : Subgroup G)) n.1) - 1) *
370 zcCompletedGroupAlgebraProjection C G i (s : ZCCompletedGroupAlgebra C G)
371 rw [zcCompletedGroupAlgebraProjection_mul]
372 simp only [zcCompletedGroupAlgebraProjection_sub, zcCompletedGroupAlgebraProjection_groupLike,
373 MonoidAlgebra.of_apply, zcCompletedGroupAlgebraProjection_one, QuotientGroup.mk'_apply]
375/--
376Projecting an element of \(I(\ker\psi)I(G)\) to an open-image finite stage lands in the
377corresponding finite-stage kernel-augmentation product.
378-/
379theorem zcCompletedGAKernelAugmentationIdealMulStandard_proj_mem_openImageStage
382 (hForm : ProCGroups.FiniteGroupClass.Formation C)
383 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
384 (hfopen : IsOpenMap psi)
385 (i : ZCCompletedGroupAlgebraIndex C G)
386 {x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G}
387 (hx : x ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi) :
388 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x ∈
389 zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
390 C hC hForm psi hpsi hfopen i := by
391 let T :=
392 zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
393 C hC hForm psi hpsi hfopen i
394 refine Submodule.span_induction
395 (p := fun x _ => zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x ∈ T)
396 ?_ ?_ ?_ ?_ hx
397 · rintro _ ⟨p, rfl⟩
398 rcases p with ⟨n, s⟩
399 rw [zcCompletedGroupAlgebraStandardAugmentationIdealProjection_kernel_generator_smul
400 C hC hForm psi hpsi hfopen i n s]
401 exact
402 zcCompletedGAOpenImageKernelAugmentationIdealMulStageStandard_generator_mem
403 C hC hForm psi hpsi hfopen i
404 (zcCompletedGroupAlgebraOpenImageKernelClass C hC hForm psi hpsi hfopen i n)
405 (zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i s)
406 · simp only [zcCompletedGroupAlgebraStandardAugmentationIdealProjection_zero, zero_mem, T]
407 · intro x y _ _ hx hy
408 simpa [T] using T.add_mem hx hy
409 · intro a x _ hx
410 rw [zcCompletedGroupAlgebraStandardAugmentationIdealProjection_smul]
411 exact T.smul_mem (zcCompletedGroupAlgebraProjection C G i a) hx
413/--
414Every element of the finite-stage open-image kernel-augmentation product is the projection of an
415element of the algebraic product \(I(\ker\psi)I(G)\).
416-/
417theorem zcCompletedGAOpenImageKernelAugmentationIdealMulStageStandard_mem_proj
420 (hForm : ProCGroups.FiniteGroupClass.Formation C)
421 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
422 (hfopen : IsOpenMap psi)
423 (i : ZCCompletedGroupAlgebraIndex C G)
424 (x : zcCompletedGroupAlgebraStageAugmentationIdeal C G i)
425 (hx :
426 x ∈ zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
427 C hC hForm psi hpsi hfopen i) :
428 ∃ y ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi,
429 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i y = x := by
430 let T :=
431 zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
432 C hC hForm psi hpsi hfopen i
433 let P : zcCompletedGroupAlgebraStageAugmentationIdeal C G i → Prop := fun x =>
434 ∃ y ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi,
435 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i y = x
436 refine Submodule.span_induction (p := fun x _ => P x) ?_ ?_ ?_ ?_ hx
437 · rintro _ ⟨p, rfl⟩
438 rcases p with ⟨q, s⟩
439 rcases
440 zcCompletedGroupAlgebraOpenImageQuotientMap_kernel_lift
441 C hC hForm psi hpsi hfopen i q with
442 ⟨n, hn⟩
443 rcases
444 zcCompletedGroupAlgebraStageAugmentationIdeal_mem_projection_standard
445 (C := C) (H := G) i s with
446 ⟨s', hs', hs'proj⟩
447 let sStd : zcCompletedGroupAlgebraStandardAugmentationIdeal C G := ⟨s', hs'⟩
448 refine ⟨(zcGroupLike C G n.1 - 1) • sStd,
449 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_generator_mem C psi n sStd, ?_⟩
450 rw [zcCompletedGroupAlgebraStandardAugmentationIdealProjection_kernel_generator_smul
451 C hC hForm psi hpsi hfopen i n sStd]
452 apply Subtype.ext
453 change
454 (MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
455 (CompletedGroupAlgebraQuotientInClass G C i.2)
456 (QuotientGroup.mk'
457 ((((OrderDual.ofDual i.2).1 : OpenNormalSubgroup G) : Subgroup G)) n.1) - 1) *
458 zcCompletedGroupAlgebraProjection C G i s' =
459 (MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
460 (CompletedGroupAlgebraQuotientInClass G C i.2) q.1 - 1) * s.1
461 rw [hn, hs'proj]
462 · refine ⟨0, (zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi).zero_mem, ?_⟩
463 simp only [zcCompletedGroupAlgebraStandardAugmentationIdealProjection_zero]
464 · intro x y _ _ hx hy
465 rcases hx with ⟨x', hx'mem, hx'proj⟩
466 rcases hy with ⟨y', hy'mem, hy'proj⟩
467 refine ⟨x' + y',
468 (zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi).add_mem hx'mem hy'mem, ?_⟩
469 simp only [zcCompletedGroupAlgebraStandardAugmentationIdealProjection_add, hx'proj, hy'proj]
470 · intro a x _ hx
471 rcases hx with ⟨x', hx'mem, hx'proj⟩
472 rcases zcCompletedGroupAlgebraProjection_surjective C G i a with ⟨a', ha'⟩
473 refine ⟨a' • x',
474 (zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi).smul_mem a' hx'mem, ?_⟩
475 rw [zcCompletedGroupAlgebraStandardAugmentationIdealProjection_smul, ha', hx'proj]
477/--
478The closed finite-stage hull of \(I(\ker \psi)I(G)\) inside the standard source augmentation
479ideal: an element belongs exactly when every finite projection belongs to the finite-stage
480open-image kernel product.
481-/
482def zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
485 (hForm : ProCGroups.FiniteGroupClass.Formation C)
486 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
487 (hfopen : IsOpenMap psi) :
488 Submodule (ZCCompletedGroupAlgebra C G)
489 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G) where
490 carrier := {x | ∀ i : ZCCompletedGroupAlgebraIndex C G,
491 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x ∈
492 zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
493 C hC hForm psi hpsi hfopen i}
494 zero_mem' := by
495 intro i
496 simp only [zcCompletedGroupAlgebraStandardAugmentationIdealProjection_zero, zero_mem]
497 add_mem' := by
498 intro x y hx hy i
499 simpa using
500 (zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
501 C hC hForm psi hpsi hfopen i).add_mem (hx i) (hy i)
502 smul_mem' := by
503 intro a x hx i
504 rw [zcCompletedGroupAlgebraStandardAugmentationIdealProjection_smul]
505 exact
506 (zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
507 C hC hForm psi hpsi hfopen i).smul_mem
508 (zcCompletedGroupAlgebraProjection C G i a) (hx i)
510/-- The algebraic product is contained in its finite-stage closed hull. -/
511theorem zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_le_closed
514 (hForm : ProCGroups.FiniteGroupClass.Formation C)
515 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
516 (hfopen : IsOpenMap psi) :
517 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi ≤
518 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
519 C hC hForm psi hpsi hfopen := by
520 intro x hx i
521 exact
522 zcCompletedGAKernelAugmentationIdealMulStandard_proj_mem_openImageStage
523 C hC hForm psi hpsi hfopen i hx
525/-- The finite-stage hull is closed in the standard source augmentation ideal. -/
526theorem isClosed_zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
529 (hForm : ProCGroups.FiniteGroupClass.Formation C)
530 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
531 (hfopen : IsOpenMap psi) :
532 IsClosed
533 ((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
534 C hC hForm psi hpsi hfopen :
535 Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G))) := by
536 change IsClosed {x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G |
537 ∀ i : ZCCompletedGroupAlgebraIndex C G,
538 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x ∈
539 zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
540 C hC hForm psi hpsi hfopen i}
541 simp only [Set.setOf_forall]
542 refine isClosed_iInter ?_
543 intro i
544 haveI : DiscreteTopology (zcCompletedGroupAlgebraStageAugmentationIdeal C G i) := by
545 infer_instance
546 exact
547 (isClosed_discrete
548 ((zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
549 C hC hForm psi hpsi hfopen i :
550 Set (zcCompletedGroupAlgebraStageAugmentationIdeal C G i)))).preimage
551 (continuous_zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i)
553/-- The closure of the algebraic product is contained in the finite-stage closed hull. -/
554theorem closure_zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_le_closed
557 (hForm : ProCGroups.FiniteGroupClass.Formation C)
558 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
559 (hfopen : IsOpenMap psi) :
560 closure
561 ((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :
562 Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G))) ⊆
563 (zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
564 C hC hForm psi hpsi hfopen :
565 Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)) := by
566 exact
567 closure_minimal
568 (by
569 intro x hx
570 exact
571 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_le_closed
572 C hC hForm psi hpsi hfopen hx)
573 (isClosed_zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
574 C hC hForm psi hpsi hfopen)
576/-- The finite-stage closed hull is contained in the closure of the algebraic product. -/
577theorem zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed_le_closure
580 (hForm : ProCGroups.FiniteGroupClass.Formation C)
581 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
582 (hfopen : IsOpenMap psi) :
583 (zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
584 C hC hForm psi hpsi hfopen :
585 Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)) ⊆
586 closure
587 ((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :
588 Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G))) := by
589 intro x hx
590 let R := ZCCompletedGroupAlgebra C G
591 let Ssys := zcCompletedGroupAlgebraSystem C G
592 let Ystd : Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :=
593 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi
594 let Yamb : Set R := Subtype.val '' Ystd
595 have hxAmb : (x : R) ∈ closure Yamb := by
596 letI : Nonempty (ZCCompletedGroupAlgebraIndex C G) :=
597 ⟨(ProCGroups.Completion.ProCIntegerIndex.terminal (C := C) inferInstance,
598 zcCompletedGroupAlgebraTopIndex C G)⟩
599 letI : ∀ i : ZCCompletedGroupAlgebraIndex C G, TopologicalSpace (Ssys.X i) :=
600 Ssys.topologicalSpace
601 letI : ∀ i : ZCCompletedGroupAlgebraIndex C G, CompactSpace (Ssys.X i) := fun i => by
602 dsimp [Ssys, zcCompletedGroupAlgebraSystem]
603 change @CompactSpace (ZCCompletedGroupAlgebraStage C G i) ⊥
604 letI : Fact (0 < i.1.modulus) := ⟨i.1.positive⟩
605 letI : Finite (ZCCompletedGroupAlgebraStage C G i) :=
606 finite_modNCompletedGroupAlgebraStageInClass
607 (n := i.1.modulus) (G := G) C i.2
608 exact Finite.compactSpace
609 letI : ∀ i : ZCCompletedGroupAlgebraIndex C G, T2Space (Ssys.X i) := fun i => by
610 dsimp [Ssys, zcCompletedGroupAlgebraSystem]
611 change @T2Space (ZCCompletedGroupAlgebraStage C G i) ⊥
612 exact @DiscreteTopology.toT2Space _ ⊥ ⟨rfl⟩
613 have hdir : Directed (· ≤ ·) (id : ZCCompletedGroupAlgebraIndex C G →
614 ZCCompletedGroupAlgebraIndex C G) :=
615 directed_zcCompletedGroupAlgebraIndex_of_formation (C := C) (H := G) hForm
616 have hxLim :
617 (show Ssys.inverseLimit from (x : R)) ∈
618 closure (show Set Ssys.inverseLimit from Yamb) := by
619 rw [Ssys.mem_isClosed_iff_forall_projection_mem hdir isClosed_closure]
620 intro i
621 rcases
622 zcCompletedGAOpenImageKernelAugmentationIdealMulStageStandard_mem_proj
623 C hC hForm psi hpsi hfopen i
624 (zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x)
625 (hx i) with
626 ⟨y, hy, hyproj⟩
627 refine ⟨(y : R), subset_closure ?_, ?_⟩
628 · exact ⟨y, by simpa [Ystd] using hy, rfl⟩
629 · simpa [Ssys, zcCompletedGroupAlgebraSystem,
630 zcCompletedGroupAlgebraStandardAugmentationIdealProjection_val] using
631 congrArg Subtype.val hyproj
632 convert hxLim using 1 <;> rfl
633 have hclosure :
634 closure Ystd =
635 (Subtype.val : zcCompletedGroupAlgebraStandardAugmentationIdeal C G → R) ⁻¹'
636 closure Yamb := by
637 exact Topology.IsEmbedding.subtypeVal.closure_eq_preimage_closure_image Ystd
638 change x ∈ closure Ystd
639 rw [hclosure]
640 exact hxAmb
642/-- The closure of \(I(\ker \psi)I(G)\) is exactly the finite-stage kernel-product condition. -/
643theorem closure_zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_eq_closed
646 (hForm : ProCGroups.FiniteGroupClass.Formation C)
647 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
648 (hfopen : IsOpenMap psi) :
649 closure
650 ((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :
651 Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G))) =
652 (zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
653 C hC hForm psi hpsi hfopen :
654 Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)) := by
655 exact Set.Subset.antisymm
656 (closure_zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_le_closed
657 C hC hForm psi hpsi hfopen)
658 (zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed_le_closure
659 C hC hForm psi hpsi hfopen)
661/--
662The algebraic product is closed exactly when it already equals the finite-stage closed hull.
663This is the precise non-circular closedness frontier.
664-/
665theorem isClosed_zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_iff_eq_closed
666 {C : ProCGroups.FiniteGroupClass.{u}}
669 (hForm : ProCGroups.FiniteGroupClass.Formation C)
670 {psi : ContinuousMonoidHom G H} (hpsi : Function.Surjective psi)
671 (hfopen : IsOpenMap psi) :
672 IsClosed
673 ((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :
674 Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G))) ↔
675 (zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :
676 Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)) =
677 (zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
678 C hC hForm psi hpsi hfopen :
679 Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)) := by
680 constructor
681 · intro hclosed
682 apply Set.Subset.antisymm
683 · exact zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_le_closed
684 C hC hForm psi hpsi hfopen
685 · intro x hx
686 have hxclosure :
687 x ∈ closure
688 ((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :
689 Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G))) := by
690 rw [closure_zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_eq_closed
691 C hC hForm psi hpsi hfopen]
692 exact hx
693 rwa [hclosed.closure_eq] at hxclosure
694 · intro hEq
695 rw [hEq]
696 exact isClosed_zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
697 C hC hForm psi hpsi hfopen
700/--
701The source-side augmentation quotient \(I\mathbb{Z}_C\llbracket G\rrbracket / I(\ker \psi)
702I\mathbb{Z}_C\llbracket G\rrbracket\), before descending scalars to \(\mathbb{Z}_C\llbracket
703H\rrbracket\).
704-/
705abbrev KernelAugmentationIdealQuotient
706 (psi : ContinuousMonoidHom G H) : Type u :=
707 zcCompletedGroupAlgebraStandardAugmentationIdeal C G ⧸
708 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi
710/--
711The closed source-side augmentation quotient, using the finite-stage closed hull of \(I(\ker
712\psi)I(G)\) as denominator.
713-/
714abbrev KernelAugmentationIdealClosedQuotient
717 (hForm : ProCGroups.FiniteGroupClass.Formation C)
718 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
719 (hfopen : IsOpenMap psi) : Type u :=
720 zcCompletedGroupAlgebraStandardAugmentationIdeal C G ⧸
721 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
722 C hC hForm psi hpsi hfopen
724/-- The quotient map to the closed source augmentation quotient is a quotient map. -/
725theorem isQuotientMap_kernelAugmentationIdealClosedQuotient_mkQ
728 (hForm : ProCGroups.FiniteGroupClass.Formation C)
729 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
730 (hfopen : IsOpenMap psi) :
731 Topology.IsQuotientMap
732 (zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
733 C hC hForm psi hpsi hfopen).mkQ := by
734 rw [Topology.isQuotientMap_iff]
735 constructor
736 · exact ⟨rfl⟩
737 · exact
738 Submodule.Quotient.mk_surjective
739 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
740 C hC hForm psi hpsi hfopen)
742/--
743Continuity out of the closed source augmentation quotient can be tested after precomposing with
744the quotient map from the standard augmentation ideal.
745-/
746theorem continuous_kernelAugmentationIdealClosedQuotient_iff_comp_mkQ
747 {C : ProCGroups.FiniteGroupClass.{u}}
750 {hForm : ProCGroups.FiniteGroupClass.Formation C}
751 {psi : ContinuousMonoidHom G H} {hpsi : Function.Surjective psi}
752 {hfopen : IsOpenMap psi}
753 {A : Type u} [TopologicalSpace A]
754 {f : KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen → A} :
755 Continuous f ↔
756 Continuous (fun x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
757 f ((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
758 C hC hForm psi hpsi hfopen).mkQ x)) := by
759 simpa [Function.comp_def] using
760 (isQuotientMap_kernelAugmentationIdealClosedQuotient_mkQ
761 C hC hForm psi hpsi hfopen).continuous_iff (g := f)
763/-- The closed source augmentation quotient is \(T_1\) for the quotient topology. -/
764theorem t1Space_kernelAugmentationIdealClosedQuotient
767 (hForm : ProCGroups.FiniteGroupClass.Formation C)
768 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
769 (hfopen : IsOpenMap psi) :
770 T1Space (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) := by
771 letI : IsClosed
772 ((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
773 C hC hForm psi hpsi hfopen :
774 Submodule (ZCCompletedGroupAlgebra C G)
775 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)) :
776 Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)) :=
777 isClosed_zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
778 C hC hForm psi hpsi hfopen
779 infer_instance
781/-- The zero class is closed in the closed source augmentation quotient. -/
782theorem isClosed_zero_kernelAugmentationIdealClosedQuotient
785 (hForm : ProCGroups.FiniteGroupClass.Formation C)
786 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
787 (hfopen : IsOpenMap psi) :
788 IsClosed
789 ({0} : Set
790 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)) := by
791 letI : T1Space
792 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
793 t1Space_kernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen
794 exact isClosed_singleton
796/-- The finite-stage quotient of the source augmentation ideal by the open-image kernel product. -/
797abbrev KernelAugmentationIdealClosedStageQuotient
800 (hForm : ProCGroups.FiniteGroupClass.Formation C)
801 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
802 (hfopen : IsOpenMap psi)
803 (i : ZCCompletedGroupAlgebraIndex C G) : Type u :=
804 zcCompletedGroupAlgebraStageAugmentationIdeal C G i ⧸
805 zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
806 C hC hForm psi hpsi hfopen i
808/-- The finite-stage coordinate of the closed source augmentation quotient. -/
809def kernelAugmentationIdealClosedQuotientStageProjection
812 (hForm : ProCGroups.FiniteGroupClass.Formation C)
813 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
814 (hfopen : IsOpenMap psi)
815 (i : ZCCompletedGroupAlgebraIndex C G) :
816 KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen →ₛₗ[
817 zcCompletedGroupAlgebraProjectionRingHom C G i]
818 KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i :=
819 (zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
820 C hC hForm psi hpsi hfopen).mapQ
821 (zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
822 C hC hForm psi hpsi hfopen i)
823 (zcCompletedGroupAlgebraStandardAugmentationIdealProjectionLinear C i)
824 (by
825 intro x hx
826 exact hx i)
828/--
829The stage projection of a closed kernel-augmentation quotient class is represented by the
830corresponding finite-stage projection.
831-/
832@[simp 900]
833theorem kernelAugmentationIdealClosedQuotientStageProjection_mk
836 (hForm : ProCGroups.FiniteGroupClass.Formation C)
837 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
838 (hfopen : IsOpenMap psi)
839 (i : ZCCompletedGroupAlgebraIndex C G)
840 (x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
841 kernelAugmentationIdealClosedQuotientStageProjection
842 C hC hForm psi hpsi hfopen i
843 ((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
844 C hC hForm psi hpsi hfopen).mkQ x) =
845 ((zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
846 C hC hForm psi hpsi hfopen i).mkQ
847 (zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x) :
848 KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) := by
849 exact
850 Submodule.mapQ_apply
851 (zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
852 C hC hForm psi hpsi hfopen)
853 (zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
854 C hC hForm psi hpsi hfopen i)
855 (zcCompletedGroupAlgebraStandardAugmentationIdealProjectionLinear C i)
856 x
858/-- Each finite-stage coordinate of the closed source augmentation quotient is continuous. -/
859theorem continuous_kernelAugmentationIdealClosedQuotientStageProjection
862 (hForm : ProCGroups.FiniteGroupClass.Formation C)
863 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
864 (hfopen : IsOpenMap psi)
865 (i : ZCCompletedGroupAlgebraIndex C G) :
866 Continuous
867 (kernelAugmentationIdealClosedQuotientStageProjection
868 C hC hForm psi hpsi hfopen i) := by
869 rw [continuous_kernelAugmentationIdealClosedQuotient_iff_comp_mkQ
870 (C := C) (G := G) (H := H) (hC := hC) (hForm := hForm)
871 (psi := psi) (hpsi := hpsi) (hfopen := hfopen)]
872 have hproj :
873 Continuous (zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i) :=
874 continuous_zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i
875 have hq :
876 Continuous (fun y : zcCompletedGroupAlgebraStageAugmentationIdeal C G i =>
877 ((zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
878 C hC hForm psi hpsi hfopen i).mkQ y :
879 KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i)) :=
880 continuous_quotient_mk'
881 have hcomp := hq.comp hproj
882 have hfun :
883 ((fun y : zcCompletedGroupAlgebraStageAugmentationIdeal C G i =>
884 ((zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
885 C hC hForm psi hpsi hfopen i).mkQ y :
886 KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i)) ∘
887 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i) =
888 (fun x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
889 kernelAugmentationIdealClosedQuotientStageProjection
890 C hC hForm psi hpsi hfopen i
891 ((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
892 C hC hForm psi hpsi hfopen).mkQ x)) := by
893 funext x
894 exact
895 (kernelAugmentationIdealClosedQuotientStageProjection_mk
896 C hC hForm psi hpsi hfopen i x).symm
897 exact hfun ▸ hcomp
899/-- The product of all finite-stage coordinates of the closed source augmentation quotient. -/
900def kernelAugmentationIdealClosedQuotientStageProjectionProduct
903 (hForm : ProCGroups.FiniteGroupClass.Formation C)
904 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
905 (hfopen : IsOpenMap psi) :
906 KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen →
907 ∀ i : ZCCompletedGroupAlgebraIndex C G,
908 KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i :=
909 fun x i => kernelAugmentationIdealClosedQuotientStageProjection
910 C hC hForm psi hpsi hfopen i x
912/--
913Evaluating the product projection of the closed kernel-augmentation quotient at `i` returns its
914stage projection at `i`.
915-/
916@[simp]
917theorem kernelAugmentationIdealClosedQuotientStageProjectionProduct_apply
920 (hForm : ProCGroups.FiniteGroupClass.Formation C)
921 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
922 (hfopen : IsOpenMap psi)
923 (x : KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
924 (i : ZCCompletedGroupAlgebraIndex C G) :
925 kernelAugmentationIdealClosedQuotientStageProjectionProduct
926 C hC hForm psi hpsi hfopen x i =
927 kernelAugmentationIdealClosedQuotientStageProjection
928 C hC hForm psi hpsi hfopen i x :=
929 rfl
931/-- The finite-stage coordinate product of the closed source augmentation quotient is continuous. -/
932theorem continuous_kernelAugmentationIdealClosedQuotientStageProjectionProduct
935 (hForm : ProCGroups.FiniteGroupClass.Formation C)
936 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
937 (hfopen : IsOpenMap psi) :
938 Continuous
939 (kernelAugmentationIdealClosedQuotientStageProjectionProduct
940 C hC hForm psi hpsi hfopen) := by
941 exact continuous_pi fun i =>
942 continuous_kernelAugmentationIdealClosedQuotientStageProjection
943 C hC hForm psi hpsi hfopen i
945/-- Finite-stage coordinates separate points in the closed source augmentation quotient. -/
946theorem kernelAugmentationIdealClosedQuotientStageProjectionProduct_injective
949 (hForm : ProCGroups.FiniteGroupClass.Formation C)
950 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
951 (hfopen : IsOpenMap psi) :
952 Function.Injective
953 (kernelAugmentationIdealClosedQuotientStageProjectionProduct
954 C hC hForm psi hpsi hfopen) := by
955 let S :=
956 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
957 C hC hForm psi hpsi hfopen
958 intro qx qy hxy
959 revert qy
960 refine Submodule.Quotient.induction_on (p := S) qx ?_
961 intro x qy hxy
962 revert hxy
963 refine Submodule.Quotient.induction_on (p := S) qy ?_
964 intro y hxy
965 apply (Submodule.Quotient.eq S).2
966 change x - y ∈ S
967 intro i
968 let T :=
969 zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
970 C hC hForm psi hpsi hfopen i
971 have hi :
972 kernelAugmentationIdealClosedQuotientStageProjection
973 C hC hForm psi hpsi hfopen i (Submodule.Quotient.mk (p := S) x) =
974 kernelAugmentationIdealClosedQuotientStageProjection
975 C hC hForm psi hpsi hfopen i (Submodule.Quotient.mk (p := S) y) := by
976 exact congrFun hxy i
977 have hstage :
978 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x -
979 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i y ∈ T := by
980 apply (Submodule.Quotient.eq T).1
981 change
982 kernelAugmentationIdealClosedQuotientStageProjection
983 C hC hForm psi hpsi hfopen i (S.mkQ x) =
984 kernelAugmentationIdealClosedQuotientStageProjection
985 C hC hForm psi hpsi hfopen i (S.mkQ y) at hi
986 rw [kernelAugmentationIdealClosedQuotientStageProjection_mk,
987 kernelAugmentationIdealClosedQuotientStageProjection_mk] at hi
988 exact hi
989 simpa [S, T] using hstage
991/-- Extensionality for the closed source augmentation quotient by finite-stage coordinates. -/
992theorem kernelAugmentationIdealClosedQuotientStageProjection_ext
995 (hForm : ProCGroups.FiniteGroupClass.Formation C)
996 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
997 (hfopen : IsOpenMap psi)
998 {x y : KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen}
999 (h : ∀ i : ZCCompletedGroupAlgebraIndex C G,
1000 kernelAugmentationIdealClosedQuotientStageProjection
1001 C hC hForm psi hpsi hfopen i x =
1002 kernelAugmentationIdealClosedQuotientStageProjection
1003 C hC hForm psi hpsi hfopen i y) :
1004 x = y := by
1005 exact
1006 kernelAugmentationIdealClosedQuotientStageProjectionProduct_injective
1007 C hC hForm psi hpsi hfopen
1008 (by
1009 funext i
1010 exact h i)
1012/--
1013The quotient topology on the closed source augmentation quotient is exactly the topology induced
1014by all finite closed-quotient coordinates.
1015-/
1016theorem kernelAugmentationIdealClosedQuotient_topology_eq_induced_stageProjProduct
1019 (hForm : ProCGroups.FiniteGroupClass.Formation C)
1020 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
1021 (hfopen : IsOpenMap psi) :
1022 (inferInstance :
1023 TopologicalSpace
1024 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)) =
1025 TopologicalSpace.induced
1026 (kernelAugmentationIdealClosedQuotientStageProjectionProduct
1027 C hC hForm psi hpsi hfopen) inferInstance := by
1028 let Sclosed :=
1029 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
1030 C hC hForm psi hpsi hfopen
1031 let Q := KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen
1032 let stageProduct :=
1033 kernelAugmentationIdealClosedQuotientStageProjectionProduct
1034 C hC hForm psi hpsi hfopen
1035 ext U
1036 constructor
1037 · intro hU
1038 let Tind : TopologicalSpace Q :=
1039 TopologicalSpace.induced stageProduct inferInstance
1040 rw [@isOpen_iff_forall_mem_open Q Tind U]
1041 intro qx hqxU
1042 refine Submodule.Quotient.induction_on
1043 (p := Sclosed)
1044 (C := fun qx =>
1045 qx ∈ U → ∃ t, t ⊆ U ∧ @IsOpen Q Tind t ∧ qx ∈ t)
1046 qx ?_ hqxU
1047 intro x hxU
1048 let q : zcCompletedGroupAlgebraStandardAugmentationIdeal C G → Q :=
1049 Sclosed.mkQ
1050 have hUquot :
1051 @IsOpen Q (QuotientModule.Quotient.topologicalSpace Sclosed) U := hU
1052 have hpreOpen :
1053 @IsOpen
1054 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
1055 inferInstance (q ⁻¹' U) := by
1056 change
1057 @IsOpen
1058 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
1059 inferInstance (q ⁻¹' U) at hUquot
1060 exact hUquot
1061 rcases isOpen_induced_iff.mp hpreOpen with ⟨V, hVopen, hVeq⟩
1062 have hxV : (x : ZCCompletedGroupAlgebra C G) ∈ V := by
1063 have hxpre : x ∈ q ⁻¹' U := hxU
1064 rwa [← hVeq] at hxpre
1065 let Ssys := zcCompletedGroupAlgebraSystem C G
1066 letI : Nonempty (ZCCompletedGroupAlgebraIndex C G) :=
1067 ⟨(ProCGroups.Completion.ProCIntegerIndex.terminal (C := C) inferInstance,
1068 zcCompletedGroupAlgebraTopIndex C G)⟩
1069 letI : ∀ i : ZCCompletedGroupAlgebraIndex C G, TopologicalSpace (Ssys.X i) :=
1070 Ssys.topologicalSpace
1071 letI : ∀ i : ZCCompletedGroupAlgebraIndex C G, CompactSpace (Ssys.X i) := fun i => by
1072 dsimp [Ssys, zcCompletedGroupAlgebraSystem]
1073 change @CompactSpace (ZCCompletedGroupAlgebraStage C G i) ⊥
1074 letI : Fact (0 < i.1.modulus) := ⟨i.1.positive⟩
1075 letI : Finite (ZCCompletedGroupAlgebraStage C G i) :=
1076 finite_modNCompletedGroupAlgebraStageInClass
1077 (n := i.1.modulus) (G := G) C i.2
1078 exact Finite.compactSpace
1079 letI : ∀ i : ZCCompletedGroupAlgebraIndex C G, T2Space (Ssys.X i) := fun i => by
1080 dsimp [Ssys, zcCompletedGroupAlgebraSystem]
1081 change @T2Space (ZCCompletedGroupAlgebraStage C G i) ⊥
1082 exact @DiscreteTopology.toT2Space _ ⊥ ⟨rfl⟩
1083 have hdir : Directed (· ≤ ·) (id : ZCCompletedGroupAlgebraIndex C G →
1084 ZCCompletedGroupAlgebraIndex C G) :=
1085 directed_zcCompletedGroupAlgebraIndex_of_formation (C := C) (H := G) hForm
1086 rcases Ssys.exists_projection_preimage_subset hdir hVopen hxV with
1087 ⟨i, W, hWopen, hxW, hWV⟩
1088 let t : Set Q :=
1089 {z | kernelAugmentationIdealClosedQuotientStageProjection
1090 C hC hForm psi hpsi hfopen i z =
1091 kernelAugmentationIdealClosedQuotientStageProjection
1092 C hC hForm psi hpsi hfopen i (q x)}
1093 refine ⟨t, ?_, ?_, ?_⟩
1094 · intro z hz
1095 refine Submodule.Quotient.induction_on
1096 (p := Sclosed)
1097 (C := fun z => z ∈ t → z ∈ U)
1098 z ?_ hz
1099 intro y hy
1100 let T :=
1101 zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
1102 C hC hForm psi hpsi hfopen i
1103 have hyStage :
1104 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i y -
1105 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x ∈ T := by
1106 apply (Submodule.Quotient.eq T).1
1107 have hy' := hy
1108 change
1109 kernelAugmentationIdealClosedQuotientStageProjection
1110 C hC hForm psi hpsi hfopen i (Sclosed.mkQ y) =
1111 kernelAugmentationIdealClosedQuotientStageProjection
1112 C hC hForm psi hpsi hfopen i (Sclosed.mkQ x) at hy'
1113 rw [kernelAugmentationIdealClosedQuotientStageProjection_mk,
1114 kernelAugmentationIdealClosedQuotientStageProjection_mk] at hy'
1115 exact hy'
1116 rcases
1117 zcCompletedGAOpenImageKernelAugmentationIdealMulStageStandard_mem_proj
1118 C hC hForm psi hpsi hfopen i
1119 (zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i y -
1120 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x)
1121 hyStage with
1122 ⟨r, hr, hrproj⟩
1123 have hq : q (y - r) = q y := by
1124 apply (Submodule.Quotient.eq Sclosed).2
1125 change (y - r) - y ∈ Sclosed
1126 have hdiff : (y - r) - y = -r := by
1127 abel
1128 rw [hdiff]
1129 exact Sclosed.neg_mem
1130 (zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_le_closed
1131 C hC hForm psi hpsi hfopen hr)
1132 have hproj_eq :
1133 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i (y - r) =
1134 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x := by
1135 rw [zcCompletedGroupAlgebraStandardAugmentationIdealProjection_sub, hrproj]
1136 abel
1137 have hyW : Ssys.projection i ((y - r : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
1138 ZCCompletedGroupAlgebra C G) ∈ W := by
1139 have hval := congrArg Subtype.val hproj_eq
1140 change
1141 zcCompletedGroupAlgebraProjection C G i
1142 ((y - r : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
1143 ZCCompletedGroupAlgebra C G) =
1144 zcCompletedGroupAlgebraProjection C G i
1145 (x : ZCCompletedGroupAlgebra C G) at hval
1146 have hxW' :
1147 zcCompletedGroupAlgebraProjection C G i
1148 (x : ZCCompletedGroupAlgebra C G) ∈ W := by
1149 simpa [Ssys, zcCompletedGroupAlgebraSystem] using hxW
1150 change
1151 zcCompletedGroupAlgebraProjection C G i
1152 (((y - r : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
1153 ZCCompletedGroupAlgebra C G)) ∈ W
1154 rw [hval]
1155 exact hxW'
1156 have hyV :
1157 (((y - r : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
1158 ZCCompletedGroupAlgebra C G)) ∈ V :=
1159 hWV hyW
1160 have hyU : q (y - r) ∈ U := by
1161 have hyPre :
1162 (y - r : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
1163 (Subtype.val : zcCompletedGroupAlgebraStandardAugmentationIdeal C G →
1164 ZCCompletedGroupAlgebra C G) ⁻¹' V := hyV
1165 rwa [hVeq] at hyPre
1166 rwa [hq] at hyU
1167 · letI : TopologicalSpace Q := Tind
1168 have hprod : Continuous stageProduct :=
1169 continuous_induced_dom
1170 have hcoord :
1171 Continuous (fun z : Q =>
1172 kernelAugmentationIdealClosedQuotientStageProjection
1173 C hC hForm psi hpsi hfopen i z) := by
1174 have hi := (continuous_apply i).comp hprod
1175 change Continuous
1176 (kernelAugmentationIdealClosedQuotientStageProjection
1177 C hC hForm psi hpsi hfopen i) at hi
1178 exact hi
1179 have hsingle_open :
1180 IsOpen
1181 ({kernelAugmentationIdealClosedQuotientStageProjection
1182 C hC hForm psi hpsi hfopen i (q x)} :
1183 Set (KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i)) := by
1184 let T :=
1185 zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
1186 C hC hForm psi hpsi hfopen i
1187 haveI : DiscreteTopology (zcCompletedGroupAlgebraStageAugmentationIdeal C G i) := by
1188 infer_instance
1189 change
1190 @IsOpen
1191 (KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i)
1192 (TopologicalSpace.coinduced T.mkQ inferInstance)
1193 ({kernelAugmentationIdealClosedQuotientStageProjection
1194 C hC hForm psi hpsi hfopen i (q x)} :
1195 Set (KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i))
1196 rw [isOpen_coinduced]
1197 exact isOpen_discrete _
1198 exact
1199 hsingle_open.preimage hcoord
1200 · exact rfl
1201 · intro hU
1202 rcases isOpen_induced_iff.mp hU with ⟨V, hVopen, hVU⟩
1203 rw [← hVU]
1204 exact hVopen.preimage
1205 (continuous_kernelAugmentationIdealClosedQuotientStageProjectionProduct
1206 C hC hForm psi hpsi hfopen)
1208/--
1209The finite-stage source-to-open-image group-algebra map used to descend the source-stage action
1210on the closed finite augmentation quotient to the matching target stage.
1211-/
1212def zcCompletedGroupAlgebraOpenImageStageRingHom
1214 (hForm : ProCGroups.FiniteGroupClass.Formation C)
1215 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
1216 (hfopen : IsOpenMap psi)
1217 (i : ZCCompletedGroupAlgebraIndex C G) :
1218 ZCCompletedGroupAlgebraStage C G i →+*
1219 ZCCompletedGroupAlgebraStage C H
1220 (zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i) :=
1221 MonoidAlgebra.mapDomainRingHom (ModNCompletedCoeff i.1.modulus)
1222 (zcCompletedGroupAlgebraOpenImageQuotientMap C hC hForm psi hpsi hfopen i)
1224/--
1225The open-image stage ring homomorphism sends generators according to the finite-stage Fox
1226coordinate formula.
1227-/
1228@[simp]
1229theorem zcCompletedGroupAlgebraOpenImageStageRingHom_of
1232 (hForm : ProCGroups.FiniteGroupClass.Formation C)
1233 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
1234 (hfopen : IsOpenMap psi)
1235 (i : ZCCompletedGroupAlgebraIndex C G)
1236 (q : CompletedGroupAlgebraQuotientInClass G C i.2) :
1237 zcCompletedGroupAlgebraOpenImageStageRingHom C hC hForm psi hpsi hfopen i
1238 (MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
1239 (CompletedGroupAlgebraQuotientInClass G C i.2) q) =
1240 MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
1241 (CompletedGroupAlgebraQuotientInClass H C
1242 (zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i).2)
1243 (zcCompletedGroupAlgebraOpenImageQuotientMap
1244 C hC hForm psi hpsi hfopen i q) := by
1245 simp only [zcCompletedGroupAlgebraOpenImageStageRingHom, MonoidAlgebra.of_apply]
1246 exact MonoidAlgebra.mapDomain_single
1248/-- The finite-stage source-to-open-image group-algebra map is surjective. -/
1249theorem zcCompletedGroupAlgebraOpenImageStageRingHom_surjective
1252 (hForm : ProCGroups.FiniteGroupClass.Formation C)
1253 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
1254 (hfopen : IsOpenMap psi)
1255 (i : ZCCompletedGroupAlgebraIndex C G) :
1256 Function.Surjective
1257 (zcCompletedGroupAlgebraOpenImageStageRingHom
1258 C hC hForm psi hpsi hfopen i) := by
1259 intro y
1260 change
1261 MonoidAlgebra (ModNCompletedCoeff i.1.modulus)
1262 (CompletedGroupAlgebraQuotientInClass H C
1263 (zcCompletedGroupAlgebraOpenImageIndexInClass C hForm psi hpsi hfopen i)) at y
1264 rcases
1265 Finsupp.mapDomain_surjective (M := ModNCompletedCoeff i.1.modulus)
1266 (zcCompletedGroupAlgebraOpenImageQuotientMap_surjective
1267 C hC hForm psi hpsi hfopen i) y.coeff with
1268 ⟨x, hx⟩
1269 refine ⟨MonoidAlgebra.ofCoeff x, ?_⟩
1270 change MonoidAlgebra.mapDomain
1271 (zcCompletedGroupAlgebraOpenImageQuotientMap
1272 C hC hForm psi hpsi hfopen i) (MonoidAlgebra.ofCoeff x) = y
1273 apply MonoidAlgebra.coeff_injective
1274 simpa [MonoidAlgebra.mapDomain] using hx
1276/--
1277Source-stage elements in the kernel of the open-image stage map multiply the finite augmentation
1278stage into the finite kernel-product denominator.
1279-/
1280theorem zcCompletedGroupAlgebraOpenImageStageRingHom_ker_smul_mem
1283 (hForm : ProCGroups.FiniteGroupClass.Formation C)
1284 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
1285 (hfopen : IsOpenMap psi)
1286 (i : ZCCompletedGroupAlgebraIndex C G)
1287 (k : ZCCompletedGroupAlgebraStage C G i)
1288 (hk : k ∈ RingHom.ker
1289 (zcCompletedGroupAlgebraOpenImageStageRingHom
1290 C hC hForm psi hpsi hfopen i))
1291 (s : zcCompletedGroupAlgebraStageAugmentationIdeal C G i) :
1292 k • s ∈
1293 zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
1294 C hC hForm psi hpsi hfopen i := by
1295 let f := zcCompletedGroupAlgebraOpenImageQuotientMap
1296 C hC hForm psi hpsi hfopen i
1297 let T :=
1298 zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
1299 C hC hForm psi hpsi hfopen i
1300 have hkIdeal :
1301 k ∈ groupAlgebraMapDomainKernelAugmentationIdeal
1302 (R := ModNCompletedCoeff i.1.modulus) f := by
1303 have hk' :
1304 k ∈ RingHom.ker
1305 (MonoidAlgebra.mapDomainRingHom
1306 (ModNCompletedCoeff i.1.modulus) f) := by
1307 change MonoidAlgebra.mapDomain f k = 0 at hk
1308 exact hk
1309 rwa [groupAlgebraMapDomainRingHom_ker_eq_kernelAugmentationIdeal_of_surjective
1310 (R := ModNCompletedCoeff i.1.modulus) f
1311 (zcCompletedGroupAlgebraOpenImageQuotientMap_surjective
1312 C hC hForm psi hpsi hfopen i)] at hk'
1313 change k • s ∈ T
1314 rw [groupAlgebraMapDomainKernelAugmentationIdeal] at hkIdeal
1315 refine Submodule.span_induction
1316 (p := fun k _ => k • s ∈ T) ?hgen ?hzero ?hadd ?hsmul hkIdeal
1317 · rintro _ ⟨q, rfl⟩
1318 exact
1319 zcCompletedGAOpenImageKernelAugmentationIdealMulStageStandard_generator_mem
1320 C hC hForm psi hpsi hfopen i q s
1321 · simp only [zero_smul, zero_mem]
1322 · intro a b _ _ ha hb
1323 simpa [add_smul] using T.add_mem ha hb
1324 · intro a b _ hb
1325 simpa [mul_smul] using T.smul_mem a hb
1327/-- Source-stage kernels act trivially on the closed finite augmentation quotient. -/
1328theorem kernelAugmentationIdealClosedStageQuotient_openImageStageRingHom_ker_smul_eq_zero
1331 (hForm : ProCGroups.FiniteGroupClass.Formation C)
1332 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
1333 (hfopen : IsOpenMap psi)
1334 (i : ZCCompletedGroupAlgebraIndex C G)
1335 (k : ZCCompletedGroupAlgebraStage C G i)
1336 (hk : k ∈ RingHom.ker
1337 (zcCompletedGroupAlgebraOpenImageStageRingHom
1338 C hC hForm psi hpsi hfopen i))
1339 (x : KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) :
1340 k • x = 0 := by
1341 let T :=
1342 zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
1343 C hC hForm psi hpsi hfopen i
1344 refine Submodule.Quotient.induction_on (p := T) x ?_
1345 intro s
1346 apply (Submodule.Quotient.mk_eq_zero (p := T)).2
1347 exact
1348 zcCompletedGroupAlgebraOpenImageStageRingHom_ker_smul_mem
1349 C hC hForm psi hpsi hfopen i k hk s
1351/--
1352The finite closed augmentation quotient as a module over the matching open-image target
1353group-algebra stage.
1354-/
1355@[implicit_reducible]
1356def kernelAugmentationIdealClosedStageQuotientTargetStageModule
1359 (hForm : ProCGroups.FiniteGroupClass.Formation C)
1360 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
1361 (hfopen : IsOpenMap psi)
1362 (i : ZCCompletedGroupAlgebraIndex C G) :
1363 Module
1364 (ZCCompletedGroupAlgebraStage C H
1365 (zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i))
1366 (KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) := by
1367 let φ := zcCompletedGroupAlgebraOpenImageStageRingHom
1368 C hC hForm psi hpsi hfopen i
1369 let hφ := zcCompletedGroupAlgebraOpenImageStageRingHom_surjective
1370 C hC hForm psi hpsi hfopen i
1371 letI : SMul
1372 (ZCCompletedGroupAlgebraStage C H
1373 (zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i))
1374 (KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) :=
1375 ⟨fun a x => Function.surjInv hφ a • x⟩
1376 refine hφ.moduleLeft φ ?_
1377 intro a x
1378 change Function.surjInv hφ (φ a) • x = a • x
1379 have hdiff : Function.surjInv hφ (φ a) - a ∈ RingHom.ker φ := by
1380 rw [RingHom.mem_ker, map_sub, Function.surjInv_eq hφ, sub_self]
1381 have hzero :=
1382 kernelAugmentationIdealClosedStageQuotient_openImageStageRingHom_ker_smul_eq_zero
1383 C hC hForm psi hpsi hfopen i
1384 (Function.surjInv hφ (φ a) - a) hdiff x
1385 rw [sub_smul] at hzero
1386 exact sub_eq_zero.mp hzero
1388/-- The stage quotient target module map is compatible with scalar multiplication. -/
1389theorem kernelAugmentationIdealClosedStageQuotientTargetStageModule_map_smul
1392 (hForm : ProCGroups.FiniteGroupClass.Formation C)
1393 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
1394 (hfopen : IsOpenMap psi)
1395 (i : ZCCompletedGroupAlgebraIndex C G)
1396 (a : ZCCompletedGroupAlgebraStage C G i)
1397 (x : KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) :
1398 letI : Module
1399 (ZCCompletedGroupAlgebraStage C H
1400 (zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i))
1401 (KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) :=
1402 kernelAugmentationIdealClosedStageQuotientTargetStageModule
1403 C hC hForm psi hpsi hfopen i
1404 zcCompletedGroupAlgebraOpenImageStageRingHom C hC hForm psi hpsi hfopen i a • x =
1405 a • x := by
1406 let φ := zcCompletedGroupAlgebraOpenImageStageRingHom
1407 C hC hForm psi hpsi hfopen i
1408 let hφ := zcCompletedGroupAlgebraOpenImageStageRingHom_surjective
1409 C hC hForm psi hpsi hfopen i
1410 letI : Module
1411 (ZCCompletedGroupAlgebraStage C H
1412 (zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i))
1413 (KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) :=
1414 kernelAugmentationIdealClosedStageQuotientTargetStageModule
1415 C hC hForm psi hpsi hfopen i
1416 change Function.surjInv hφ (φ a) • x = a • x
1417 have hdiff : Function.surjInv hφ (φ a) - a ∈ RingHom.ker φ := by
1418 rw [RingHom.mem_ker, map_sub, Function.surjInv_eq hφ, sub_self]
1419 have hzero :=
1420 kernelAugmentationIdealClosedStageQuotient_openImageStageRingHom_ker_smul_eq_zero
1421 C hC hForm psi hpsi hfopen i
1422 (Function.surjInv hφ (φ a) - a) hdiff x
1423 rw [sub_smul] at hzero
1424 exact sub_eq_zero.mp hzero
1426/--
1427The finite source quotient paired with the open-image target stage for a source group-algebra
1428coordinate.
1429-/
1430@[implicit_reducible]
1431def zcCompletedDifferentialModuleOpenImageIndex
1432 (hForm : ProCGroups.FiniteGroupClass.Formation C)
1433 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
1434 (hfopen : IsOpenMap psi)
1435 (i : ZCCompletedGroupAlgebraIndex C G) :
1436 ZCCompletedDifferentialModuleIndex C psi.toMonoidHom where
1437 source := OrderDual.ofDual i.2
1438 target := zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i
1439 compatible := by
1440 intro g hg
1441 change psi g ∈
1442 ((((OrderDual.ofDual
1443 (zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i).2).1 :
1444 OpenNormalSubgroup H) : Subgroup H))
1445 exact ⟨g, hg, rfl⟩
1447/--
1448The open-image stage condition for the \(\mathbb{Z}_C\)-completed differential module is
1449equivalent to the finite-stage coordinate condition.
1450-/
1451@[simp 900]
1452theorem zcCompletedDifferentialModuleOpenImageIndex_stageScalar
1455 (hForm : ProCGroups.FiniteGroupClass.Formation C)
1456 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
1457 (hfopen : IsOpenMap psi)
1458 (i : ZCCompletedGroupAlgebraIndex C G)
1459 (q : zcCompletedDifferentialModuleStageSource C psi.toMonoidHom
1460 (zcCompletedDifferentialModuleOpenImageIndex C hForm psi hpsi hfopen i)) :
1461 zcCompletedDifferentialModuleStageScalar C psi.toMonoidHom
1462 (zcCompletedDifferentialModuleOpenImageIndex C hForm psi hpsi hfopen i) q =
1463 zcCompletedGroupAlgebraOpenImageStageRingHom C hC hForm psi hpsi hfopen i
1464 (MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
1465 (CompletedGroupAlgebraQuotientInClass G C i.2) q) := by
1466 refine QuotientGroup.induction_on q ?_
1467 intro g
1468 simp only [zcCompletedDifferentialModuleStageScalar_coe,
1469 zcCompletedGroupAlgebraOpenImageStageRingHom_of]
1470 have hq :=
1471 zcCompletedGroupAlgebraOpenImageQuotientMap_mk
1472 C hC hForm psi hpsi hfopen i g
1473 change
1474 zcCompletedGroupAlgebraOpenImageQuotientMap C hC hForm psi hpsi hfopen i
1475 (QuotientGroup.mk g :
1476 CompletedGroupAlgebraQuotientInClass G C i.2) =
1477 (QuotientGroup.mk (psi g) :
1478 CompletedGroupAlgebraQuotientInClass H C
1479 (zcCompletedGroupAlgebraOpenImageTargetIndex
1480 C hForm psi hpsi hfopen i).2) at hq
1481 symm
1482 convert
1483 congrArg
1484 (MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
1485 (CompletedGroupAlgebraQuotientInClass H C
1486 (zcCompletedGroupAlgebraOpenImageTargetIndex
1487 C hForm psi hpsi hfopen i).2))
1488 hq using 1 <;> rfl
1490/-- The finite source-boundary coordinate over the matching open-image target stage. -/
1491def kernelAugmentationIdealClosedStageQuotientBoundary
1494 (hForm : ProCGroups.FiniteGroupClass.Formation C)
1495 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
1496 (hfopen : IsOpenMap psi)
1497 (i : ZCCompletedGroupAlgebraIndex C G) :
1498 let j := zcCompletedDifferentialModuleOpenImageIndex C hForm psi hpsi hfopen i
1499 letI : Module (zcCompletedDifferentialModuleStageRing C psi.toMonoidHom j)
1500 (KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) :=
1501 kernelAugmentationIdealClosedStageQuotientTargetStageModule
1502 C hC hForm psi hpsi hfopen i
1503 ScalarCrossedHom
1504 (zcCompletedDifferentialModuleStageScalar C psi.toMonoidHom j)
1505 (KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) := by
1506 let j := zcCompletedDifferentialModuleOpenImageIndex C hForm psi hpsi hfopen i
1507 let T :=
1508 zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
1509 C hC hForm psi hpsi hfopen i
1510 letI : Module (zcCompletedDifferentialModuleStageRing C psi.toMonoidHom j)
1511 (KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) :=
1512 kernelAugmentationIdealClosedStageQuotientTargetStageModule
1513 C hC hForm psi hpsi hfopen i
1514 refine
1515 { toFun := fun q =>
1516 Submodule.Quotient.mk
1517 (p := zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
1518 C hC hForm psi hpsi hfopen i)
1519 (zcCompletedGroupAlgebraStageAugmentationGeneratorSubtype C G i q)
1520 map_mul' := ?_ }
1521 intro q r
1522 rw [scalarCrossedAction_apply]
1523 rw [zcCompletedDifferentialModuleOpenImageIndex_stageScalar
1524 (C := C) (hC := hC) (hForm := hForm) (psi := psi)
1525 (hpsi := hpsi) (hfopen := hfopen) (i := i)]
1526 rw [kernelAugmentationIdealClosedStageQuotientTargetStageModule_map_smul
1527 (C := C) (hC := hC) (hForm := hForm) (psi := psi)
1528 (hpsi := hpsi) (hfopen := hfopen) (i := i)
1529 (a := MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
1530 (CompletedGroupAlgebraQuotientInClass G C i.2) q)
1531 (x := Submodule.Quotient.mk
1532 (p := zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
1533 C hC hForm psi hpsi hfopen i)
1534 (zcCompletedGroupAlgebraStageAugmentationGeneratorSubtype C G i r))]
1535 change
1536 Submodule.Quotient.mk (p := T)
1537 (zcCompletedGroupAlgebraStageAugmentationGeneratorSubtype C G i (q * r)) =
1538 Submodule.Quotient.mk (p := T)
1539 (zcCompletedGroupAlgebraStageAugmentationGeneratorSubtype C G i q) +
1540 MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
1541 (CompletedGroupAlgebraQuotientInClass G C i.2) q •
1542 Submodule.Quotient.mk (p := T)
1543 (zcCompletedGroupAlgebraStageAugmentationGeneratorSubtype C G i r)
1544 rw [← Submodule.Quotient.mk_smul, ← Submodule.Quotient.mk_add]
1545 apply congrArg (fun s : zcCompletedGroupAlgebraStageAugmentationIdeal C G i =>
1546 Submodule.Quotient.mk (p := T) s)
1547 apply Subtype.ext
1548 change
1549 MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
1550 (CompletedGroupAlgebraQuotientInClass G C i.2) (q * r) - 1 =
1551 (MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
1552 (CompletedGroupAlgebraQuotientInClass G C i.2) q - 1) +
1553 MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
1554 (CompletedGroupAlgebraQuotientInClass G C i.2) q *
1555 (MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
1556 (CompletedGroupAlgebraQuotientInClass G C i.2) r - 1)
1557 change CompletedGroupAlgebraQuotientInClass G C i.2 at q r
1558 change
1559 MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
1560 (CompletedGroupAlgebraQuotientInClass G C i.2) (q * r) - 1 =
1561 (MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
1562 (CompletedGroupAlgebraQuotientInClass G C i.2) q - 1) +
1563 MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
1564 (CompletedGroupAlgebraQuotientInClass G C i.2) q *
1565 (MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
1566 (CompletedGroupAlgebraQuotientInClass G C i.2) r - 1)
1567 rw [map_mul, mul_sub, mul_one]
1568 abel
1570/-- The finite-stage lift induced by the finite closed source-boundary coordinate. -/
1571def kernelAugmentationIdealClosedStageQuotientBoundaryLift
1574 (hForm : ProCGroups.FiniteGroupClass.Formation C)
1575 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
1576 (hfopen : IsOpenMap psi)
1577 (i : ZCCompletedGroupAlgebraIndex C G) :
1578 let j := zcCompletedDifferentialModuleOpenImageIndex C hForm psi hpsi hfopen i
1579 letI : Module (zcCompletedDifferentialModuleStageRing C psi.toMonoidHom j)
1580 (KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) :=
1581 kernelAugmentationIdealClosedStageQuotientTargetStageModule
1582 C hC hForm psi hpsi hfopen i
1583 CrossedDifferentialPreModule
1584 (zcCompletedDifferentialModuleStageRing C psi.toMonoidHom j)
1585 (zcCompletedDifferentialModuleStageSource C psi.toMonoidHom j) →ₗ[
1586 zcCompletedDifferentialModuleStageRing C psi.toMonoidHom j]
1587 KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i := by
1588 let j := zcCompletedDifferentialModuleOpenImageIndex C hForm psi hpsi hfopen i
1589 letI : Module (zcCompletedDifferentialModuleStageRing C psi.toMonoidHom j)
1590 (KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) :=
1591 kernelAugmentationIdealClosedStageQuotientTargetStageModule
1592 C hC hForm psi hpsi hfopen i
1593 exact
1594 crossedDifferentialModuleLiftLinear
1595 (R := zcCompletedDifferentialModuleStageRing C psi.toMonoidHom j)
1596 (kernelAugmentationIdealClosedStageQuotientBoundary
1597 C hC hForm psi hpsi hfopen i)
1599/--
1600The boundary lift lands in the finite-stage closed augmentation quotient by the Fox-differential
1601kernel condition.
1602-/
1603@[simp 900]
1604theorem kernelAugmentationIdealClosedStageQuotientBoundaryLift_single
1607 (hForm : ProCGroups.FiniteGroupClass.Formation C)
1608 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
1609 (hfopen : IsOpenMap psi)
1610 (i : ZCCompletedGroupAlgebraIndex C G)
1611 (q : zcCompletedDifferentialModuleStageSource C psi.toMonoidHom
1612 (zcCompletedDifferentialModuleOpenImageIndex C hForm psi hpsi hfopen i))
1613 (a : zcCompletedDifferentialModuleStageRing C psi.toMonoidHom
1614 (zcCompletedDifferentialModuleOpenImageIndex C hForm psi hpsi hfopen i)) :
1615 let j := zcCompletedDifferentialModuleOpenImageIndex C hForm psi hpsi hfopen i
1616 letI : Module (zcCompletedDifferentialModuleStageRing C psi.toMonoidHom j)
1617 (KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) :=
1618 kernelAugmentationIdealClosedStageQuotientTargetStageModule
1619 C hC hForm psi hpsi hfopen i
1620 kernelAugmentationIdealClosedStageQuotientBoundaryLift
1621 C hC hForm psi hpsi hfopen i (Finsupp.single q a) =
1622 a • kernelAugmentationIdealClosedStageQuotientBoundary
1623 C hC hForm psi hpsi hfopen i q := by
1624 let j := zcCompletedDifferentialModuleOpenImageIndex C hForm psi hpsi hfopen i
1625 letI : Module (zcCompletedDifferentialModuleStageRing C psi.toMonoidHom j)
1626 (KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) :=
1627 kernelAugmentationIdealClosedStageQuotientTargetStageModule
1628 C hC hForm psi hpsi hfopen i
1629 simp only [ContinuousMonoidHom.coe_toMonoidHom, Lean.Elab.WF.paramLet,
1630 kernelAugmentationIdealClosedStageQuotientBoundaryLift,
1631 crossedDifferentialModuleLiftLinear_single]
1633/--
1634The canonical quotient map from the algebraic kernel-product quotient to its closed finite-stage
1635quotient.
1636-/
1637def zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient
1640 (hForm : ProCGroups.FiniteGroupClass.Formation C)
1641 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
1642 (hfopen : IsOpenMap psi) :
1643 KernelAugmentationIdealQuotient C psi →ₗ[ZCCompletedGroupAlgebra C G]
1644 KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen :=
1645 (zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi).mapQ
1646 (zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
1647 C hC hForm psi hpsi hfopen)
1648 LinearMap.id
1649 (by
1650 intro x hx
1651 simpa using
1652 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_le_closed
1653 C hC hForm psi hpsi hfopen hx)
1655/--
1656The map from the algebraic kernel-product quotient to the closed quotient sends a representative
1657to its closed quotient class.
1658-/
1659@[simp]
1660theorem zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient_mk
1663 (hForm : ProCGroups.FiniteGroupClass.Formation C)
1664 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
1665 (hfopen : IsOpenMap psi)
1666 (x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
1667 zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient
1668 C hC hForm psi hpsi hfopen (Submodule.Quotient.mk x) =
1669 Submodule.Quotient.mk x := by
1670 rw [zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient,
1671 Submodule.mapQ_apply]
1672 rfl
1674/--
1675The canonical map from the algebraic quotient \(I(G) / I(\ker \psi)I(G)\) to the closed
1676finite-stage quotient is injective exactly when the algebraic product already equals its
1677finite-stage closed hull.
1678-/
1679theorem zcCompletedGAKerAugQuotToClosedQuotient_inj_iff_eq_closed
1680 {C : ProCGroups.FiniteGroupClass.{u}}
1683 {hForm : ProCGroups.FiniteGroupClass.Formation C}
1684 {psi : ContinuousMonoidHom G H} {hpsi : Function.Surjective psi}
1685 {hfopen : IsOpenMap psi} :
1686 Function.Injective
1687 (zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient
1688 C hC hForm psi hpsi hfopen) ↔
1689 (zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :
1690 Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)) =
1691 (zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
1692 C hC hForm psi hpsi hfopen :
1693 Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)) := by
1694 let S := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi
1695 let T :=
1696 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
1697 C hC hForm psi hpsi hfopen
1698 let Q :=
1699 zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient
1700 C hC hForm psi hpsi hfopen
1701 constructor
1702 · intro hQ
1703 apply Set.Subset.antisymm
1704 · intro x hx
1705 exact zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_le_closed
1706 C hC hForm psi hpsi hfopen hx
1707 · intro x hx
1708 have hmap :
1709 Q (Submodule.Quotient.mk (p := S) x) = 0 := by
1710 rw [zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient_mk
1711 (C := C) (hC := hC) (hForm := hForm) (psi := psi)
1712 (hpsi := hpsi) (hfopen := hfopen) x]
1713 exact (Submodule.Quotient.mk_eq_zero (p := T) (x := x)).2 hx
1714 have hzero :
1715 (Submodule.Quotient.mk (p := S) x :
1716 KernelAugmentationIdealQuotient C psi) = 0 := by
1717 apply hQ
1718 rw [hmap, map_zero]
1719 exact (Submodule.Quotient.mk_eq_zero (p := S) (x := x)).1 hzero
1720 · intro hEq a b hxy
1721 revert b
1722 refine Submodule.Quotient.induction_on
1723 (p := S) a ?_
1724 intro x y hxy
1725 revert hxy
1726 refine Submodule.Quotient.induction_on
1727 (p := S) y ?_
1728 intro y hxy
1729 apply (Submodule.Quotient.eq S).2
1730 have hmemT : x - y ∈ T := by
1731 apply (Submodule.Quotient.eq T).1
1732 simpa [Q, zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient_mk
1733 (C := C) (hC := hC) (hForm := hForm) (psi := psi)
1734 (hpsi := hpsi) (hfopen := hfopen)] using hxy
1735 have hEqST : (S : Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)) = T := by
1736 simpa [S, T] using hEq
1737 have hmemTset :
1738 (x - y) ∈
1739 (T : Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)) := hmemT
1740 have hmemSset :
1741 (x - y) ∈
1742 (S : Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)) := by
1743 rw [hEqST]
1744 exact hmemTset
1745 exact hmemSset
1747/--
1748Closedness of \(I(\ker \psi)I(G)\) is equivalently the injectivity of the canonical map from the
1749algebraic source augmentation quotient to the closed finite-stage quotient.
1750-/
1751theorem isClosed_zcCompletedGAKernelAugmentationIdealMulStandard_iff_toClosedQuotient_inj
1752 {C : ProCGroups.FiniteGroupClass.{u}}
1755 (hForm : ProCGroups.FiniteGroupClass.Formation C)
1756 {psi : ContinuousMonoidHom G H} (hpsi : Function.Surjective psi)
1757 (hfopen : IsOpenMap psi) :
1758 IsClosed
1759 ((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :
1760 Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G))) ↔
1761 Function.Injective
1762 (zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient
1763 C hC hForm psi hpsi hfopen) := by
1764 rw [isClosed_zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_iff_eq_closed
1765 (C := C) (psi := psi) hC hForm hpsi hfopen]
1766 exact
1767 (zcCompletedGAKerAugQuotToClosedQuotient_inj_iff_eq_closed
1768 (C := C) (hC := hC) (hForm := hForm) (psi := psi)
1769 (hpsi := hpsi) (hfopen := hfopen)).symm
1771/-- The source Fox boundary, valued in the source augmentation quotient. -/
1772def zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient
1773 (psi : ContinuousMonoidHom G H) :
1774 ScalarCrossedHom
1775 (zcCompletedGroupAlgebraScalar C (MonoidHom.id G))
1776 (KernelAugmentationIdealQuotient C psi) :=
1777 (zcCompletedGroupAlgebraBoundaryToStandardAugmentationIdeal
1778 C G (MonoidHom.id G)).mapLinear
1779 (Submodule.mkQ
1780 (zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi))
1782/-- The source Fox boundary, valued in the closed source augmentation quotient. -/
1783def zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
1786 (hForm : ProCGroups.FiniteGroupClass.Formation C)
1787 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
1788 (hfopen : IsOpenMap psi) :
1789 ScalarCrossedHom
1790 (zcCompletedGroupAlgebraScalar C (MonoidHom.id G))
1791 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
1792 (zcCompletedGroupAlgebraBoundaryToStandardAugmentationIdeal
1793 C G (MonoidHom.id G)).mapLinear
1794 (Submodule.mkQ
1795 (zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
1796 C hC hForm psi hpsi hfopen))
1798/--
1799The map from the algebraic source augmentation quotient to the closed quotient sends source
1800boundary classes to the corresponding closed source boundary classes.
1801-/
1802@[simp]
1803theorem zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient_sourceBoundary
1806 (hForm : ProCGroups.FiniteGroupClass.Formation C)
1807 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
1808 (hfopen : IsOpenMap psi) (g : G) :
1809 zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient
1810 C hC hForm psi hpsi hfopen
1811 (zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient C psi g) =
1812 zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
1813 C hC hForm psi hpsi hfopen g := by
1814 simp only [zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient,
1815 zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient,
1816 ScalarCrossedHom.mapLinear_apply,
1817 Submodule.mkQ_apply]
1818 rw [
1819 zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient_mk]
1821/-- The source Fox boundary into the closed source augmentation quotient is continuous. -/
1822theorem continuous_zcCompletedGASourceBoundaryToKerAugClosedQuot
1825 (hForm : ProCGroups.FiniteGroupClass.Formation C)
1826 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
1827 (hfopen : IsOpenMap psi) :
1828 Continuous
1829 (zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
1830 C hC hForm psi hpsi hfopen) := by
1831 have hstd :
1832 Continuous
1833 (zcCompletedGroupAlgebraBoundaryToStandardAugmentationIdeal
1834 C G (MonoidHom.id G)) := by
1835 have hval :
1836 Continuous (fun g : G =>
1837 (zcCompletedGroupAlgebraBoundaryToStandardAugmentationIdeal
1838 C G (MonoidHom.id G) g : ZCCompletedGroupAlgebra C G)) :=
1839 continuous_zcCompletedGroupAlgebraBoundary
1840 (C := C) (G := G) (MonoidHom.id G) continuous_id
1841 exact Continuous.subtype_mk hval
1842 (fun g =>
1843 (zcCompletedGroupAlgebraBoundaryToStandardAugmentationIdeal
1844 C G (MonoidHom.id G) g).2)
1845 have hq :
1846 Continuous (fun x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
1847 (Submodule.Quotient.mk x :
1848 KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)) :=
1849 continuous_quotient_mk'
1850 exact hq.comp hstd
1852omit [IsTopologicalGroup H] in
1853/-- Products \((n-1)s\) vanish in the source augmentation quotient. -/
1854@[simp]
1855theorem zcCompletedGroupAlgebraKernelAugmentationIdealQuotient_mk_generator_smul
1856 (psi : ContinuousMonoidHom G H) (n : ProfiniteKernelSubgroup psi)
1857 (s : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
1858 Submodule.Quotient.mk
1859 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi)
1860 ((zcGroupLike C G n.1 - 1) • s) = 0 := by
1861 apply (Submodule.Quotient.mk_eq_zero
1862 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi)).2
1863 exact zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_generator_mem C psi n s
1865omit [IsTopologicalGroup H] in
1866/--
1867Source group-like actions with the same image under psi agree on the source augmentation
1868quotient. This is the algebraic descent statement needed before a completed target scalar action
1869can be installed.
1870-/
1871theorem zcCompletedGroupAlgebraKernelAugmentationQuotient_groupLike_smul_eq_of_map_eq
1872 (psi : ContinuousMonoidHom G H) {g₁ g₂ : G} (h : psi g₁ = psi g₂)
1873 (x : KernelAugmentationIdealQuotient C psi) :
1874 zcGroupLike C G g₁ • x = zcGroupLike C G g₂ • x := by
1875 refine Submodule.Quotient.induction_on
1876 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi) x ?_
1877 intro y
1878 apply (Submodule.Quotient.eq
1879 (zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi)).2
1880 let n : ProfiniteKernelSubgroup psi :=
1881 ⟨g₂⁻¹ * g₁, by
1882 change psi (g₂⁻¹ * g₁) = 1
1883 rw [map_mul, map_inv, h]
1884 simp only [inv_mul_cancel]⟩
1885 have hgen :
1886 (zcGroupLike C G n.1 - 1) • y ∈
1887 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :=
1888 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_generator_mem C psi n y
1889 have hmem :
1890 zcGroupLike C G g₂ • ((zcGroupLike C G n.1 - 1) • y) ∈
1891 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :=
1892 (zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi).smul_mem
1893 (zcGroupLike C G g₂) hgen
1894 convert hmem using 1
1895 apply Subtype.ext
1896 change zcGroupLike C G g₁ * (y : ZCCompletedGroupAlgebra C G) -
1897 zcGroupLike C G g₂ * (y : ZCCompletedGroupAlgebra C G) =
1898 zcGroupLike C G g₂ *
1899 ((zcGroupLike C G n.1 - 1) * (y : ZCCompletedGroupAlgebra C G))
1900 rw [sub_mul, one_mul, mul_sub, ← mul_assoc, ← map_mul]
1901 simp only [mul_inv_cancel_left, n]
1903omit [IsTopologicalGroup H] in
1904/--
1905The additive endomorphism of the source augmentation quotient induced by any chosen lift of a
1906target element.
1907-/
1908def zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupLikeEndOfSurjective
1909 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi) (h : H) :
1910 AddMonoid.End (KernelAugmentationIdealQuotient C psi) where
1911 toFun x := zcGroupLike C G (Function.surjInv hpsi h) • x
1912 map_zero' := smul_zero _
1913 map_add' := by
1914 intro x y
1915 rw [smul_add]
1917omit [IsTopologicalGroup H] in
1918/--
1919The chosen-lift target group-like endomorphism acts by scalar multiplication with the completed
1920group-like element of the selected source lift.
1921-/
1922@[simp]
1923theorem zcCompletedGAKerAugQuotTargetGroupLikeEndOfSurjective_apply
1924 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi) (h : H)
1925 (x : KernelAugmentationIdealQuotient C psi) :
1926 zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupLikeEndOfSurjective
1927 C psi hpsi h x =
1928 zcGroupLike C G (Function.surjInv hpsi h) • x :=
1929 rfl
1931omit [IsTopologicalGroup H] in
1932/--
1933The chosen-lift target action agrees with scalar multiplication by any source lift of the same
1934target element.
1935-/
1936theorem zcCompletedGAKerAugQuotTargetGroupLikeEndOfSurjective_eq_smul_of_map_eq
1937 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi) {h : H} {g : G}
1938 (hg : psi g = h) (x : KernelAugmentationIdealQuotient C psi) :
1939 zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupLikeEndOfSurjective
1940 C psi hpsi h x =
1941 zcGroupLike C G g • x := by
1942 have hlift : psi (Function.surjInv hpsi h) = psi g := by
1943 rw [Function.surjInv_eq hpsi h, hg]
1944 exact zcCompletedGroupAlgebraKernelAugmentationQuotient_groupLike_smul_eq_of_map_eq
1945 C psi hlift x
1947omit [IsTopologicalGroup H] in
1948/--
1949The chosen-lift target group-like endomorphism for the identity target element acts as the
1950identity.
1951-/
1952@[simp]
1953theorem zcCompletedGAKerAugQuotTargetGroupLikeEndOfSurjective_one_apply
1954 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
1955 (x : KernelAugmentationIdealQuotient C psi) :
1956 zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupLikeEndOfSurjective
1957 C psi hpsi (1 : H) x = x := by
1958 have hmap : psi (Function.surjInv hpsi (1 : H)) = psi (1 : G) := by
1959 rw [Function.surjInv_eq hpsi (1 : H), map_one]
1960 have hsmul :=
1961 zcCompletedGroupAlgebraKernelAugmentationQuotient_groupLike_smul_eq_of_map_eq
1962 C psi hmap x
1963 simpa using hsmul
1965omit [IsTopologicalGroup H] in
1966/-- The chosen-lift target group-like endomorphisms multiply pointwise on the quotient. -/
1967theorem zcCompletedGAKerAugQuotTargetGroupLikeEndOfSurjective_mul_apply
1968 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi) (h₁ h₂ : H)
1969 (x : KernelAugmentationIdealQuotient C psi) :
1970 zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupLikeEndOfSurjective
1971 C psi hpsi (h₁ * h₂) x =
1972 zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupLikeEndOfSurjective
1973 C psi hpsi h₁
1974 (zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupLikeEndOfSurjective
1975 C psi hpsi h₂ x) := by
1976 let s₁ : G := Function.surjInv hpsi h₁
1977 let s₂ : G := Function.surjInv hpsi h₂
1978 let s₁₂ : G := Function.surjInv hpsi (h₁ * h₂)
1979 have hs : psi s₁₂ = psi (s₁ * s₂) := by
1980 rw [map_mul]
1981 simp only [Function.surjInv_eq hpsi, s₁₂, s₁, s₂]
1982 have hsmul :=
1983 zcCompletedGroupAlgebraKernelAugmentationQuotient_groupLike_smul_eq_of_map_eq
1984 C psi hs x
1985 change zcGroupLike C G s₁₂ • x =
1986 zcGroupLike C G s₁ • (zcGroupLike C G s₂ • x)
1987 rw [← mul_smul]
1988 rw [← (zcGroupLike C G).map_mul s₁ s₂]
1989 exact hsmul
1991omit [IsTopologicalGroup H] in
1992/--
1993The descended group-like target action on the source augmentation quotient, for a surjective
1994\(\psi\).
1995-/
1996def zcCompletedGAKerAugQuotTargetGroupLikeActionOfSurjective
1997 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi) :
1998 H →* AddMonoid.End (KernelAugmentationIdealQuotient C psi) where
1999 toFun h :=
2000 zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupLikeEndOfSurjective
2001 C psi hpsi h
2002 map_one' := by
2003 refine AddMonoidHom.ext ?_
2004 intro x
2005 exact
2006 zcCompletedGAKerAugQuotTargetGroupLikeEndOfSurjective_one_apply
2007 C psi hpsi x
2008 map_mul' h₁ h₂ := by
2009 refine AddMonoidHom.ext ?_
2010 intro x
2011 change
2012 zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupLikeEndOfSurjective
2013 C psi hpsi (h₁ * h₂) x =
2014 zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupLikeEndOfSurjective
2015 C psi hpsi h₁
2016 (zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupLikeEndOfSurjective
2017 C psi hpsi h₂ x)
2018 exact
2019 zcCompletedGAKerAugQuotTargetGroupLikeEndOfSurjective_mul_apply
2020 C psi hpsi h₁ h₂ x
2022omit [IsTopologicalGroup H] in
2023/--
2024The descended target group-like action for a surjective map acts through the selected source
2025lift.
2026-/
2027@[simp]
2028theorem zcCompletedGAKerAugQuotTargetGroupLikeActionOfSurjective_apply
2029 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi) (h : H)
2030 (x : KernelAugmentationIdealQuotient C psi) :
2031 zcCompletedGAKerAugQuotTargetGroupLikeActionOfSurjective
2032 C psi hpsi h x =
2033 zcGroupLike C G (Function.surjInv hpsi h) • x :=
2034 rfl
2036omit [IsTopologicalGroup H] in
2037/--
2038Coefficients from \(\mathbb{Z}_C\) act on the source augmentation quotient through the source
2039completed group algebra.
2040-/
2041def zcCompletedGroupAlgebraKernelAugmentationQuotientTargetCoeffEnd
2042 (psi : ContinuousMonoidHom G H) (a : ZCCoeff C) :
2043 AddMonoid.End (KernelAugmentationIdealQuotient C psi) where
2044 toFun x := zcCompletedGroupAlgebraCoeffMap C G a • x
2045 map_zero' := smul_zero _
2046 map_add' := by
2047 intro x y
2048 rw [smul_add]
2050omit [IsTopologicalGroup H] in
2051/--
2052The target coefficient endomorphism acts on the kernel-augmentation quotient by multiplication with
2053the coefficient-map image of `a`.
2054-/
2055@[simp]
2056theorem zcCompletedGroupAlgebraKernelAugmentationQuotientTargetCoeffEnd_apply
2057 (psi : ContinuousMonoidHom G H) (a : ZCCoeff C)
2058 (x : KernelAugmentationIdealQuotient C psi) :
2059 zcCompletedGroupAlgebraKernelAugmentationQuotientTargetCoeffEnd C psi a x =
2060 zcCompletedGroupAlgebraCoeffMap C G a • x :=
2061 rfl
2063omit [IsTopologicalGroup H] in
2064/-- The coefficient action of \(\mathbb{Z}_C\) on the source augmentation quotient. -/
2065def zcCompletedGroupAlgebraKernelAugmentationQuotientTargetCoeffAction
2066 (psi : ContinuousMonoidHom G H) :
2067 ZCCoeff C →+* AddMonoid.End (KernelAugmentationIdealQuotient C psi) where
2068 toFun a := zcCompletedGroupAlgebraKernelAugmentationQuotientTargetCoeffEnd C psi a
2069 map_zero' := by
2070 refine AddMonoidHom.ext ?_
2071 intro x
2072 change zcCompletedGroupAlgebraCoeffMap C G (0 : ZCCoeff C) • x = 0
2073 rw [map_zero, zero_smul]
2074 map_one' := by
2075 refine AddMonoidHom.ext ?_
2076 intro x
2077 change zcCompletedGroupAlgebraCoeffMap C G (1 : ZCCoeff C) • x = x
2078 rw [map_one, one_smul]
2079 map_add' a b := by
2080 refine AddMonoidHom.ext ?_
2081 intro x
2082 change zcCompletedGroupAlgebraCoeffMap C G (a + b) • x =
2083 zcCompletedGroupAlgebraCoeffMap C G a • x +
2084 zcCompletedGroupAlgebraCoeffMap C G b • x
2085 rw [map_add, add_smul]
2086 map_mul' a b := by
2087 refine AddMonoidHom.ext ?_
2088 intro x
2089 change zcCompletedGroupAlgebraCoeffMap C G (a * b) • x =
2090 zcCompletedGroupAlgebraCoeffMap C G a •
2091 (zcCompletedGroupAlgebraCoeffMap C G b • x)
2092 rw [map_mul, mul_smul]
2094omit [IsTopologicalGroup H] in
2095/--
2096Coefficient elements are central with respect to group-like elements in \(\mathbb{Z}_C\llbracket
2097G\rrbracket\).
2098-/
2099theorem zcCompletedGroupAlgebraCoeffMap_mul_groupLike_eq_groupLike_mul_coeffMap
2100 (a : ZCCoeff C) (g : G) :
2101 zcCompletedGroupAlgebraCoeffMap C G a * zcGroupLike C G g =
2102 zcGroupLike C G g * zcCompletedGroupAlgebraCoeffMap C G a := by
2103 apply Subtype.ext
2104 funext i
2105 letI : Fact (0 < i.1.modulus) := ⟨i.1.positive⟩
2106 change
2107 zcCompletedGroupAlgebraProjection C G i
2108 (zcCompletedGroupAlgebraCoeffMap C G a * zcGroupLike C G g) =
2109 zcCompletedGroupAlgebraProjection C G i
2110 (zcGroupLike C G g * zcCompletedGroupAlgebraCoeffMap C G a)
2111 rw [zcCompletedGroupAlgebraProjection_mul, zcCompletedGroupAlgebraProjection_coeffMap,
2112 zcCompletedGroupAlgebraProjection_groupLike]
2113 simp [mul_comm]
2115omit [IsTopologicalGroup H] in
2116/--
2117The algebraic target group algebra \(\mathbb{Z}_C[H]\) acts on the source augmentation quotient.
2118This is the dense algebraic part of the eventual completed \(\mathbb{Z}_C\llbracket
2119H\rrbracket\) scalar action.
2120-/
2121def kerAugQuotTargetGAActionOfSurj
2122 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi) :
2123 MonoidAlgebra (ZCCoeff C) H →+*
2124 AddMonoid.End (KernelAugmentationIdealQuotient C psi) :=
2125 MonoidAlgebra.liftNCRingHom
2126 (zcCompletedGroupAlgebraKernelAugmentationQuotientTargetCoeffAction C psi)
2127 (zcCompletedGAKerAugQuotTargetGroupLikeActionOfSurjective
2128 C psi hpsi)
2129 (by
2130 intro a h
2131 rw [Commute]
2132 apply AddMonoidHom.ext
2133 intro x
2134 change
2135 zcCompletedGroupAlgebraCoeffMap C G a •
2136 (zcGroupLike C G (Function.surjInv hpsi h) • x) =
2137 zcGroupLike C G (Function.surjInv hpsi h) •
2138 (zcCompletedGroupAlgebraCoeffMap C G a • x)
2139 rw [← mul_smul, ← mul_smul,
2140 zcCompletedGroupAlgebraCoeffMap_mul_groupLike_eq_groupLike_mul_coeffMap])
2142omit [IsTopologicalGroup H] in
2143/--
2144The induced target group-algebra action evaluates on group-like elements by lifting through the
2145chosen surjection.
2146-/
2147@[simp]
2148theorem kerAugQuotTargetGAActionOfSurj_of
2149 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi) (h : H) :
2150 kerAugQuotTargetGAActionOfSurj
2151 C psi hpsi (MonoidAlgebra.of (ZCCoeff C) H h) =
2152 zcCompletedGAKerAugQuotTargetGroupLikeActionOfSurjective
2153 C psi hpsi h := by
2154 apply AddMonoidHom.ext
2155 intro x
2156 simp only [kerAugQuotTargetGAActionOfSurj, MonoidAlgebra.of_apply,
2157 MonoidAlgebra.liftNCRingHom_single,
2158 map_one, one_mul]
2160omit [IsTopologicalGroup H] in
2161/-- The descended algebraic `Z_C[H]`-module structure on the source augmentation quotient. -/
2162noncomputable abbrev
2163 zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupAlgebraModuleOfSurjective
2164 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi) :
2165 Module (MonoidAlgebra (ZCCoeff C) H) (KernelAugmentationIdealQuotient C psi) :=
2166 Module.compHom (KernelAugmentationIdealQuotient C psi)
2167 (kerAugQuotTargetGAActionOfSurj
2168 C psi hpsi)
2170omit [IsTopologicalGroup H] in
2171/-- The target group-algebra module structure acts by the induced quotient action. -/
2172@[simp]
2173theorem kerAugQuotTargetGAModuleOfSurj_smul
2174 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
2175 (a : MonoidAlgebra (ZCCoeff C) H) (x : KernelAugmentationIdealQuotient C psi) :
2176 letI : Module (MonoidAlgebra (ZCCoeff C) H) (KernelAugmentationIdealQuotient C psi) :=
2177 zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupAlgebraModuleOfSurjective
2178 C psi hpsi
2179 a • x =
2180 kerAugQuotTargetGAActionOfSurj
2181 C psi hpsi a x := by
2182 rfl
2184omit [IsTopologicalGroup H] in
2185/-- A group-like basis element acts on the target quotient module by the induced quotient action. -/
2186@[simp]
2187theorem kerAugQuotTargetGAModuleOfSurj_of_smul
2188 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi) (h : H)
2189 (x : KernelAugmentationIdealQuotient C psi) :
2190 letI : Module (MonoidAlgebra (ZCCoeff C) H) (KernelAugmentationIdealQuotient C psi) :=
2191 zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupAlgebraModuleOfSurjective
2192 C psi hpsi
2193 MonoidAlgebra.of (ZCCoeff C) H h • x =
2194 zcGroupLike C G (Function.surjInv hpsi h) • x := by
2195 letI : Module (MonoidAlgebra (ZCCoeff C) H) (KernelAugmentationIdealQuotient C psi) :=
2196 zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupAlgebraModuleOfSurjective
2197 C psi hpsi
2198 change
2199 kerAugQuotTargetGAActionOfSurj
2200 C psi hpsi (MonoidAlgebra.of (ZCCoeff C) H h) x =
2201 zcGroupLike C G (Function.surjInv hpsi h) • x
2202 rw [kerAugQuotTargetGAActionOfSurj_of]
2203 rfl
2205omit [IsTopologicalGroup H] in
2206/--
2207The algebraic target group-algebra action by \([h]\) agrees with source multiplication by any
2208lift of \(h\).
2209-/
2210theorem kerAugQuotTargetGAModuleOfSurj_of_smul_eq_source_groupLike_smul
2211 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi) {h : H} {g : G}
2212 (hg : psi g = h) (x : KernelAugmentationIdealQuotient C psi) :
2213 letI : Module (MonoidAlgebra (ZCCoeff C) H) (KernelAugmentationIdealQuotient C psi) :=
2214 zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupAlgebraModuleOfSurjective
2215 C psi hpsi
2216 MonoidAlgebra.of (ZCCoeff C) H h • x = zcGroupLike C G g • x := by
2217 letI : Module (MonoidAlgebra (ZCCoeff C) H) (KernelAugmentationIdealQuotient C psi) :=
2218 zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupAlgebraModuleOfSurjective
2219 C psi hpsi
2220 rw [kerAugQuotTargetGAModuleOfSurj_of_smul]
2221 have hlift : psi (Function.surjInv hpsi h) = psi g := by
2222 rw [Function.surjInv_eq hpsi h, hg]
2223 exact zcCompletedGroupAlgebraKernelAugmentationQuotient_groupLike_smul_eq_of_map_eq
2224 C psi hlift x
2226omit [IsTopologicalGroup H] in
2227/--
2228The source boundary is a crossed differential for the descended algebraic target group-algebra
2229coefficients.
2230-/
2231def zcCompletedGASourceBoundaryToKerAugQuotTargetGACrossedHomOfSurjective
2232 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi) :
2233 letI : Module (MonoidAlgebra (ZCCoeff C) H) (KernelAugmentationIdealQuotient C psi) :=
2234 zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupAlgebraModuleOfSurjective
2235 C psi hpsi
2236 ScalarCrossedHom
2237 ((MonoidAlgebra.of (ZCCoeff C) H).comp psi.toMonoidHom)
2238 (KernelAugmentationIdealQuotient C psi) := by
2239 letI : Module (MonoidAlgebra (ZCCoeff C) H) (KernelAugmentationIdealQuotient C psi) :=
2240 zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupAlgebraModuleOfSurjective
2241 C psi hpsi
2242 exact
2243 { toFun := zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient C psi
2244 map_mul' := by
2245 intro g h
2246 rw [(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient
2247 C psi).map_mul]
2248 congr 1
2249 exact
2250 (kerAugQuotTargetGAModuleOfSurj_of_smul_eq_source_groupLike_smul
2251 C psi hpsi (h := psi g) (g := g) rfl
2252 (zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient
2253 C psi h)).symm }
2255omit [IsTopologicalGroup H] in
2256/--
2257The algebraic target-coefficient universal differential module maps to the source augmentation
2258quotient by \(dg \mapsto\) \([g]-1\).
2259-/
2260def zcAlgebraicDifferentialModuleToKernelAugmentationQuotientOfSurjective
2261 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi) :
2262 letI : Module (MonoidAlgebra (ZCCoeff C) H) (KernelAugmentationIdealQuotient C psi) :=
2263 zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupAlgebraModuleOfSurjective
2264 C psi hpsi
2265 CrossedDifferentialModule ((MonoidAlgebra.of (ZCCoeff C) H).comp psi.toMonoidHom) →ₗ[
2266 MonoidAlgebra (ZCCoeff C) H] KernelAugmentationIdealQuotient C psi := by
2267 letI : Module (MonoidAlgebra (ZCCoeff C) H) (KernelAugmentationIdealQuotient C psi) :=
2268 zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupAlgebraModuleOfSurjective
2269 C psi hpsi
2270 exact
2271 crossedHomModuleLift
2272 (A := KernelAugmentationIdealQuotient C psi)
2273 ((MonoidAlgebra.of (ZCCoeff C) H).comp psi.toMonoidHom)
2274 (zcCompletedGASourceBoundaryToKerAugQuotTargetGACrossedHomOfSurjective
2275 C psi hpsi)
2277omit [IsTopologicalGroup H] in
2278/--
2279The algebraic differential module maps universally to the kernel-augmentation quotient in the
2280surjective case.
2281-/
2282@[simp]
2283theorem zcAlgebraicDifferentialModuleToKernelAugmentationQuotientOfSurjective_universal
2284 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi) (g : G) :
2285 letI : Module (MonoidAlgebra (ZCCoeff C) H) (KernelAugmentationIdealQuotient C psi) :=
2286 zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupAlgebraModuleOfSurjective
2287 C psi hpsi
2288 zcAlgebraicDifferentialModuleToKernelAugmentationQuotientOfSurjective C psi hpsi
2289 (universalCrossedDifferential
2290 ((MonoidAlgebra.of (ZCCoeff C) H).comp psi.toMonoidHom) g) =
2291 zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient C psi g := by
2292 letI : Module (MonoidAlgebra (ZCCoeff C) H) (KernelAugmentationIdealQuotient C psi) :=
2293 zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupAlgebraModuleOfSurjective
2294 C psi hpsi
2295 exact
2296 crossedHomModuleLift_universal
2297 (A := KernelAugmentationIdealQuotient C psi)
2298 ((MonoidAlgebra.of (ZCCoeff C) H).comp psi.toMonoidHom)
2299 (zcCompletedGASourceBoundaryToKerAugQuotTargetGACrossedHomOfSurjective
2300 C psi hpsi) g
2302omit [IsTopologicalGroup H] in
2303/--
2304Multiplying an algebraic kernel-augmentation element by a standard augmentation element lands in
2305the algebraic product \(I(\ker \psi)I(G)\). The remaining completed-target descent problem is
2306exactly replacing the first hypothesis by membership in the completed map kernel.
2307-/
2308theorem mulStandard_mul_mem_of_mem_kernelAugIdealMul
2309 (psi : ContinuousMonoidHom G H)
2310 {k : ZCCompletedGroupAlgebra C G}
2311 (hk : k ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMul C psi)
2312 (y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
2313 (⟨k * (y : ZCCompletedGroupAlgebra C G),
2314 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
2315 zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
2316 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi := by
2317 let S := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi
2318 refine Submodule.span_induction
2319 (p := fun k _ =>
2320 ∀ y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
2321 (⟨k * (y : ZCCompletedGroupAlgebra C G),
2322 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
2323 zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈ S) ?_ ?_ ?_ ?_ hk y
2324 · rintro _ ⟨n, rfl⟩ y
2325 exact zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_generator_mem C psi n y
2326 · intro y
2327 convert S.zero_mem using 1
2328 ext
2329 simp only [zero_mul, zcCompletedGroupAlgebraProjection_zero, ZeroMemClass.coe_zero]
2330 · intro a b _ _ ha hb y
2331 have hsum : (⟨a * (y : ZCCompletedGroupAlgebra C G),
2332 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left a y.2⟩ :
2333 zcCompletedGroupAlgebraStandardAugmentationIdeal C G) +
2334 (⟨b * (y : ZCCompletedGroupAlgebra C G),
2335 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left b y.2⟩ :
2336 zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈ S :=
2337 S.add_mem (ha y) (hb y)
2338 convert hsum using 1
2339 ext
2340 simp only [add_mul, zcCompletedGroupAlgebraProjection_add,
2341 zcCompletedGroupAlgebraProjection_mul,
2342 MonoidAlgebra.coeff_add, AddMemClass.mk_add_mk]
2343 · intro a b _ hb y
2344 have hsmul : a •
2345 (⟨b * (y : ZCCompletedGroupAlgebra C G),
2346 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left b y.2⟩ :
2347 zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈ S :=
2348 S.smul_mem a (hb y)
2349 convert hsmul using 1
2350 ext
2351 simp only [smul_eq_mul, mul_assoc, zcCompletedGroupAlgebraProjection_mul, SetLike.mk_smul_mk]
2353/--
2354Under the finite-stage open-map kernel theorem, a completed-kernel scalar times a standard
2355augmentation element lies in the closure of the algebraic product; no closedness hypothesis is
2356assumed.
2357-/
2358theorem zcCompletedGAKernelAugmentationIdealMulStandard_mul_mem_closure_of_mem_ker_map
2361 (hForm : ProCGroups.FiniteGroupClass.Formation C)
2362 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
2363 (hfopen : IsOpenMap psi)
2364 {k : ZCCompletedGroupAlgebra C G}
2365 (hk : k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi))
2366 (y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
2367 (⟨k * (y : ZCCompletedGroupAlgebra C G),
2368 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
2369 zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
2370 closure
2371 ((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :
2372 Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G))) := by
2373 let R := ZCCompletedGroupAlgebra C G
2374 let I := zcCompletedGroupAlgebraKernelAugmentationIdealMul C psi
2375 let S := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi
2376 let f : R → zcCompletedGroupAlgebraStandardAugmentationIdeal C G := fun a =>
2377 ⟨a * (y : R), (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left a y.2⟩
2378 have hf : Continuous f := by
2379 have hmul : Continuous (fun a : R => a * (y : R)) :=
2380 continuous_id.mul continuous_const
2381 exact Continuous.subtype_mk hmul
2382 (fun a => (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left a y.2)
2383 have hkClosure : k ∈ closure ((I : Set R)) := by
2384 have hEq :=
2385 closure_zcCompletedGAKernelAugmentationIdealMul_eq_ker_map_of_openMap_surj
2386 C hC hForm psi hpsi hfopen
2387 rw [hEq]
2388 exact hk
2389 have hmemImage : f k ∈ f '' closure ((I : Set R)) := ⟨k, hkClosure, rfl⟩
2390 have hclosureImage : f k ∈ closure (f '' ((I : Set R))) :=
2391 image_closure_subset_closure_image hf hmemImage
2392 have himage_subset : f '' ((I : Set R)) ⊆ (S : Set _) := by
2393 rintro _ ⟨a, ha, rfl⟩
2394 exact
2395 mulStandard_mul_mem_of_mem_kernelAugIdealMul
2396 C psi ha y
2397 exact closure_mono himage_subset hclosureImage
2399/--
2400If \(k\) lies in the kernel of the completed group-algebra map, then \(k y\) lies in the
2401finite-stage closed hull of \(I(\ker\psi)I(G)\) for every standard augmentation element \(y\).
2402-/
2403theorem zcCompletedGAKernelAugmentationIdealMulStandard_mul_mem_closed_of_mem_ker_map
2406 (hForm : ProCGroups.FiniteGroupClass.Formation C)
2407 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
2408 (hfopen : IsOpenMap psi)
2409 {k : ZCCompletedGroupAlgebra C G}
2410 (hk : k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi))
2411 (y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
2412 (⟨k * (y : ZCCompletedGroupAlgebra C G),
2413 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
2414 zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
2415 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
2416 C hC hForm psi hpsi hfopen := by
2417 have hclosure :=
2418 zcCompletedGAKernelAugmentationIdealMulStandard_mul_mem_closure_of_mem_ker_map
2419 C hC hForm psi hpsi hfopen hk y
2420 rwa [closure_zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_eq_closed
2421 C hC hForm psi hpsi hfopen] at hclosure
2423/--
2424Completed-kernel scalars send the standard source augmentation ideal into the finite-stage
2425closed hull of \(I(\ker \psi)I(G)\).
2426-/
2427theorem zcCompletedGAKernelAugmentationIdealMulStandard_kernelMulStandard_le_closed
2430 (hForm : ProCGroups.FiniteGroupClass.Formation C)
2431 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
2432 (hfopen : IsOpenMap psi) :
2433 ∀ k : ZCCompletedGroupAlgebra C G,
2434 k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) →
2435 ∀ y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
2436 (⟨k * (y : ZCCompletedGroupAlgebra C G),
2437 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
2438 zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
2439 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
2440 C hC hForm psi hpsi hfopen := by
2441 intro k hk y
2442 exact
2443 zcCompletedGAKernelAugmentationIdealMulStandard_mul_mem_closed_of_mem_ker_map
2444 C hC hForm psi hpsi hfopen hk y
2446/--
2447If the algebraic product \(I(\ker \psi)I(G)\) is closed in the standard augmentation ideal, then
2448completed-kernel scalars send standard augmentation elements into that product.
2449-/
2450theorem zcCompletedGAKernelAugmentationIdealMulStandard_kernelMulStandard_le_of_isClosed
2453 (hForm : ProCGroups.FiniteGroupClass.Formation C)
2454 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
2455 (hfopen : IsOpenMap psi)
2456 (hclosed :
2457 IsClosed
2458 ((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :
2459 Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)))) :
2460 ∀ k : ZCCompletedGroupAlgebra C G,
2461 k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) →
2462 ∀ y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
2463 (⟨k * (y : ZCCompletedGroupAlgebra C G),
2464 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
2465 zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
2466 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi := by
2467 intro k hk y
2468 have hclosure :=
2469 zcCompletedGAKernelAugmentationIdealMulStandard_mul_mem_closure_of_mem_ker_map
2470 C hC hForm psi hpsi hfopen hk y
2471 rwa [hclosed.closure_eq] at hclosure
2473/--
2474If the canonical map from the algebraic source augmentation quotient to the closed finite-stage
2475quotient is injective, then completed-kernel scalars multiply standard augmentation elements
2476into the algebraic product \(I(\ker \psi)I(G)\). This descent step uses finite-stage closed
2477membership and converts it back to algebraic membership through injectivity of the quotient
2478comparison map.
2479-/
2480theorem kernelMulStandard_le_of_toClosedQuotient_inj
2483 (hForm : ProCGroups.FiniteGroupClass.Formation C)
2484 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
2485 (hfopen : IsOpenMap psi)
2486 (hinj :
2487 Function.Injective
2488 (zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient
2489 C hC hForm psi hpsi hfopen)) :
2490 ∀ k : ZCCompletedGroupAlgebra C G,
2491 k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) →
2492 ∀ y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
2493 (⟨k * (y : ZCCompletedGroupAlgebra C G),
2494 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
2495 zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
2496 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi := by
2497 intro k hk y
2498 let S := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi
2499 let T :=
2500 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
2501 C hC hForm psi hpsi hfopen
2502 let x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G :=
2503 ⟨k * (y : ZCCompletedGroupAlgebra C G),
2504 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩
2505 have hxT : x ∈ T :=
2506 zcCompletedGAKernelAugmentationIdealMulStandard_mul_mem_closed_of_mem_ker_map
2507 C hC hForm psi hpsi hfopen hk y
2508 have hxmap :
2509 zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient
2510 C hC hForm psi hpsi hfopen
2511 (Submodule.Quotient.mk (p := S) x) = 0 := by
2512 rw [zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient_mk
2513 (C := C) (hC := hC) (hForm := hForm) (psi := psi)
2514 (hpsi := hpsi) (hfopen := hfopen) x]
2515 exact (Submodule.Quotient.mk_eq_zero (p := T) (x := x)).2 hxT
2516 have hxzero :
2517 (Submodule.Quotient.mk (p := S) x :
2518 KernelAugmentationIdealQuotient C psi) = 0 := by
2519 apply hinj
2520 rw [hxmap, map_zero]
2521 exact (Submodule.Quotient.mk_eq_zero (p := S) (x := x)).1 hxzero
2523/--
2524If every completed-kernel scalar sends the standard source augmentation ideal into \(I(\ker
2525\psi)I(G)\), then the completed kernel acts trivially on the quotient.
2526-/
2527theorem zcCompletedGAKerAugQuot_ker_map_smul_eq_zero_of_kernelMulStandard_le
2528 (hC : ProCGroups.FiniteGroupClass.Hereditary C) (psi : ContinuousMonoidHom G H)
2529 (hker_mul :
2530 ∀ k : ZCCompletedGroupAlgebra C G,
2531 k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) →
2532 ∀ y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
2533 (⟨k * (y : ZCCompletedGroupAlgebra C G),
2534 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
2535 zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
2536 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi)
2537 (k : ZCCompletedGroupAlgebra C G)
2538 (hk : k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi))
2539 (x : KernelAugmentationIdealQuotient C psi) :
2540 k • x = 0 := by
2541 refine Submodule.Quotient.induction_on
2542 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi) x ?_
2543 intro y
2544 apply (Submodule.Quotient.mk_eq_zero
2545 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi)).2
2546 change
2547 (⟨k * (y : ZCCompletedGroupAlgebra C G),
2548 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
2549 zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
2550 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi
2551 exact hker_mul k hk y
2553/--
2554Conditional descent of the source action to a completed target \(\mathbb{Z}_C\llbracket
2555H\rrbracket\)-module. The extra hypothesis is exactly the missing kernel-product statement; it
2556is kept explicit so that closure membership is not used as algebraic equality.
2557-/
2558def zcCompletedGroupAlgebraTargetLiftOfSurjective
2561 (hForm : ProCGroups.FiniteGroupClass.Formation C)
2562 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi) :
2563 ZCCompletedGroupAlgebra C H → ZCCompletedGroupAlgebra C G :=
2564 Function.surjInv
2565 (zcCompletedGroupAlgebraMap_surjective_of_surjective
2566 (C := C) (hC := hC) hForm psi hpsi)
2568/--
2569For a surjective presentation, the completed group-algebra map sends the chosen target lift of `a`
2570back to `a`.
2571-/
2572@[simp 900]
2573theorem zcCompletedGroupAlgebraMap_targetLiftOfSurjective
2576 (hForm : ProCGroups.FiniteGroupClass.Formation C)
2577 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
2578 (a : ZCCompletedGroupAlgebra C H) :
2579 zcCompletedGroupAlgebraMap C hC psi
2580 (zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi a) = a :=
2581 Function.surjInv_eq
2582 (zcCompletedGroupAlgebraMap_surjective_of_surjective
2583 (C := C) (hC := hC) hForm psi hpsi) a
2585/--
2586The kernel-multiple containment hypothesis makes the source augmentation quotient a module over
2587the target completed group algebra, using chosen lifts along the surjective presentation.
2588-/
2589@[implicit_reducible]
2590def zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
2593 (hForm : ProCGroups.FiniteGroupClass.Formation C)
2594 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
2595 (hker_mul :
2596 ∀ k : ZCCompletedGroupAlgebra C G,
2597 k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) →
2598 ∀ y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
2599 (⟨k * (y : ZCCompletedGroupAlgebra C G),
2600 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
2601 zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
2602 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi) :
2603 Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi) := by
2604 letI : SMul (ZCCompletedGroupAlgebra C H)
2605 (KernelAugmentationIdealQuotient C psi) :=
2606 ⟨fun a x => zcCompletedGroupAlgebraTargetLiftOfSurjective
2607 C hC hForm psi hpsi a • x⟩
2608 refine (zcCompletedGroupAlgebraMap_surjective_of_surjective
2609 (C := C) (hC := hC) hForm psi hpsi).moduleLeft
2610 (zcCompletedGroupAlgebraMap C hC psi) ?_
2611 intro a x
2612 change zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi
2613 (zcCompletedGroupAlgebraMap C hC psi a) • x =
2614 a • x
2615 have hdiff :
2616 zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi
2617 (zcCompletedGroupAlgebraMap C hC psi a) - a ∈
2618 RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) := by
2619 change zcCompletedGroupAlgebraMap C hC psi
2620 (zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi
2621 (zcCompletedGroupAlgebraMap C hC psi a) - a) = 0
2622 rw [map_sub, zcCompletedGroupAlgebraMap_targetLiftOfSurjective, sub_self]
2623 have hzero :=
2624 zcCompletedGAKerAugQuot_ker_map_smul_eq_zero_of_kernelMulStandard_le
2625 C hC psi hker_mul
2626 (zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi
2627 (zcCompletedGroupAlgebraMap C hC psi a) - a) hdiff x
2628 rw [sub_smul] at hzero
2629 exact sub_eq_zero.mp hzero
2631/--
2632Closedness of the kernel-multiple submodule supplies the containment needed to descend the target
2633completed-group-algebra module structure to the source augmentation quotient.
2634-/
2635@[implicit_reducible]
2636def zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_closed_kernelMulStandard
2639 (hForm : ProCGroups.FiniteGroupClass.Formation C)
2640 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
2641 (hfopen : IsOpenMap psi)
2642 (hclosed :
2643 IsClosed
2644 ((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :
2645 Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)))) :
2646 Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi) :=
2647 zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
2648 C hC hForm psi hpsi
2649 (zcCompletedGAKernelAugmentationIdealMulStandard_kernelMulStandard_le_of_isClosed
2650 C hC hForm psi hpsi hfopen hclosed)
2652/-- The completed kernel acts trivially on the closed source augmentation quotient. -/
2653theorem zcCompletedGroupAlgebraKernelAugmentationClosedQuotient_ker_map_smul_eq_zero
2656 (hForm : ProCGroups.FiniteGroupClass.Formation C)
2657 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
2658 (hfopen : IsOpenMap psi)
2659 (k : ZCCompletedGroupAlgebra C G)
2660 (hk : k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi))
2661 (x : KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :
2662 k • x = 0 := by
2663 refine Submodule.Quotient.induction_on
2664 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
2665 C hC hForm psi hpsi hfopen) x ?_
2666 intro y
2667 apply (Submodule.Quotient.mk_eq_zero
2668 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
2669 C hC hForm psi hpsi hfopen)).2
2670 change
2671 (⟨k * (y : ZCCompletedGroupAlgebra C G),
2672 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
2673 zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
2674 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
2675 C hC hForm psi hpsi hfopen
2676 exact
2677 zcCompletedGAKernelAugmentationIdealMulStandard_kernelMulStandard_le_closed
2678 C hC hForm psi hpsi hfopen k hk y
2680/--
2681Unconditional descent of the source action to a completed target \(\mathbb{Z}_C\llbracket
2682H\rrbracket\)-module on the closed source augmentation quotient.
2683-/
2684@[implicit_reducible]
2685def kerAugClosedQuotTargetCompletedModuleOfSurj
2688 (hForm : ProCGroups.FiniteGroupClass.Formation C)
2689 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
2690 (hfopen : IsOpenMap psi) :
2691 Module (ZCCompletedGroupAlgebra C H)
2692 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) := by
2693 letI : SMul (ZCCompletedGroupAlgebra C H)
2694 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
2695 ⟨fun a x => zcCompletedGroupAlgebraTargetLiftOfSurjective
2696 C hC hForm psi hpsi a • x⟩
2697 refine (zcCompletedGroupAlgebraMap_surjective_of_surjective
2698 (C := C) (hC := hC) hForm psi hpsi).moduleLeft
2699 (zcCompletedGroupAlgebraMap C hC psi) ?_
2700 intro a x
2701 change zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi
2702 (zcCompletedGroupAlgebraMap C hC psi a) • x =
2703 a • x
2704 have hdiff :
2705 zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi
2706 (zcCompletedGroupAlgebraMap C hC psi a) - a ∈
2707 RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) := by
2708 change zcCompletedGroupAlgebraMap C hC psi
2709 (zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi
2710 (zcCompletedGroupAlgebraMap C hC psi a) - a) = 0
2711 rw [map_sub, zcCompletedGroupAlgebraMap_targetLiftOfSurjective, sub_self]
2712 have hzero :=
2713 zcCompletedGroupAlgebraKernelAugmentationClosedQuotient_ker_map_smul_eq_zero
2714 C hC hForm psi hpsi hfopen
2715 (zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi
2716 (zcCompletedGroupAlgebraMap C hC psi a) - a) hdiff x
2717 rw [sub_smul] at hzero
2718 exact sub_eq_zero.mp hzero
2720/-- The closed quotient target module action is compatible with mapped scalars. -/
2721theorem kerAugClosedQuotTargetCompletedModuleOfSurj_map_smul
2724 (hForm : ProCGroups.FiniteGroupClass.Formation C)
2725 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
2726 (hfopen : IsOpenMap psi)
2727 (a : ZCCompletedGroupAlgebra C G)
2728 (x : KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :
2729 letI : Module (ZCCompletedGroupAlgebra C H)
2730 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
2731 kerAugClosedQuotTargetCompletedModuleOfSurj
2732 C hC hForm psi hpsi hfopen
2733 zcCompletedGroupAlgebraMap C hC psi a • x = a • x := by
2734 letI : Module (ZCCompletedGroupAlgebra C H)
2735 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
2736 kerAugClosedQuotTargetCompletedModuleOfSurj
2737 C hC hForm psi hpsi hfopen
2738 change zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi
2739 (zcCompletedGroupAlgebraMap C hC psi a) • x =
2740 a • x
2741 have hdiff :
2742 zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi
2743 (zcCompletedGroupAlgebraMap C hC psi a) - a ∈
2744 RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) := by
2745 change zcCompletedGroupAlgebraMap C hC psi
2746 (zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi
2747 (zcCompletedGroupAlgebraMap C hC psi a) - a) = 0
2748 rw [map_sub, zcCompletedGroupAlgebraMap_targetLiftOfSurjective, sub_self]
2749 have hzero :=
2750 zcCompletedGroupAlgebraKernelAugmentationClosedQuotient_ker_map_smul_eq_zero
2751 C hC hForm psi hpsi hfopen
2752 (zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi
2753 (zcCompletedGroupAlgebraMap C hC psi a) - a) hdiff x
2754 rw [sub_smul] at hzero
2755 exact sub_eq_zero.mp hzero
2757/--
2758Source scalar multiplication on the closed source augmentation quotient is continuous in the
2759source scalar, for a fixed quotient element.
2760-/
2761theorem continuous_zcCompletedGAKerAugClosedQuot_source_smul_const
2764 (hForm : ProCGroups.FiniteGroupClass.Formation C)
2765 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
2766 (hfopen : IsOpenMap psi)
2767 (x : KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :
2768 Continuous (fun a : ZCCompletedGroupAlgebra C G => a • x) := by
2769 refine Submodule.Quotient.induction_on
2770 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
2771 C hC hForm psi hpsi hfopen) x ?_
2772 intro y
2773 have hpre :
2774 Continuous (fun a : ZCCompletedGroupAlgebra C G =>
2775 (⟨a * (y : ZCCompletedGroupAlgebra C G),
2776 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left a y.2⟩ :
2777 zcCompletedGroupAlgebraStandardAugmentationIdeal C G)) := by
2778 have hmul : Continuous (fun a : ZCCompletedGroupAlgebra C G =>
2779 a * (y : ZCCompletedGroupAlgebra C G)) :=
2780 continuous_id.mul continuous_const
2781 exact Continuous.subtype_mk hmul
2782 (fun a => (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left a y.2)
2783 have hq :
2784 Continuous (fun z : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
2785 (Submodule.Quotient.mk
2786 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
2787 C hC hForm psi hpsi hfopen) z :
2788 KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)) :=
2789 continuous_quotient_mk'
2790 change Continuous (fun a : ZCCompletedGroupAlgebra C G =>
2791 (Submodule.Quotient.mk
2792 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
2793 C hC hForm psi hpsi hfopen)
2794 ⟨a * (y : ZCCompletedGroupAlgebra C G),
2795 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left a y.2⟩ :
2796 KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen))
2797 exact hq.comp hpre
2799/--
2800Descended target scalar multiplication on the closed source augmentation quotient is continuous
2801in the target scalar, for a fixed quotient element.
2802-/
2803theorem continuous_kerAugClosedQuotTargetCompletedModuleOfSurj_smul_const
2806 (hForm : ProCGroups.FiniteGroupClass.Formation C)
2807 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
2808 (hfopen : IsOpenMap psi)
2809 (x : KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :
2810 letI : Module (ZCCompletedGroupAlgebra C H)
2811 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
2812 kerAugClosedQuotTargetCompletedModuleOfSurj
2813 C hC hForm psi hpsi hfopen
2814 Continuous (fun a : ZCCompletedGroupAlgebra C H => a • x) := by
2815 letI : Module (ZCCompletedGroupAlgebra C H)
2816 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
2817 kerAugClosedQuotTargetCompletedModuleOfSurj
2818 C hC hForm psi hpsi hfopen
2819 let q := zcCompletedGroupAlgebraMap C hC psi
2820 have hq : Topology.IsQuotientMap q :=
2821 isQuotientMap_zcCompletedGroupAlgebraMap_of_surjective C hC hForm psi hpsi
2822 rw [hq.continuous_iff]
2823 change Continuous (fun a : ZCCompletedGroupAlgebra C G => q a • x)
2824 have hsource :=
2825 continuous_zcCompletedGAKerAugClosedQuot_source_smul_const
2826 C hC hForm psi hpsi hfopen x
2827 have hEq :
2828 (fun a : ZCCompletedGroupAlgebra C G => q a • x) =
2829 (fun a : ZCCompletedGroupAlgebra C G => a • x) := by
2830 funext a
2831 exact
2832 kerAugClosedQuotTargetCompletedModuleOfSurj_map_smul
2833 C hC hForm psi hpsi hfopen a x
2834 simpa [hEq] using hsource
2836/--
2837Source scalar multiplication on the closed source augmentation quotient is jointly continuous.
2838-/
2839theorem continuous_zcCompletedGroupAlgebraKernelAugmentationClosedQuotient_source_smul
2842 (hForm : ProCGroups.FiniteGroupClass.Formation C)
2843 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
2844 (hfopen : IsOpenMap psi) :
2845 Continuous (fun p : ZCCompletedGroupAlgebra C G ×
2846 KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen =>
2847 p.1 • p.2) := by
2848 let S :=
2849 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
2850 C hC hForm psi hpsi hfopen
2851 have hquot :
2852 IsOpenQuotientMap
2853 (Prod.map (id : ZCCompletedGroupAlgebra C G → ZCCompletedGroupAlgebra C G)
2854 (fun z : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
2855 (Submodule.Quotient.mk (p := S) z :
2856 KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen))) :=
2857 IsOpenQuotientMap.id.prodMap S.isOpenQuotientMap_mkQ
2858 rw [← hquot.continuous_comp_iff]
2859 have hpre :
2860 Continuous (fun p : ZCCompletedGroupAlgebra C G ×
2861 zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
2862 (⟨p.1 * (p.2 : ZCCompletedGroupAlgebra C G),
2863 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left p.1 p.2.2⟩ :
2864 zcCompletedGroupAlgebraStandardAugmentationIdeal C G)) := by
2865 have hmul : Continuous (fun p : ZCCompletedGroupAlgebra C G ×
2866 zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
2867 p.1 * (p.2 : ZCCompletedGroupAlgebra C G)) :=
2868 continuous_fst.mul (continuous_subtype_val.comp continuous_snd)
2869 exact Continuous.subtype_mk hmul
2870 (fun p => (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left p.1 p.2.2)
2871 have hmk :
2872 Continuous (fun z : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
2873 (Submodule.Quotient.mk (p := S) z :
2874 KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)) :=
2875 continuous_quotient_mk'
2876 change Continuous (fun p : ZCCompletedGroupAlgebra C G ×
2877 zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
2878 (Submodule.Quotient.mk (p := S)
2879 ⟨p.1 * (p.2 : ZCCompletedGroupAlgebra C G),
2880 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left p.1 p.2.2⟩ :
2881 KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen))
2882 exact hmk.comp hpre
2884/--
2885Descended target scalar multiplication on the closed source augmentation quotient is jointly
2886continuous.
2887-/
2888theorem continuous_kerAugClosedQuotTargetCompletedModuleOfSurj_smul
2891 (hForm : ProCGroups.FiniteGroupClass.Formation C)
2892 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
2893 (hfopen : IsOpenMap psi) :
2894 letI : Module (ZCCompletedGroupAlgebra C H)
2895 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
2896 kerAugClosedQuotTargetCompletedModuleOfSurj
2897 C hC hForm psi hpsi hfopen
2898 Continuous (fun p : ZCCompletedGroupAlgebra C H ×
2899 KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen =>
2900 p.1 • p.2) := by
2901 letI : Module (ZCCompletedGroupAlgebra C H)
2902 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
2903 kerAugClosedQuotTargetCompletedModuleOfSurj
2904 C hC hForm psi hpsi hfopen
2905 let q := zcCompletedGroupAlgebraMap C hC psi
2906 have hq : IsOpenQuotientMap q :=
2907 isOpenQuotientMap_zcCompletedGroupAlgebraMap_of_surjective C hC hForm psi hpsi
2908 have hquot :
2909 IsOpenQuotientMap
2910 (Prod.map q
2911 (id :
2912 KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen →
2913 KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)) :=
2914 hq.prodMap IsOpenQuotientMap.id
2915 rw [← hquot.continuous_comp_iff]
2916 have hsource :=
2917 continuous_zcCompletedGroupAlgebraKernelAugmentationClosedQuotient_source_smul
2918 C hC hForm psi hpsi hfopen
2919 have hEq :
2920 (fun p : ZCCompletedGroupAlgebra C G ×
2921 KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen =>
2922 q p.1 • p.2) =
2923 (fun p : ZCCompletedGroupAlgebra C G ×
2924 KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen =>
2925 p.1 • p.2) := by
2926 funext p
2927 exact
2928 kerAugClosedQuotTargetCompletedModuleOfSurj_map_smul
2929 C hC hForm psi hpsi hfopen p.1 p.2
2930 simpa [Function.comp_def, Prod.map, hEq] using hsource
2932/--
2933The closed source augmentation quotient is a topological module for the source completed group
2934algebra.
2935-/
2936theorem continuousSMul_zcCompletedGroupAlgebraKernelAugmentationClosedQuotient_source
2939 (hForm : ProCGroups.FiniteGroupClass.Formation C)
2940 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
2941 (hfopen : IsOpenMap psi) :
2942 ContinuousSMul (ZCCompletedGroupAlgebra C G)
2943 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) where
2944 continuous_smul :=
2945 continuous_zcCompletedGroupAlgebraKernelAugmentationClosedQuotient_source_smul
2946 C hC hForm psi hpsi hfopen
2948/--
2949The descended target module structure on the closed source augmentation quotient is topological.
2950-/
2951theorem continuousSMul_kerAugClosedQuotTargetCompletedModuleOfSurj
2954 (hForm : ProCGroups.FiniteGroupClass.Formation C)
2955 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
2956 (hfopen : IsOpenMap psi) :
2957 letI : Module (ZCCompletedGroupAlgebra C H)
2958 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
2959 kerAugClosedQuotTargetCompletedModuleOfSurj
2960 C hC hForm psi hpsi hfopen
2961 ContinuousSMul (ZCCompletedGroupAlgebra C H)
2962 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) := by
2963 letI : Module (ZCCompletedGroupAlgebra C H)
2964 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
2965 kerAugClosedQuotTargetCompletedModuleOfSurj
2966 C hC hForm psi hpsi hfopen
2967 exact
2968 ⟨continuous_kerAugClosedQuotTargetCompletedModuleOfSurj_smul
2969 C hC hForm psi hpsi hfopen⟩
2971/-- Group-like elements act on the closed quotient target module by the induced target action. -/
2972theorem kerAugClosedQuotTargetCompletedModuleOfSurj_groupLike_smul
2975 (hForm : ProCGroups.FiniteGroupClass.Formation C)
2976 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
2977 (hfopen : IsOpenMap psi)
2978 (g : G) (x : KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :
2979 letI : Module (ZCCompletedGroupAlgebra C H)
2980 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
2981 kerAugClosedQuotTargetCompletedModuleOfSurj
2982 C hC hForm psi hpsi hfopen
2983 zcGroupLike C H (psi g) • x = zcGroupLike C G g • x := by
2984 letI : Module (ZCCompletedGroupAlgebra C H)
2985 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
2986 kerAugClosedQuotTargetCompletedModuleOfSurj
2987 C hC hForm psi hpsi hfopen
2988 rw [← zcCompletedGroupAlgebraMap_groupLike (C := C) (hC := hC) psi g]
2989 exact
2990 kerAugClosedQuotTargetCompletedModuleOfSurj_map_smul
2991 C hC hForm psi hpsi hfopen (zcGroupLike C G g) x
2993/--
2994The source boundary to the closed source augmentation quotient is a crossed differential for the
2995descended completed target scalars.
2996-/
2997def zcCompletedGASourceBoundaryToKerAugClosedQuotTargetCompletedCrossedHomOfSurjective
3000 (hForm : ProCGroups.FiniteGroupClass.Formation C)
3001 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
3002 (hfopen : IsOpenMap psi) :
3003 letI : Module (ZCCompletedGroupAlgebra C H)
3004 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
3005 kerAugClosedQuotTargetCompletedModuleOfSurj
3006 C hC hForm psi hpsi hfopen
3007 ScalarCrossedHom
3008 (zcCompletedGroupAlgebraScalar C psi.toMonoidHom)
3009 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) := by
3010 letI : Module (ZCCompletedGroupAlgebra C H)
3011 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
3012 kerAugClosedQuotTargetCompletedModuleOfSurj
3013 C hC hForm psi hpsi hfopen
3014 exact
3015 { toFun :=
3016 zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
3017 C hC hForm psi hpsi hfopen
3018 map_mul' := by
3019 intro g h
3020 rw [(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
3021 C hC hForm psi hpsi hfopen).map_mul]
3022 congr 1
3023 change zcGroupLike C G g •
3024 zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
3025 C hC hForm psi hpsi hfopen h =
3026 zcGroupLike C H (psi g) •
3027 zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
3028 C hC hForm psi hpsi hfopen h
3029 exact
3030 (kerAugClosedQuotTargetCompletedModuleOfSurj_groupLike_smul
3031 C hC hForm psi hpsi hfopen g
3032 (zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
3033 C hC hForm psi hpsi hfopen h)).symm }
3035/--
3036Projecting a chosen completed source lift of a target coefficient to a source stage and then
3037passing to the open-image stage recovers the corresponding target finite-stage projection.
3038-/
3039theorem zcCompletedGroupAlgebraOpenImageStageRingHom_projection_targetLift
3042 (hForm : ProCGroups.FiniteGroupClass.Formation C)
3043 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
3044 (hfopen : IsOpenMap psi)
3045 (i : ZCCompletedGroupAlgebraIndex C G)
3046 (a : ZCCompletedGroupAlgebra C H) :
3047 zcCompletedGroupAlgebraOpenImageStageRingHom C hC hForm psi hpsi hfopen i
3048 (zcCompletedGroupAlgebraProjection C G i
3049 (zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi a)) =
3050 zcCompletedGroupAlgebraProjection C H
3051 (zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i) a := by
3052 let b := zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi a
3053 have hsource :
3054 (i.1,
3055 completedGroupAlgebraComapIndexInClass (G := G) (H := H) C hC psi
3056 (zcCompletedGroupAlgebraOpenImageIndexInClass C hForm psi hpsi hfopen i)) ≤ i :=
3057 ⟨le_rfl, zcCompletedGroupAlgebraOpenImage_comapIndex_le C hC hForm psi hpsi hfopen i⟩
3058 have hstage :
3059 zcCompletedGroupAlgebraOpenImageStageRingHom C hC hForm psi hpsi hfopen i
3060 (zcCompletedGroupAlgebraProjection C G i b) =
3061 zcCompletedGroupAlgebraMapStage C hC psi
3062 (zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i)
3063 (zcCompletedGroupAlgebraTransition C G hsource
3064 (zcCompletedGroupAlgebraProjection C G i b)) := by
3065 exact congrFun
3066 (congrArg DFunLike.coe
3067 (zcCompletedGroupAlgebraOpenImageQuotientMap_stage_eq
3068 C hC hForm psi hpsi hfopen i))
3069 (zcCompletedGroupAlgebraProjection C G i b)
3070 rw [hstage, zcCompletedGroupAlgebraTransition_projection]
3071 change
3072 zcCompletedGroupAlgebraMapStage C hC psi
3073 (zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i)
3074 (zcCompletedGroupAlgebraProjection C G
3075 ((zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i).1,
3076 completedGroupAlgebraComapIndexInClass (G := G) (H := H) C hC psi
3077 (zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i).2) b) =
3078 zcCompletedGroupAlgebraProjection C H
3079 (zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i) a
3080 have hprojmap :=
3081 zcCompletedGroupAlgebraProjection_map
3082 (C := C) (hC := hC) (H := G) (K := H) (φ := psi)
3083 (i := zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i)
3084 (x := b)
3085 have hmap : zcCompletedGroupAlgebraMap C hC psi b = a := by
3086 dsimp [b]
3087 exact zcCompletedGroupAlgebraMap_targetLiftOfSurjective C hC hForm psi hpsi a
3088 have hmain :
3089 zcCompletedGroupAlgebraProjection C H
3090 (zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i)
3091 (zcCompletedGroupAlgebraMap C hC psi b) =
3092 zcCompletedGroupAlgebraProjection C H
3093 (zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i) a :=
3094 congrArg
3095 (zcCompletedGroupAlgebraProjection C H
3096 (zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i))
3097 hmap
3098 exact hprojmap.symm.trans hmain
3100/--
3101The i-th closed augmentation quotient coordinate of the completed source-boundary lift factors
3102through the corresponding open-image finite pre-stage.
3103-/
3104theorem kerAugIdealClosedQuotStageProj_liftLinear_eq_boundaryLift_preStageMap
3107 (hForm : ProCGroups.FiniteGroupClass.Formation C)
3108 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
3109 (hfopen : IsOpenMap psi)
3110 (i : ZCCompletedGroupAlgebraIndex C G)
3111 (x : CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G) :
3112 letI : Module (ZCCompletedGroupAlgebra C H)
3113 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
3114 kerAugClosedQuotTargetCompletedModuleOfSurj
3115 C hC hForm psi hpsi hfopen
3116 kernelAugmentationIdealClosedQuotientStageProjection
3117 C hC hForm psi hpsi hfopen i
3118 (crossedDifferentialModuleLiftLinear
3119 (R := ZCCompletedGroupAlgebra C H)
3120 (zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
3121 C hC hForm psi hpsi hfopen) x) =
3122 kernelAugmentationIdealClosedStageQuotientBoundaryLift
3123 C hC hForm psi hpsi hfopen i
3124 (zcCompletedDifferentialModulePreStageMap C psi.toMonoidHom
3125 (zcCompletedDifferentialModuleOpenImageIndex C hForm psi hpsi hfopen i) x) := by
3126 letI : Module (ZCCompletedGroupAlgebra C H)
3127 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
3128 kerAugClosedQuotTargetCompletedModuleOfSurj
3129 C hC hForm psi hpsi hfopen
3130 let j := zcCompletedDifferentialModuleOpenImageIndex C hForm psi hpsi hfopen i
3131 letI : Module (zcCompletedDifferentialModuleStageRing C psi.toMonoidHom j)
3132 (KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) :=
3133 kernelAugmentationIdealClosedStageQuotientTargetStageModule
3134 C hC hForm psi hpsi hfopen i
3135 letI : Module
3136 (ZCCompletedGroupAlgebraStage C H
3137 (zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i))
3138 (KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) :=
3139 kernelAugmentationIdealClosedStageQuotientTargetStageModule
3140 C hC hForm psi hpsi hfopen i
3141 refine Finsupp.induction_linear x ?zero ?add ?single
3142 · simp only [crossedDifferentialModuleLiftLinear, map_zero, ContinuousMonoidHom.coe_toMonoidHom,
3143 Lean.Elab.WF.paramLet, kernelAugmentationIdealClosedStageQuotientBoundaryLift,
3144 zcCompletedDifferentialModulePreStageMap]
3145 · intro x y hx hy
3146 simp only [map_add, hx, ContinuousMonoidHom.coe_toMonoidHom, Lean.Elab.WF.paramLet, hy]
3147 · intro g a
3148 rw [crossedDifferentialModuleLiftLinear_single,
3149 zcCompletedDifferentialModulePreStageMap_single,
3150 kernelAugmentationIdealClosedStageQuotientBoundaryLift_single]
3151 let b := zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi a
3152 have htarget :
3153 a • zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
3154 C hC hForm psi hpsi hfopen g =
3155 b • zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
3156 C hC hForm psi hpsi hfopen g := by
3157 rfl
3158 rw [htarget]
3159 change
3160 kernelAugmentationIdealClosedQuotientStageProjection
3161 C hC hForm psi hpsi hfopen i
3162 (b • zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
3163 C hC hForm psi hpsi hfopen g) =
3164 zcCompletedGroupAlgebraProjection C H
3165 (zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i) a •
3166 kernelAugmentationIdealClosedStageQuotientBoundary
3167 C hC hForm psi hpsi hfopen i
3168 (zcCompletedDifferentialModuleStageSourceProj C psi.toMonoidHom j g)
3169 rw [map_smulₛₗ]
3170 rw [← zcCompletedGroupAlgebraOpenImageStageRingHom_projection_targetLift
3171 C hC hForm psi hpsi hfopen i a]
3172 rw [kernelAugmentationIdealClosedStageQuotientTargetStageModule_map_smul
3173 (C := C) (hC := hC) (hForm := hForm) (psi := psi)
3174 (hpsi := hpsi) (hfopen := hfopen) (i := i)
3175 (a := zcCompletedGroupAlgebraProjection C G i b)
3176 (x := kernelAugmentationIdealClosedStageQuotientBoundary
3177 C hC hForm psi hpsi hfopen i
3178 (zcCompletedDifferentialModuleStageSourceProj C psi.toMonoidHom j g))]
3179 have hstage_boundary :
3180 kernelAugmentationIdealClosedQuotientStageProjection
3181 C hC hForm psi hpsi hfopen i
3182 (zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
3183 C hC hForm psi hpsi hfopen g) =
3184 kernelAugmentationIdealClosedStageQuotientBoundary
3185 C hC hForm psi hpsi hfopen i
3186 (zcCompletedDifferentialModuleStageSourceProj C psi.toMonoidHom j g) := by
3187 rw [zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient]
3188 let s : zcCompletedGroupAlgebraStandardAugmentationIdeal C G :=
3189 ⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
3190 zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
3191 C G (MonoidHom.id G) g⟩
3192 change
3193 kernelAugmentationIdealClosedQuotientStageProjection
3194 C hC hForm psi hpsi hfopen i
3195 (Submodule.Quotient.mk
3196 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
3197 C hC hForm psi hpsi hfopen) s) =
3198 kernelAugmentationIdealClosedStageQuotientBoundary
3199 C hC hForm psi hpsi hfopen i
3200 (zcCompletedDifferentialModuleStageSourceProj C psi.toMonoidHom j g)
3201 let T :=
3202 zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
3203 C hC hForm psi hpsi hfopen i
3204 calc
3205 kernelAugmentationIdealClosedQuotientStageProjection
3206 C hC hForm psi hpsi hfopen i
3207 (Submodule.Quotient.mk
3208 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
3209 C hC hForm psi hpsi hfopen) s) =
3210 (Submodule.Quotient.mk
3211 (p := T)
3212 (zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i s) :
3213 KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) := by
3214 exact
3215 kernelAugmentationIdealClosedQuotientStageProjection_mk
3216 C hC hForm psi hpsi hfopen i s
3217 _ =
3218 kernelAugmentationIdealClosedStageQuotientBoundary
3219 C hC hForm psi hpsi hfopen i
3220 (zcCompletedDifferentialModuleStageSourceProj C psi.toMonoidHom j g) := by
3221 simp only [kernelAugmentationIdealClosedStageQuotientBoundary]
3222 change
3223 Submodule.Quotient.mk (p := T)
3224 (zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i s) =
3225 Submodule.Quotient.mk (p := T)
3226 (zcCompletedGroupAlgebraStageAugmentationGeneratorSubtype C G i
3227 (zcCompletedDifferentialModuleStageSourceProj
3228 C psi.toMonoidHom j g))
3229 apply congrArg (fun y : zcCompletedGroupAlgebraStageAugmentationIdeal C G i =>
3230 (Submodule.Quotient.mk (p := T) y :
3231 KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i))
3232 apply Subtype.ext
3233 change
3234 zcCompletedGroupAlgebraProjection C G i
3235 (zcGroupLike C G g - 1) =
3236 MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
3237 (CompletedGroupAlgebraQuotientInClass G C i.2)
3238 (QuotientGroup.mk g) - 1
3239 rw [zcCompletedGroupAlgebraProjection_sub,
3240 zcCompletedGroupAlgebraProjection_groupLike,
3241 zcCompletedGroupAlgebraProjection_one]
3242 rw [hstage_boundary]
3243 rfl
3245/--
3246Each finite closed-augmentation coordinate of the pre-quotient source-boundary lift is
3247continuous for the finite-stage pre-module topology.
3248-/
3249theorem continuous_kernelAugmentationIdealClosedQuotientStageProjection_liftLinear
3252 (hForm : ProCGroups.FiniteGroupClass.Formation C)
3253 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
3254 (hfopen : IsOpenMap psi)
3255 (i : ZCCompletedGroupAlgebraIndex C G) :
3256 letI : Module (ZCCompletedGroupAlgebra C H)
3257 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
3258 kerAugClosedQuotTargetCompletedModuleOfSurj
3259 C hC hForm psi hpsi hfopen
3260 @Continuous
3261 (CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)
3262 (KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i)
3263 (zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom)
3264 inferInstance
3265 (fun x =>
3266 kernelAugmentationIdealClosedQuotientStageProjection
3267 C hC hForm psi hpsi hfopen i
3268 (crossedDifferentialModuleLiftLinear
3269 (R := ZCCompletedGroupAlgebra C H)
3270 (zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
3271 C hC hForm psi hpsi hfopen) x)) := by
3272 letI : Module (ZCCompletedGroupAlgebra C H)
3273 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
3274 kerAugClosedQuotTargetCompletedModuleOfSurj
3275 C hC hForm psi hpsi hfopen
3276 let j := zcCompletedDifferentialModuleOpenImageIndex C hForm psi hpsi hfopen i
3277 letI : Module (zcCompletedDifferentialModuleStageRing C psi.toMonoidHom j)
3278 (KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) :=
3279 kernelAugmentationIdealClosedStageQuotientTargetStageModule
3280 C hC hForm psi hpsi hfopen i
3281 letI : TopologicalSpace
3282 (CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G) :=
3283 zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom
3284 letI : TopologicalSpace
3285 (CrossedDifferentialPreModule
3286 (zcCompletedDifferentialModuleStageRing C psi.toMonoidHom j)
3287 (zcCompletedDifferentialModuleStageSource C psi.toMonoidHom j)) :=
3288 ⊥
3289 letI : DiscreteTopology
3290 (CrossedDifferentialPreModule
3291 (zcCompletedDifferentialModuleStageRing C psi.toMonoidHom j)
3292 (zcCompletedDifferentialModuleStageSource C psi.toMonoidHom j)) :=
3293 ⟨rfl⟩
3294 have hpre :
3295 @Continuous
3296 (CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)
3297 (CrossedDifferentialPreModule
3298 (zcCompletedDifferentialModuleStageRing C psi.toMonoidHom j)
3299 (zcCompletedDifferentialModuleStageSource C psi.toMonoidHom j))
3300 (zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom)
3301 inferInstance
3302 (zcCompletedDifferentialModulePreStageMap C psi.toMonoidHom j) :=
3303 continuous_zcCompletedDifferentialModulePreStageMap_naturalTopology
3304 C psi.toMonoidHom j
3305 have hfinite :
3306 Continuous
3307 (kernelAugmentationIdealClosedStageQuotientBoundaryLift
3308 C hC hForm psi hpsi hfopen i) :=
3309 continuous_of_discreteTopology
3310 have hfactor :
3311 (fun x : CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G =>
3312 kernelAugmentationIdealClosedQuotientStageProjection
3313 C hC hForm psi hpsi hfopen i
3314 (crossedDifferentialModuleLiftLinear
3315 (R := ZCCompletedGroupAlgebra C H)
3316 (zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
3317 C hC hForm psi hpsi hfopen) x)) =
3318 fun x =>
3319 kernelAugmentationIdealClosedStageQuotientBoundaryLift
3320 C hC hForm psi hpsi hfopen i
3321 (zcCompletedDifferentialModulePreStageMap C psi.toMonoidHom j x) := by
3322 funext x
3323 exact
3324 kerAugIdealClosedQuotStageProj_liftLinear_eq_boundaryLift_preStageMap
3325 C hC hForm psi hpsi hfopen i x
3326 rw [hfactor]
3327 exact hfinite.comp hpre
3329/--
3330The pre-quotient source-boundary lift to the closed source augmentation quotient is continuous
3331for the finite-stage pre-module topology.
3332-/
3333theorem continuous_crossedDiffModuleLiftLinear_sourceBoundaryToKerAugClosedQuot
3336 (hForm : ProCGroups.FiniteGroupClass.Formation C)
3337 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
3338 (hfopen : IsOpenMap psi) :
3339 letI : Module (ZCCompletedGroupAlgebra C H)
3340 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
3341 kerAugClosedQuotTargetCompletedModuleOfSurj
3342 C hC hForm psi hpsi hfopen
3343 @Continuous
3344 (CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)
3345 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
3346 (zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom)
3347 inferInstance
3348 (crossedDifferentialModuleLiftLinear
3349 (R := ZCCompletedGroupAlgebra C H)
3350 (zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
3351 C hC hForm psi hpsi hfopen)) := by
3352 letI : Module (ZCCompletedGroupAlgebra C H)
3353 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
3354 kerAugClosedQuotTargetCompletedModuleOfSurj
3355 C hC hForm psi hpsi hfopen
3356 letI : TopologicalSpace
3357 (CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G) :=
3358 zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom
3359 rw [kernelAugmentationIdealClosedQuotient_topology_eq_induced_stageProjProduct
3360 C hC hForm psi hpsi hfopen]
3361 rw [continuous_induced_rng]
3362 exact continuous_pi fun i =>
3363 continuous_kernelAugmentationIdealClosedQuotientStageProjection_liftLinear
3364 C hC hForm psi hpsi hfopen i
3366/--
3367The separated universal differential is also a crossed differential for source completed
3368group-algebra scalars after restricting scalars along \(\mathbb{Z}_C\llbracket G\rrbracket \to
3369\mathbb{Z}_C\llbracket H\rrbracket\).
3370-/
3371def zcSeparatedUniversalDifferentialSourceCompletedCrossedHom
3373 (psi : ContinuousMonoidHom G H) :
3374 letI : Module (ZCCompletedGroupAlgebra C G)
3375 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
3376 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
3377 ScalarCrossedHom
3378 (zcCompletedGroupAlgebraScalar C (MonoidHom.id G))
3379 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) := by
3380 letI : Module (ZCCompletedGroupAlgebra C G)
3381 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
3382 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
3383 exact
3384 { toFun := zcSeparatedUniversalDifferential C psi.toMonoidHom
3385 map_mul' := by
3386 intro g h
3387 rw [(zcSeparatedUniversalDifferential C psi.toMonoidHom).map_mul]
3388 congr 1
3389 change zcGroupLike C H (psi g) •
3390 zcSeparatedUniversalDifferential C psi.toMonoidHom h =
3391 zcCompletedGroupAlgebraMap C hC psi (zcGroupLike C G g) •
3392 zcSeparatedUniversalDifferential C psi.toMonoidHom h
3393 rw [zcCompletedGroupAlgebraMap_groupLike] }
3395/--
3396The source-identity completed differential module maps to the separated module for \(\psi\) by
3397\(dg \mapsto\) \(d_{\mathrm{sep}}\) g, with source scalars restricted through
3398\(\mathbb{Z}_C\llbracket G\rrbracket\) \(\to\) \(\mathbb{Z}_C\llbracket H\rrbracket\).
3399-/
3400def zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
3403 (psi : ContinuousMonoidHom G H) :
3404 letI : Module (ZCCompletedGroupAlgebra C G)
3405 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
3406 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
3407 ZCCompletedDifferentialModule C (MonoidHom.id G) →ₗ[ZCCompletedGroupAlgebra C G]
3408 ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom := by
3409 letI : Module (ZCCompletedGroupAlgebra C G)
3410 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
3411 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
3412 exact
3413 crossedHomModuleLift
3414 (A := ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)
3415 (zcCompletedGroupAlgebraScalar C (MonoidHom.id G))
3416 (zcSeparatedUniversalDifferentialSourceCompletedCrossedHom C hC psi)
3418/--
3419The comparison map from the identity differential module to the separated completed module sends
3420the universal differential to the separated universal differential.
3421-/
3422@[simp]
3423theorem zcDiffModuleIdToZCSepDiffModule_universal
3426 (psi : ContinuousMonoidHom G H) (g : G) :
3427 zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule C hC psi
3428 (zcUniversalDifferential C (MonoidHom.id G) g) =
3429 zcSeparatedUniversalDifferential C psi.toMonoidHom g := by
3430 letI : Module (ZCCompletedGroupAlgebra C G)
3431 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
3432 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
3433 exact
3434 crossedHomModuleLift_universal
3435 (A := ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)
3436 (zcCompletedGroupAlgebraScalar C (MonoidHom.id G))
3437 (zcSeparatedUniversalDifferentialSourceCompletedCrossedHom C hC psi) g
3439attribute [local implicit_reducible] zcCompletedDifferentialModuleIdentitySourceIndex
3441/--
3442The finite source-identity coefficient map agrees with first projecting a completed source
3443coefficient down to the source-identity stage and then applying the finite target map.
3444-/
3445theorem zcCompletedDifferentialModuleIdentitySourceStageRingHom_transition_mapStage
3447 (psi : ContinuousMonoidHom G H)
3448 (i : ZCCompletedDifferentialModuleIndex C psi.toMonoidHom) :
3449 let sourceIndex : ZCCompletedGroupAlgebraIndex C G :=
3450 (i.target.1, completedGroupAlgebraComapIndexInClass
3451 (G := G) (H := H) C hC psi i.target.2)
3452 let idIndex := zcCompletedDifferentialModuleIdentitySourceIndex C psi.toMonoidHom i
3453 let hle : sourceIndex ≤ idIndex.target := by
3454 constructor
3455 · exact le_rfl
3456 · exact i.compatible
3457 ∀ x : ZCCompletedGroupAlgebraStage C G idIndex.target,
3458 zcCompletedGroupAlgebraMapStage C hC psi i.target
3459 (zcCompletedGroupAlgebraTransition C G hle x) =
3460 zcCompletedDifferentialModuleIdentitySourceStageRingHom C psi.toMonoidHom i x := by
3461 intro sourceIndex idIndex hle x
3462 refine MonoidAlgebra.induction_on
3463 (p := fun x : ZCCompletedGroupAlgebraStage C G idIndex.target =>
3464 zcCompletedGroupAlgebraMapStage C hC psi i.target
3465 (zcCompletedGroupAlgebraTransition C G hle x) =
3466 zcCompletedDifferentialModuleIdentitySourceStageRingHom C psi.toMonoidHom i x)
3467 x ?single ?add ?smul
3468 · intro q
3469 refine QuotientGroup.induction_on q ?_
3470 intro g
3471 dsimp only [sourceIndex, idIndex, hle,
3472 zcCompletedDifferentialModuleIdentitySourceIndex] at ⊢
3473 rw [zcCompletedGroupAlgebraTransition_of]
3474 simp only [MonoidAlgebra.of_apply]
3475 simp only [zcCompletedGroupAlgebraMapStage,
3476 zcCompletedDifferentialModuleIdentitySourceStageRingHom]
3477 change MonoidAlgebra.mapDomain _ _ = MonoidAlgebra.mapDomain _ _
3478 dsimp only [OpenNormalSubgroupInClass.map]
3479 rw [QuotientGroup.map_mk]
3480 dsimp only [CompletedGroupAlgebraQuotientInClass,
3481 openNormalSubgroupInClassSystem] at ⊢
3482 rw [MonoidAlgebra.mapDomain_single, MonoidAlgebra.mapDomain_single]
3483 change MonoidAlgebra.single
3484 ((completedGroupAlgebraComapQuotientMapInClass (G := G) (H := H) C hC psi i.target.2)
3485 (QuotientGroup.mk'
3486 ((((OrderDual.ofDual (completedGroupAlgebraComapIndexInClass
3487 (G := G) (H := H) C hC psi i.target.2)).1 :
3488 OpenNormalSubgroup G) : Subgroup G)) g)) 1 =
3489 MonoidAlgebra.single
3490 ((zcCompletedDifferentialModuleStagePsi C psi.toMonoidHom i)
3491 (QuotientGroup.mk' (i.source.1 : Subgroup G) g)) 1
3492 rw [completedGroupAlgebraComapQuotientMapInClass_mk]
3493 change MonoidAlgebra.single (QuotientGroup.mk'
3494 ((((OrderDual.ofDual i.target.2).1 : OpenNormalSubgroup H) : Subgroup H))
3495 (psi g)) 1 =
3496 MonoidAlgebra.single
3497 ((QuotientGroup.map (i.source.1 : Subgroup G)
3498 ((((OrderDual.ofDual i.target.2).1 : OpenNormalSubgroup H) : Subgroup H))
3499 psi.toMonoidHom i.compatible)
3500 (QuotientGroup.mk' (i.source.1 : Subgroup G) g)) 1
3501 rw [QuotientGroup.map_mk']
3502 rfl
3503 · intro x y hx hy
3504 calc
3505 zcCompletedGroupAlgebraMapStage C hC psi i.target
3506 (zcCompletedGroupAlgebraTransition C G hle (x + y)) =
3507 zcCompletedGroupAlgebraMapStage C hC psi i.target
3508 (zcCompletedGroupAlgebraTransition C G hle x +
3509 zcCompletedGroupAlgebraTransition C G hle y) := by
3510 rw [map_add]
3511 _ =
3512 zcCompletedGroupAlgebraMapStage C hC psi i.target
3513 (zcCompletedGroupAlgebraTransition C G hle x) +
3514 zcCompletedGroupAlgebraMapStage C hC psi i.target
3515 (zcCompletedGroupAlgebraTransition C G hle y) := by
3516 rw [map_add]
3517 _ =
3518 zcCompletedDifferentialModuleIdentitySourceStageRingHom
3519 C psi.toMonoidHom i x +
3520 zcCompletedDifferentialModuleIdentitySourceStageRingHom
3521 C psi.toMonoidHom i y := by
3522 rw [hx, hy]
3523 _ =
3524 zcCompletedDifferentialModuleIdentitySourceStageRingHom
3525 C psi.toMonoidHom i (x + y) := by
3526 exact
3527 (map_add
3528 (zcCompletedDifferentialModuleIdentitySourceStageRingHom
3529 C psi.toMonoidHom i) x y).symm
3530 · intro r x hx
3531 rcases ZMod.intCast_surjective r with ⟨t, rfl⟩
3532 rw [Algebra.smul_def, RingHom.map_mul, RingHom.map_mul, hx]
3533 simp only [zcCompletedGroupAlgebraMapStage,
3534 zcCompletedGroupAlgebraTransition, modNCompletedGroupAlgebraStageCoeffMapInClass,
3535 modNCompletedGroupRingCoeffMap, AlgHom.toRingHom_eq_coe, map_intCast]
3536 symm
3537 calc
3538 zcCompletedDifferentialModuleIdentitySourceStageRingHom
3539 C psi.toMonoidHom i (↑t * x) =
3540 zcCompletedDifferentialModuleIdentitySourceStageRingHom
3541 C psi.toMonoidHom i (↑t) *
3542 zcCompletedDifferentialModuleIdentitySourceStageRingHom
3543 C psi.toMonoidHom i x := by
3544 exact map_mul _ _ _
3545 _ = ↑t *
3546 zcCompletedDifferentialModuleIdentitySourceStageRingHom
3547 C psi.toMonoidHom i x := by
3548 rw [map_intCast]
3550/--
3551Completed source coefficients viewed at the identity-source stage agree with target finite
3552projections after applying the completed group-algebra map.
3553-/
3554theorem zcCompletedDifferentialModuleIdentitySourceStageRingHom_projection_map
3556 (psi : ContinuousMonoidHom G H)
3557 (i : ZCCompletedDifferentialModuleIndex C psi.toMonoidHom)
3558 (a : ZCCompletedGroupAlgebra C G) :
3559 zcCompletedDifferentialModuleIdentitySourceStageRingHom C psi.toMonoidHom i
3560 (zcCompletedGroupAlgebraProjection C G
3561 (zcCompletedDifferentialModuleIdentitySourceIndex C psi.toMonoidHom i).target a) =
3562 zcCompletedGroupAlgebraProjection C H i.target
3563 (zcCompletedGroupAlgebraMap C hC psi a) := by
3564 let sourceIndex : ZCCompletedGroupAlgebraIndex C G :=
3565 (i.target.1, completedGroupAlgebraComapIndexInClass
3566 (G := G) (H := H) C hC psi i.target.2)
3567 let idIndex := zcCompletedDifferentialModuleIdentitySourceIndex C psi.toMonoidHom i
3568 have hle : sourceIndex ≤ idIndex.target := by
3569 constructor
3570 · exact le_rfl
3571 · exact i.compatible
3572 rw [zcCompletedGroupAlgebraProjection_map]
3573 rw [← zcCompletedDifferentialModuleIdentitySourceStageRingHom_transition_mapStage
3574 C hC psi i (zcCompletedGroupAlgebraProjection C G idIndex.target a)]
3575 rw [zcCompletedGroupAlgebraTransition_projection]
3577/--
3578Finite-stage projections of the identity-source lift to the separated \(\psi\)-module are
3579computed by first projecting to the matching source-identity finite stage.
3580-/
3581theorem zcDiffModuleIdToZCSepDiffModule_stageProj
3584 (psi : ContinuousMonoidHom G H)
3585 (i : ZCCompletedDifferentialModuleIndex C psi.toMonoidHom)
3586 (x : ZCCompletedDifferentialModule C (MonoidHom.id G)) :
3587 zcSeparatedCompletedDifferentialModuleStageProjectionAdd C psi.toMonoidHom i
3588 (zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule C hC psi x) =
3589 zcCompletedDifferentialModuleIdentitySourceStageToStage C psi.toMonoidHom i
3590 (zcCompletedDifferentialModuleStageProjection C (MonoidHom.id G)
3591 (zcCompletedDifferentialModuleIdentitySourceIndex C psi.toMonoidHom i) x) := by
3592 letI : Module (ZCCompletedGroupAlgebra C G)
3593 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
3594 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
3595 letI : Module (ZCCompletedGroupAlgebra C G)
3596 (ZCCompletedDifferentialModuleStage C psi.toMonoidHom i) :=
3597 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
3598 let idIndex := zcCompletedDifferentialModuleIdentitySourceIndex C psi.toMonoidHom i
3599 letI : Module
3600 (zcCompletedDifferentialModuleStageRing C (MonoidHom.id G) idIndex)
3601 (ZCCompletedDifferentialModuleStage C psi.toMonoidHom i) :=
3602 Module.compHom _ (zcCompletedDifferentialModuleIdentitySourceStageRingHom C psi.toMonoidHom i)
3603 let L :
3604 ZCCompletedDifferentialModule C (MonoidHom.id G) →ₗ[ZCCompletedGroupAlgebra C G]
3605 ZCCompletedDifferentialModuleStage C psi.toMonoidHom i :=
3606 { toFun := fun x =>
3607 zcSeparatedCompletedDifferentialModuleStageProjectionAdd C psi.toMonoidHom i
3608 (zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule C hC psi x)
3609 map_add' := by
3610 intro x y
3611 rw [map_add, map_add]
3612 map_smul' := by
3613 intro a x
3614 rw [map_smul]
3615 change
3616 zcSeparatedCompletedDifferentialModuleStageProjectionAdd C psi.toMonoidHom i
3617 (zcCompletedGroupAlgebraMap C hC psi a •
3618 zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
3619 C hC psi x) =
3620 zcCompletedGroupAlgebraMap C hC psi a •
3621 zcSeparatedCompletedDifferentialModuleStageProjectionAdd C psi.toMonoidHom i
3622 (zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
3623 C hC psi x)
3624 rw [map_smul] }
3625 let R :
3626 ZCCompletedDifferentialModule C (MonoidHom.id G) →ₗ[ZCCompletedGroupAlgebra C G]
3627 ZCCompletedDifferentialModuleStage C psi.toMonoidHom i :=
3628 { toFun := fun x =>
3629 zcCompletedDifferentialModuleIdentitySourceStageToStage C psi.toMonoidHom i
3630 (zcCompletedDifferentialModuleStageProjection C (MonoidHom.id G)
3631 (zcCompletedDifferentialModuleIdentitySourceIndex C psi.toMonoidHom i) x)
3632 map_add' := by
3633 intro x y
3634 rw [map_add, map_add]
3635 map_smul' := by
3636 intro a x
3637 rw [map_smul]
3638 change
3639 zcCompletedDifferentialModuleIdentitySourceStageToStage C psi.toMonoidHom i
3640 (zcCompletedGroupAlgebraProjection C G
3641 idIndex.target a •
3642 zcCompletedDifferentialModuleStageProjection C (MonoidHom.id G)
3643 idIndex x) =
3644 zcCompletedGroupAlgebraProjection C H i.target
3645 (zcCompletedGroupAlgebraMap C hC psi a) •
3646 zcCompletedDifferentialModuleIdentitySourceStageToStage C psi.toMonoidHom i
3647 (zcCompletedDifferentialModuleStageProjection C (MonoidHom.id G)
3648 idIndex x)
3649 rw [map_smul]
3650 change
3651 zcCompletedDifferentialModuleIdentitySourceStageRingHom C psi.toMonoidHom i
3652 (zcCompletedGroupAlgebraProjection C G idIndex.target a) •
3653 zcCompletedDifferentialModuleIdentitySourceStageToStage C psi.toMonoidHom i
3654 (zcCompletedDifferentialModuleStageProjection C (MonoidHom.id G)
3655 idIndex x) =
3656 zcCompletedGroupAlgebraProjection C H i.target
3657 (zcCompletedGroupAlgebraMap C hC psi a) •
3658 zcCompletedDifferentialModuleIdentitySourceStageToStage C psi.toMonoidHom i
3659 (zcCompletedDifferentialModuleStageProjection C (MonoidHom.id G)
3660 idIndex x)
3661 rw [zcCompletedDifferentialModuleIdentitySourceStageRingHom_projection_map C hC psi i a] }
3662 have hLR : L = R := by
3663 apply crossedDifferentialModuleHom_ext
3664 (zcCompletedGroupAlgebraScalar C (MonoidHom.id G))
3665 intro g
3666 change
3667 zcSeparatedCompletedDifferentialModuleStageProjectionAdd C psi.toMonoidHom i
3668 (zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule C hC psi
3669 (zcUniversalDifferential C (MonoidHom.id G) g)) =
3670 zcCompletedDifferentialModuleIdentitySourceStageToStage C psi.toMonoidHom i
3671 (zcCompletedDifferentialModuleStageProjection C (MonoidHom.id G)
3672 idIndex (zcUniversalDifferential C (MonoidHom.id G) g))
3673 rw [zcDiffModuleIdToZCSepDiffModule_universal,
3674 zcSeparatedCompletedDifferentialModuleStageProjectionAdd_universal]
3675 calc
3676 zcCompletedDifferentialModuleStageDifferential C psi.toMonoidHom i g =
3677 zcCompletedDifferentialModuleStageProjection C psi.toMonoidHom i
3678 (zcUniversalDifferential C psi.toMonoidHom g) := by
3679 rw [zcCompletedDifferentialModuleStageProjection_universal]
3680 _ =
3681 zcCompletedDifferentialModuleIdentitySourceStageToStage C psi.toMonoidHom i
3682 (zcCompletedDifferentialModuleStageProjection C (MonoidHom.id G)
3683 idIndex (zcUniversalDifferential C (MonoidHom.id G) g)) := by
3684 exact
3685 (zcDiffModuleIdentitySourceStageToStage_stageProj_universal
3686 C psi.toMonoidHom i g).symm
3687 exact LinearMap.congr_fun hLR x
3689/--
3690The finite comparison from the source-identity stage to the \(\psi\)-stage commutes with the
3691finite Fox boundary.
3692-/
3693theorem zcCompletedDifferentialModuleStageBoundary_identitySourceStageToStage
3694 (ψ : G →* H)
3695 (i : ZCCompletedDifferentialModuleIndex C ψ)
3696 (x : ZCCompletedDifferentialModuleStage C (MonoidHom.id G)
3697 (zcCompletedDifferentialModuleIdentitySourceIndex C ψ i)) :
3698 zcCompletedDifferentialModuleStageBoundary C ψ i
3699 (zcCompletedDifferentialModuleIdentitySourceStageToStage C ψ i x) =
3700 zcCompletedDifferentialModuleIdentitySourceStageRingHom C ψ i
3701 (zcCompletedDifferentialModuleStageBoundary C (MonoidHom.id G)
3702 (zcCompletedDifferentialModuleIdentitySourceIndex C ψ i) x) := by
3703 let j := zcCompletedDifferentialModuleIdentitySourceIndex C ψ i
3704 letI : Module (zcCompletedDifferentialModuleStageRing C (MonoidHom.id G) j)
3705 (zcCompletedDifferentialModuleStageRing C ψ i) :=
3706 Module.compHom _ (zcCompletedDifferentialModuleIdentitySourceStageRingHom C ψ i)
3707 letI : Module (zcCompletedDifferentialModuleStageRing C (MonoidHom.id G) j)
3708 (ZCCompletedDifferentialModuleStage C ψ i) :=
3709 Module.compHom _ (zcCompletedDifferentialModuleIdentitySourceStageRingHom C ψ i)
3710 let ringMapLinear :
3711 zcCompletedDifferentialModuleStageRing C (MonoidHom.id G) j →ₗ[
3712 zcCompletedDifferentialModuleStageRing C (MonoidHom.id G) j]
3713 zcCompletedDifferentialModuleStageRing C ψ i := {
3714 toFun := zcCompletedDifferentialModuleIdentitySourceStageRingHom C ψ i
3715 map_add' := by
3716 intro a b
3717 exact map_add (zcCompletedDifferentialModuleIdentitySourceStageRingHom C ψ i) a b
3718 map_smul' := by
3719 intro a b
3720 change
3721 zcCompletedDifferentialModuleIdentitySourceStageRingHom C ψ i (a * b) =
3722 zcCompletedDifferentialModuleIdentitySourceStageRingHom C ψ i a *
3723 zcCompletedDifferentialModuleIdentitySourceStageRingHom C ψ i b
3724 exact map_mul (zcCompletedDifferentialModuleIdentitySourceStageRingHom C ψ i) a b }
3725 let L :
3726 ZCCompletedDifferentialModuleStage C (MonoidHom.id G) j →ₗ[
3727 zcCompletedDifferentialModuleStageRing C (MonoidHom.id G) j]
3728 zcCompletedDifferentialModuleStageRing C ψ i := {
3729 toFun := fun x =>
3730 zcCompletedDifferentialModuleStageBoundary C ψ i
3731 (zcCompletedDifferentialModuleIdentitySourceStageToStage C ψ i x)
3732 map_add' := by
3733 intro x y
3734 rw [map_add, map_add]
3735 map_smul' := by
3736 intro a x
3737 rw [map_smul]
3738 change
3739 zcCompletedDifferentialModuleStageBoundary C ψ i
3740 (zcCompletedDifferentialModuleIdentitySourceStageRingHom C ψ i a •
3741 zcCompletedDifferentialModuleIdentitySourceStageToStage C ψ i x) =
3742 zcCompletedDifferentialModuleIdentitySourceStageRingHom C ψ i a •
3743 zcCompletedDifferentialModuleStageBoundary C ψ i
3744 (zcCompletedDifferentialModuleIdentitySourceStageToStage C ψ i x)
3745 rw [map_smul] }
3746 let R :
3747 ZCCompletedDifferentialModuleStage C (MonoidHom.id G) j →ₗ[
3748 zcCompletedDifferentialModuleStageRing C (MonoidHom.id G) j]
3749 zcCompletedDifferentialModuleStageRing C ψ i :=
3750 ringMapLinear.comp
3751 (zcCompletedDifferentialModuleStageBoundary C (MonoidHom.id G) j)
3752 have hLR : L = R := by
3753 apply crossedDifferentialModuleHom_ext
3754 (coeff := zcCompletedDifferentialModuleStageScalar C (MonoidHom.id G) j)
3755 intro q
3756 change
3757 zcCompletedDifferentialModuleStageBoundary C ψ i
3758 (zcCompletedDifferentialModuleIdentitySourceStageToStage C ψ i
3759 (universalCrossedDifferential
3760 (zcCompletedDifferentialModuleStageScalar C (MonoidHom.id G) j) q)) =
3761 zcCompletedDifferentialModuleIdentitySourceStageRingHom C ψ i
3762 (zcCompletedDifferentialModuleStageBoundary C (MonoidHom.id G) j
3763 (universalCrossedDifferential
3764 (zcCompletedDifferentialModuleStageScalar C (MonoidHom.id G) j) q))
3765 rw [zcCompletedDifferentialModuleIdentitySourceStageToStage_universal]
3766 have hboundaryTarget :
3767 zcCompletedDifferentialModuleStageBoundary C ψ i
3768 (universalCrossedDifferential
3769 (zcCompletedDifferentialModuleStageScalar C ψ i) q) =
3770 zcCompletedDifferentialModuleStageScalar C ψ i q - 1 := by
3771 rw [zcCompletedDifferentialModuleStageBoundary,
3772 crossedHomModuleLift_universal, coefficientFoxBoundaryCrossedHom_apply]
3773 have hboundarySource :
3774 zcCompletedDifferentialModuleStageBoundary C (MonoidHom.id G) j
3775 (universalCrossedDifferential
3776 (zcCompletedDifferentialModuleStageScalar C (MonoidHom.id G) j) q) =
3777 zcCompletedDifferentialModuleStageScalar C (MonoidHom.id G) j q - 1 := by
3778 rw [zcCompletedDifferentialModuleStageBoundary,
3779 crossedHomModuleLift_universal, coefficientFoxBoundaryCrossedHom_apply]
3780 rw [hboundaryTarget, hboundarySource]
3781 rw [map_sub, map_one,
3782 zcCompletedDifferentialModuleIdentitySourceStageRingHom_stageScalar]
3783 exact LinearMap.congr_fun hLR x
3785omit [IsTopologicalGroup H] in
3786/--
3787Applying the finite identity boundary after the source-identity finite projection recovers the
3788finite projection of the standard augmentation-valued completed Fox tail.
3789-/
3790theorem zcCompletedDifferentialModuleIdentitySourceStageBoundary_stageProj
3792 (psi : ContinuousMonoidHom G H)
3793 (i : ZCCompletedDifferentialModuleIndex C psi.toMonoidHom)
3794 (x : ZCCompletedDifferentialModule C (MonoidHom.id G)) :
3795 let j := zcCompletedDifferentialModuleIdentitySourceIndex C psi.toMonoidHom i
3796 zcCompletedDifferentialModuleStageBoundary C (MonoidHom.id G) j
3797 (zcCompletedDifferentialModuleStageProjection C (MonoidHom.id G) j x) =
3798 ((zcCompletedGroupAlgebraStandardAugmentationIdealProjection C j.target
3799 (zcToStdAugIdeal C G (MonoidHom.id G) x) :
3800 zcCompletedGroupAlgebraStageAugmentationIdeal C G j.target) :
3801 ZCCompletedGroupAlgebraStage C G j.target) := by
3802 intro j
3803 have hcomp := congrArg (fun f => f x)
3804 (zcDiffModuleStageBoundaryCompletedLinearMap_comp_stageProj
3805 C (MonoidHom.id G) j)
3806 simpa [LinearMap.comp_apply,
3807 zcToStdAugIdeal_val] using hcomp
3809/--
3810The identity-source lift to the separated \(\psi\)-module kills the kernel of the standard
3811augmentation-valued completed Fox tail.
3812-/
3813theorem zcDiffModuleIdToZCSepDiffModule_eq_zero_of_zcToStandard_eq_zero
3816 (psi : ContinuousMonoidHom G H)
3817 (x : ZCCompletedDifferentialModule C (MonoidHom.id G))
3818 (hx :
3819 zcToStdAugIdeal C G (MonoidHom.id G) x = 0) :
3820 zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule C hC psi x = 0 := by
3821 have hxFox : zcToCompletedGroupAlgebra C (MonoidHom.id G) x = 0 := by
3822 have hxval :=
3823 congrArg
3824 (fun y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
3825 (y : ZCCompletedGroupAlgebra C G)) hx
3826 simpa [zcToStdAugIdeal_val] using hxval
3827 apply zcSeparatedCompletedDifferentialModuleStageProjectionsSeparate C psi.toMonoidHom
3828 intro i
3829 rw [zcDiffModuleIdToZCSepDiffModule_stageProj]
3830 have hsource :
3831 zcCompletedDifferentialModuleStageProjection C (MonoidHom.id G)
3832 (zcCompletedDifferentialModuleIdentitySourceIndex C psi.toMonoidHom i) x = 0 :=
3833 zcDiffModuleIdentitySourceStageProj_eq_zero_of_zcTo_eq_zero
3834 C psi.toMonoidHom i x hxFox
3835 rw [hsource, map_zero]
3837omit [IsTopologicalGroup G] in
3838/--
3839The standard-augmentation-valued completed Fox tail is continuous for the finite-stage natural
3840topology on the completed differential module.
3841-/
3842theorem continuous_zcToStdAugIdeal_naturalTopology
3844 (psi : ContinuousMonoidHom G H) :
3845 @Continuous
3846 (ZCCompletedDifferentialModule C psi.toMonoidHom)
3847 (zcCompletedGroupAlgebraStandardAugmentationIdeal C H)
3848 (zcCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom)
3849 inferInstance
3850 (zcToStdAugIdeal C H psi.toMonoidHom) := by
3851 letI : TopologicalSpace (ZCCompletedDifferentialModule C psi.toMonoidHom) :=
3852 zcCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
3853 have hval :
3854 @Continuous
3855 (ZCCompletedDifferentialModule C psi.toMonoidHom)
3856 (ZCCompletedGroupAlgebra C H)
3857 (zcCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom)
3858 inferInstance
3859 (fun x =>
3860 (zcToStdAugIdeal
3861 C H psi.toMonoidHom x : ZCCompletedGroupAlgebra C H)) := by
3862 have hfun :
3863 (fun x : ZCCompletedDifferentialModule C psi.toMonoidHom =>
3864 (zcToStdAugIdeal C H psi.toMonoidHom x : ZCCompletedGroupAlgebra C H)) =
3865 zcToCompletedGroupAlgebra C psi.toMonoidHom := by
3866 funext x
3867 exact zcToStdAugIdeal_val C H psi.toMonoidHom x
3868 rw [hfun]
3869 exact continuous_zcToCompletedGroupAlgebra_naturalTopology C hC psi
3870 exact Continuous.subtype_mk hval
3871 (fun x => (zcToStdAugIdeal
3872 C H psi.toMonoidHom x).2)
3874/--
3875Kernel group-like source scalars act trivially on the separated module after scalar restriction
3876along \(\mathbb{Z}_C\llbracket G\rrbracket \to \mathbb{Z}_C\llbracket H\rrbracket\).
3877-/
3878theorem zcSeparatedCompletedDifferentialModule_source_kernel_groupLike_smul
3880 (psi : ContinuousMonoidHom G H) (n : ProfiniteKernelSubgroup psi)
3881 (x : ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :
3882 letI : Module (ZCCompletedGroupAlgebra C G)
3883 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
3884 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
3885 zcGroupLike C G n.1 • x = x := by
3886 letI : Module (ZCCompletedGroupAlgebra C G)
3887 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
3888 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
3889 change zcCompletedGroupAlgebraMap C hC psi (zcGroupLike C G n.1) • x = x
3890 rw [zcCompletedGroupAlgebraMap_groupLike]
3891 have hn : psi n.1 = 1 := by
3892 exact MonoidHom.mem_ker.mp
3893 (show n.1 ∈ psi.toMonoidHom.ker from n.2)
3894 rw [hn]
3895 rw [map_one]
3896 exact one_smul _ x
3898/--
3899Source scalar restriction on the separated module is exactly target scalar multiplication after
3900applying the completed group-algebra map.
3901-/
3902theorem zcSeparatedCompletedDifferentialModule_source_map_smul
3904 (psi : ContinuousMonoidHom G H)
3905 (a : ZCCompletedGroupAlgebra C G)
3906 (x : ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :
3907 letI : Module (ZCCompletedGroupAlgebra C G)
3908 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
3909 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
3910 zcCompletedGroupAlgebraMap C hC psi a • x = a • x := by
3911 letI : Module (ZCCompletedGroupAlgebra C G)
3912 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
3913 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
3914 rfl
3916/--
3917Elements of the source kernel augmentation ideal act trivially on the separated module after
3918restricting scalars along the source map.
3919-/
3920theorem zcSeparatedCompletedDifferentialModule_source_kernel_sub_one_smul
3923 (psi : ContinuousMonoidHom G H) (n : ProfiniteKernelSubgroup psi)
3924 (x : ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :
3925 letI : Module (ZCCompletedGroupAlgebra C G)
3926 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
3927 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
3928 (zcGroupLike C G n.1 - 1) • x = 0 := by
3929 letI : Module (ZCCompletedGroupAlgebra C G)
3930 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
3931 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
3932 calc
3933 (zcGroupLike C G n.1 - 1) • x =
3934 zcGroupLike C G n.1 • x - x := by
3935 rw [sub_smul, one_smul]
3936 _ = 0 := by
3937 rw [zcSeparatedCompletedDifferentialModule_source_kernel_groupLike_smul C hC psi n x]
3938 exact sub_self x
3940/--
3941The identity-source lift to the separated module kills source kernel augmentation generators
3942after scalar multiplication.
3943-/
3944theorem zcDiffModuleIdToZCSepDiffModule_kernel_sub_one_smul
3947 (psi : ContinuousMonoidHom G H) (n : ProfiniteKernelSubgroup psi)
3948 (x : ZCCompletedDifferentialModule C (MonoidHom.id G)) :
3949 zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule C hC psi
3950 ((zcGroupLike C G n.1 - 1) • x) = 0 := by
3951 letI : Module (ZCCompletedGroupAlgebra C G)
3952 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
3953 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
3954 rw [map_smul]
3955 exact
3956 zcSeparatedCompletedDifferentialModule_source_kernel_sub_one_smul
3957 C hC psi n
3958 (zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
3959 C hC psi x)
3961/-- A conditional source-standard-augmentation map to the separated module. The only remaining
3962well-definedness input is that the identity Fox tail kernel is killed by the source-identity
3963lift to the separated module. -/
3964noncomputable def
3965 stdAugIdealToZCSepDiffOfBoundaryKernel
3968 (psi : ContinuousMonoidHom G H)
3969 (hker :
3970 ∀ x : ZCCompletedDifferentialModule C (MonoidHom.id G),
3971 zcToStdAugIdeal C G (MonoidHom.id G) x = 0 →
3972 zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
3973 C hC psi x = 0) :
3974 letI : Module (ZCCompletedGroupAlgebra C G)
3975 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
3976 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
3977 zcCompletedGroupAlgebraStandardAugmentationIdeal C G →ₗ[ZCCompletedGroupAlgebra C G]
3978 ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom := by
3979 letI : Module (ZCCompletedGroupAlgebra C G)
3980 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
3981 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
3982 let f :=
3983 zcToStdAugIdeal C G (MonoidHom.id G)
3984 let L :=
3985 zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule C hC psi
3986 have hf : Function.Surjective f := by
3987 exact
3988 zcToStdAugIdeal_surjective_of_surjective
3989 C G (MonoidHom.id G) (fun g => ⟨g, rfl⟩)
3990 have hker_le : LinearMap.ker f ≤ LinearMap.ker L := by
3991 intro x hx
3992 rw [LinearMap.mem_ker] at hx ⊢
3993 exact hker x hx
3994 exact
3995 ((LinearMap.ker f).liftQ L hker_le).comp
3996 (f.quotKerEquivOfSurjective hf).symm.toLinearMap
3998/-- The source-standard-augmentation map to the separated module. -/
3999noncomputable def
4000 stdAugIdealToZCSepDiff
4003 (psi : ContinuousMonoidHom G H) :
4004 letI : Module (ZCCompletedGroupAlgebra C G)
4005 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
4006 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
4007 zcCompletedGroupAlgebraStandardAugmentationIdeal C G →ₗ[ZCCompletedGroupAlgebra C G]
4008 ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom :=
4009 stdAugIdealToZCSepDiffOfBoundaryKernel
4010 C hC psi
4011 (zcDiffModuleIdToZCSepDiffModule_eq_zero_of_zcToStandard_eq_zero
4012 C hC psi)
4014/-- The conditional map out of the standard augmentation ideal recovers the canonical map from
4015the completed differential module after composing with its quotient map. -/
4016@[simp 900]
4017theorem
4018 stdAugIdealToZCSepDiffOfBoundaryKernel_comp_zcToStdAugIdeal
4021 (psi : ContinuousMonoidHom G H)
4022 (hker :
4023 ∀ x : ZCCompletedDifferentialModule C (MonoidHom.id G),
4024 zcToStdAugIdeal C G (MonoidHom.id G) x = 0 →
4025 zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
4026 C hC psi x = 0) :
4027 letI : Module (ZCCompletedGroupAlgebra C G)
4028 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
4029 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
4030 (stdAugIdealToZCSepDiffOfBoundaryKernel
4031 C hC psi hker).comp
4032 (zcToStdAugIdeal C G (MonoidHom.id G)) =
4033 zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule C hC psi := by
4034 letI : Module (ZCCompletedGroupAlgebra C G)
4035 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
4036 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
4037 let f :=
4038 zcToStdAugIdeal C G (MonoidHom.id G)
4039 let L :=
4040 zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule C hC psi
4041 have hf : Function.Surjective f := by
4042 exact
4043 zcToStdAugIdeal_surjective_of_surjective
4044 C G (MonoidHom.id G) (fun g => ⟨g, rfl⟩)
4045 have hker_le : LinearMap.ker f ≤ LinearMap.ker L := by
4046 intro x hx
4047 rw [LinearMap.mem_ker] at hx ⊢
4048 exact hker x hx
4049 apply LinearMap.ext
4050 intro x
4051 change
4052 (((LinearMap.ker f).liftQ L hker_le).comp
4053 (f.quotKerEquivOfSurjective hf).symm.toLinearMap).comp f x = L x
4054 rw [LinearMap.comp_apply, LinearMap.comp_apply]
4055 have hsymm :
4056 (f.quotKerEquivOfSurjective hf).symm.toLinearMap (f x) =
4057 Submodule.Quotient.mk x := by
4058 exact LinearMap.quotKerEquivOfSurjective_symm_apply (f := f) hf x
4059 rw [hsymm, Submodule.liftQ_apply]
4061/-- The standard augmentation-ideal map recovers the canonical separated differential-module
4062map after precomposition with the quotient map. -/
4063@[simp]
4064theorem
4065 stdAugIdealToZCSepDiff_comp_zcToStdAugIdeal
4068 (psi : ContinuousMonoidHom G H) :
4069 letI : Module (ZCCompletedGroupAlgebra C G)
4070 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
4071 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
4072 (stdAugIdealToZCSepDiff
4073 C hC psi).comp
4074 (zcToStdAugIdeal C G (MonoidHom.id G)) =
4075 zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule C hC psi := by
4076 exact
4077 stdAugIdealToZCSepDiffOfBoundaryKernel_comp_zcToStdAugIdeal
4078 C hC psi
4079 (zcDiffModuleIdToZCSepDiffModule_eq_zero_of_zcToStandard_eq_zero
4080 C hC psi)
4082/-- Finite-stage boundary formula for the source-standard map into the separated module. -/
4083theorem
4084 stdAugIdealToZCSepDiff_stageBoundary_stageProj
4087 (psi : ContinuousMonoidHom G H)
4088 (i : ZCCompletedDifferentialModuleIndex C psi.toMonoidHom)
4089 (s : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
4090 zcCompletedDifferentialModuleStageBoundary C psi.toMonoidHom i
4091 (zcSeparatedCompletedDifferentialModuleStageProjectionAdd C psi.toMonoidHom i
4092 (stdAugIdealToZCSepDiff
4093 C hC psi s)) =
4094 zcCompletedGroupAlgebraProjection C H i.target
4095 (zcCompletedGroupAlgebraMap C hC psi
4096 (s : ZCCompletedGroupAlgebra C G)) := by
4097 letI : Module (ZCCompletedGroupAlgebra C G)
4098 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
4099 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
4100 let f :=
4101 zcToStdAugIdeal C G (MonoidHom.id G)
4102 let M :=
4103 stdAugIdealToZCSepDiff
4104 C hC psi
4105 let L :=
4106 zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule C hC psi
4107 have hf : Function.Surjective f := by
4108 exact
4109 zcToStdAugIdeal_surjective_of_surjective
4110 C G (MonoidHom.id G) (fun g => ⟨g, rfl⟩)
4111 rcases hf s with ⟨x, hx⟩
4112 rw [← hx]
4113 have hcomp :=
4114 congrArg (fun F => F x)
4115 (stdAugIdealToZCSepDiff_comp_zcToStdAugIdeal
4116 C hC psi)
4117 change M (f x) = L x at hcomp
4118 rw [hcomp]
4119 rw [zcDiffModuleIdToZCSepDiffModule_stageProj]
4120 rw [zcCompletedDifferentialModuleStageBoundary_identitySourceStageToStage]
4121 rw [zcCompletedDifferentialModuleIdentitySourceStageBoundary_stageProj]
4122 let idIndex := zcCompletedDifferentialModuleIdentitySourceIndex C psi.toMonoidHom i
4123 have hmap :=
4124 zcCompletedDifferentialModuleIdentitySourceStageRingHom_projection_map
4125 C hC psi i (f x : ZCCompletedGroupAlgebra C G)
4126 simpa [f, idIndex] using hmap
4128/-- The finite-stage projection of the source-standard map depends only on the matching
4129source-identity finite projection of the standard augmentation ideal. -/
4130theorem
4131 stdAugIdealToZCSepDiff_stageProj_eq_of_standardProj_eq
4134 (psi : ContinuousMonoidHom G H)
4135 (i : ZCCompletedDifferentialModuleIndex C psi.toMonoidHom)
4136 {s t : zcCompletedGroupAlgebraStandardAugmentationIdeal C G}
4137 (hst :
4138 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C
4139 (zcCompletedDifferentialModuleIdentitySourceIndex C psi.toMonoidHom i).target s =
4140 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C
4141 (zcCompletedDifferentialModuleIdentitySourceIndex C psi.toMonoidHom i).target t) :
4142 zcSeparatedCompletedDifferentialModuleStageProjectionAdd C psi.toMonoidHom i
4143 (stdAugIdealToZCSepDiff
4144 C hC psi s) =
4145 zcSeparatedCompletedDifferentialModuleStageProjectionAdd C psi.toMonoidHom i
4146 (stdAugIdealToZCSepDiff
4147 C hC psi t) := by
4148 letI : Module (ZCCompletedGroupAlgebra C G)
4149 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
4150 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
4151 let f :=
4152 zcToStdAugIdeal C G (MonoidHom.id G)
4153 let M :=
4154 stdAugIdealToZCSepDiff
4155 C hC psi
4156 let L :=
4157 zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule C hC psi
4158 let idIndex := zcCompletedDifferentialModuleIdentitySourceIndex C psi.toMonoidHom i
4159 have hf : Function.Surjective f := by
4160 exact
4161 zcToStdAugIdeal_surjective_of_surjective
4162 C G (MonoidHom.id G) (fun g => ⟨g, rfl⟩)
4163 rcases hf s with ⟨x, hx⟩
4164 rcases hf t with ⟨y, hy⟩
4165 have hxyProjection :
4166 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C idIndex.target (f x) =
4167 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C idIndex.target (f y) := by
4168 simpa [idIndex, hx, hy] using hst
4169 have hcomp_x :=
4170 congrArg (fun F => F x)
4171 (stdAugIdealToZCSepDiff_comp_zcToStdAugIdeal
4172 C hC psi)
4173 have hcomp_y :=
4174 congrArg (fun F => F y)
4175 (stdAugIdealToZCSepDiff_comp_zcToStdAugIdeal
4176 C hC psi)
4177 change M (f x) = L x at hcomp_x
4178 change M (f y) = L y at hcomp_y
4179 rw [← hx, ← hy, hcomp_x, hcomp_y]
4180 rw [zcDiffModuleIdToZCSepDiffModule_stageProj,
4181 zcDiffModuleIdToZCSepDiffModule_stageProj]
4182 have hsource :
4183 zcCompletedDifferentialModuleStageProjection C (MonoidHom.id G) idIndex x =
4184 zcCompletedDifferentialModuleStageProjection C (MonoidHom.id G) idIndex y := by
4185 apply sub_eq_zero.mp
4186 have hzero :
4187 zcCompletedDifferentialModuleStageProjection C (MonoidHom.id G) idIndex (x - y) = 0 :=
4188 zcDiffModuleIdentitySourceStageProj_eq_zero_of_boundary_eq_zero
4189 C psi.toMonoidHom i (x - y) (by
4190 rw [map_sub]
4191 rw [map_sub]
4192 rw [zcCompletedDifferentialModuleIdentitySourceStageBoundary_stageProj,
4193 zcCompletedDifferentialModuleIdentitySourceStageBoundary_stageProj]
4194 rw [hxyProjection, sub_self])
4195 simpa [map_sub] using hzero
4196 rw [hsource]
4198/-- Each finite-stage coordinate of the source-standard map into the separated module is
4199continuous. The coordinate factors through the matching finite standard augmentation
4200projection. -/
4201theorem
4202 continuous_stdAugIdealToZCSepDiff_stageProj
4205 (psi : ContinuousMonoidHom G H)
4206 (i : ZCCompletedDifferentialModuleIndex C psi.toMonoidHom) :
4207 Continuous
4208 (fun s : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
4209 zcSeparatedCompletedDifferentialModuleStageProjectionAdd C psi.toMonoidHom i
4210 (stdAugIdealToZCSepDiff
4211 C hC psi s)) := by
4212 let idIndex := zcCompletedDifferentialModuleIdentitySourceIndex C psi.toMonoidHom i
4213 let p :=
4214 zcCompletedGroupAlgebraStandardAugmentationIdealProjection C idIndex.target
4215 have hsurj : Function.Surjective p :=
4216 zcCompletedGroupAlgebraStandardAugmentationIdealProjection_surjective C idIndex.target
4217 let F : zcCompletedGroupAlgebraStageAugmentationIdeal C G idIndex.target →
4218 ZCCompletedDifferentialModuleStage C psi.toMonoidHom i := fun y =>
4219 zcSeparatedCompletedDifferentialModuleStageProjectionAdd C psi.toMonoidHom i
4220 (stdAugIdealToZCSepDiff
4221 C hC psi (Classical.choose (hsurj y)))
4222 have hfactor :
4223 (fun s : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
4224 zcSeparatedCompletedDifferentialModuleStageProjectionAdd C psi.toMonoidHom i
4225 (stdAugIdealToZCSepDiff
4226 C hC psi s)) =
4227 fun s => F (p s) := by
4228 funext s
4229 exact
4230 stdAugIdealToZCSepDiff_stageProj_eq_of_standardProj_eq
4231 C hC psi i
4232 (by
4233 dsimp [p, F]
4234 exact (Classical.choose_spec (hsurj (p s))).symm)
4235 rw [hfactor]
4236 haveI : DiscreteTopology (zcCompletedGroupAlgebraStageAugmentationIdeal C G idIndex.target) := by
4237 infer_instance
4238 have hF : Continuous F :=
4239 continuous_of_discreteTopology
4240 exact hF.comp (continuous_zcCompletedGroupAlgebraStandardAugmentationIdealProjection C
4241 idIndex.target)
4243/-- The product of all finite-stage coordinates of the source-standard map into the separated
4244module is continuous. -/
4245theorem
4246 continuous_stdAugIdealToZCSepDiff_stageProjProduct
4249 (psi : ContinuousMonoidHom G H) :
4250 Continuous
4251 (fun s : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
4252 zcSeparatedCompletedDifferentialModuleStageProjectionProduct C psi.toMonoidHom
4253 (stdAugIdealToZCSepDiff
4254 C hC psi s)) := by
4255 exact continuous_pi fun i =>
4256 continuous_stdAugIdealToZCSepDiff_stageProj
4257 C hC psi i
4259/-- The source-standard map to the separated completed differential module is continuous for
4260the finite-stage quotient topology. -/
4261theorem
4262 continuous_stdAugIdealToZCSepDiff
4265 (hForm : ProCGroups.FiniteGroupClass.Formation C)
4266 (psi : ContinuousMonoidHom G H) :
4267 letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
4268 zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
4269 @Continuous
4270 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
4271 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)
4272 inferInstance
4273 (zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom)
4274 (stdAugIdealToZCSepDiff
4275 C hC psi) := by
4276 letI : Nonempty (ZCCompletedDifferentialModuleIndex C psi.toMonoidHom) :=
4277 nonempty_zcCompletedDifferentialModuleIndex C hC psi
4278 rw [zcSepDiffModuleNaturalTopology_eq_induced_stageProjProduct
4279 C psi.toMonoidHom (directed_zcCompletedDifferentialModuleIndex C hForm hC psi)]
4280 rw [continuous_induced_rng]
4281 exact
4282 continuous_stdAugIdealToZCSepDiff_stageProjProduct
4283 C hC psi
4285/-- The conditional augmentation-ideal map sends a group boundary to the separated universal
4286differential of that group element. -/
4287@[simp 900]
4288theorem
4289 stdAugIdealToZCSepDiffOfBoundaryKernel_boundary
4292 (psi : ContinuousMonoidHom G H)
4293 (hker :
4294 ∀ x : ZCCompletedDifferentialModule C (MonoidHom.id G),
4295 zcToStdAugIdeal C G (MonoidHom.id G) x = 0 →
4296 zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
4297 C hC psi x = 0)
4298 (g : G) :
4299 stdAugIdealToZCSepDiffOfBoundaryKernel
4300 C hC psi hker
4301 ⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
4302 zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
4303 C G (MonoidHom.id G) g⟩ =
4304 zcSeparatedUniversalDifferential C psi.toMonoidHom g := by
4305 letI : Module (ZCCompletedGroupAlgebra C G)
4306 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
4307 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
4308 have hcomp :=
4309 congrArg
4310 (fun f =>
4311 f (zcUniversalDifferential C (MonoidHom.id G) g))
4312 (stdAugIdealToZCSepDiffOfBoundaryKernel_comp_zcToStdAugIdeal
4313 C hC psi hker)
4314 change
4315 stdAugIdealToZCSepDiffOfBoundaryKernel C hC psi hker
4316 (zcToStdAugIdeal C G (MonoidHom.id G)
4317 (zcUniversalDifferential C (MonoidHom.id G) g)) =
4318 zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
4319 C hC psi (zcUniversalDifferential C (MonoidHom.id G) g) at hcomp
4320 have hboundary :
4321 zcToStdAugIdeal C G (MonoidHom.id G)
4322 (zcUniversalDifferential C (MonoidHom.id G) g) =
4323 ⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
4324 zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
4325 C G (MonoidHom.id G) g⟩ := by
4326 exact
4327 crossedHomModuleLift_universal
4328 (A := zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
4329 (zcCompletedGroupAlgebraScalar C (MonoidHom.id G))
4330 (zcCompletedGroupAlgebraBoundaryToStandardAugmentationIdeal
4331 C G (MonoidHom.id G))
4332 g
4333 calc
4334 stdAugIdealToZCSepDiffOfBoundaryKernel
4335 C hC psi hker
4336 ⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
4337 zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
4338 C G (MonoidHom.id G) g⟩ =
4339 zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
4340 C hC psi (zcUniversalDifferential C (MonoidHom.id G) g) := by
4341 rw [← hboundary]
4342 exact hcomp
4343 _ = zcSeparatedUniversalDifferential C psi.toMonoidHom g := by
4344 exact
4345 zcDiffModuleIdToZCSepDiffModule_universal
4346 C hC psi g
4348/-- The augmentation-ideal map sends a group boundary to the separated universal
4349differential. -/
4350@[simp]
4351theorem
4352 stdAugIdealToZCSepDiff_boundary
4355 (psi : ContinuousMonoidHom G H)
4356 (g : G) :
4357 stdAugIdealToZCSepDiff
4358 C hC psi
4359 ⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
4360 zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
4361 C G (MonoidHom.id G) g⟩ =
4362 zcSeparatedUniversalDifferential C psi.toMonoidHom g := by
4363 exact
4364 stdAugIdealToZCSepDiffOfBoundaryKernel_boundary
4365 C hC psi
4366 (zcDiffModuleIdToZCSepDiffModule_eq_zero_of_zcToStandard_eq_zero
4367 C hC psi) g
4369/-- The conditional source-standard map kills each algebraic kernel-product generator. -/
4370theorem
4371 stdAugIdealToZCSepDiffOfBoundaryKernel_kernel_generator_smul
4374 (psi : ContinuousMonoidHom G H)
4375 (hker :
4376 ∀ x : ZCCompletedDifferentialModule C (MonoidHom.id G),
4377 zcToStdAugIdeal C G (MonoidHom.id G) x = 0 →
4378 zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
4379 C hC psi x = 0)
4380 (n : ProfiniteKernelSubgroup psi)
4381 (s : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
4382 stdAugIdealToZCSepDiffOfBoundaryKernel
4383 C hC psi hker ((zcGroupLike C G n.1 - 1) • s) = 0 := by
4384 letI : Module (ZCCompletedGroupAlgebra C G)
4385 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
4386 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
4387 let f :=
4388 zcToStdAugIdeal C G (MonoidHom.id G)
4389 let M :=
4390 stdAugIdealToZCSepDiffOfBoundaryKernel
4391 C hC psi hker
4392 let L :=
4393 zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule C hC psi
4394 have hf : Function.Surjective f := by
4395 exact
4396 zcToStdAugIdeal_surjective_of_surjective
4397 C G (MonoidHom.id G) (fun g => ⟨g, rfl⟩)
4398 rcases hf s with ⟨y, hy⟩
4399 have hcomp :=
4400 congrArg
4401 (fun F =>
4402 F ((zcGroupLike C G n.1 - 1) • y))
4403 (stdAugIdealToZCSepDiffOfBoundaryKernel_comp_zcToStdAugIdeal
4404 C hC psi hker)
4405 change
4406 M (f ((zcGroupLike C G n.1 - 1) • y)) =
4407 L ((zcGroupLike C G n.1 - 1) • y) at hcomp
4408 have harg :
4409 (zcGroupLike C G n.1 - 1) • s =
4410 f ((zcGroupLike C G n.1 - 1) • y) := by
4411 rw [map_smul, hy]
4412 calc
4413 M ((zcGroupLike C G n.1 - 1) • s) =
4414 M (f ((zcGroupLike C G n.1 - 1) • y)) := by
4415 rw [harg]
4416 _ = L ((zcGroupLike C G n.1 - 1) • y) := by
4417 exact hcomp
4418 _ = 0 := by
4419 exact
4420 zcDiffModuleIdToZCSepDiffModule_kernel_sub_one_smul
4421 C hC psi n y
4423/-- The conditional source-standard map kills the algebraic product `I(ker psi) I(G)`. -/
4424theorem
4425 stdAugIdealToZCSepDiffOfBoundaryKernel_kills_kernelMulStandard
4428 (psi : ContinuousMonoidHom G H)
4429 (hker :
4430 ∀ x : ZCCompletedDifferentialModule C (MonoidHom.id G),
4431 zcToStdAugIdeal C G (MonoidHom.id G) x = 0 →
4432 zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
4433 C hC psi x = 0)
4434 {x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G}
4435 (hx : x ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi) :
4436 stdAugIdealToZCSepDiffOfBoundaryKernel
4437 C hC psi hker x = 0 := by
4438 letI : Module (ZCCompletedGroupAlgebra C G)
4439 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
4440 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
4441 let M :=
4442 stdAugIdealToZCSepDiffOfBoundaryKernel
4443 C hC psi hker
4444 rw [zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard] at hx
4445 refine Submodule.span_induction
4446 (p := fun y _ => M y = 0) ?_ ?_ ?_ ?_ hx
4447 · rintro _ ⟨p, rfl⟩
4448 exact
4449 stdAugIdealToZCSepDiffOfBoundaryKernel_kernel_generator_smul
4450 C hC psi hker p.1 p.2
4451 · exact map_zero M
4452 · intro y z _ _ hy hz
4453 rw [map_add, hy, hz, add_zero]
4454 · intro a y _ hy
4455 rw [map_smul, hy, smul_zero]
4457/-- The source-standard map kills the algebraic product `I(ker psi) I(G)`. -/
4458theorem
4459 stdAugIdealToZCSepDiff_kills_kernelMulStandard
4462 (psi : ContinuousMonoidHom G H)
4463 {x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G}
4464 (hx : x ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi) :
4465 stdAugIdealToZCSepDiff
4466 C hC psi x = 0 :=
4467 stdAugIdealToZCSepDiffOfBoundaryKernel_kills_kernelMulStandard
4468 C hC psi
4469 (zcDiffModuleIdToZCSepDiffModule_eq_zero_of_zcToStandard_eq_zero
4470 C hC psi) hx
4472/-- If the conditional source-standard map is continuous for the separated quotient topology,
4473then killing the algebraic denominator implies that it kills the finite-stage closed denominator.
4474-/
4475theorem
4476 stdAugIdealToZCSepDiffOfBoundaryKernel_kills_kernelMulStandardClosed_of_continuous
4479 (hForm : ProCGroups.FiniteGroupClass.Formation C)
4480 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
4481 (hfopen : IsOpenMap psi)
4482 (hker :
4483 ∀ x : ZCCompletedDifferentialModule C (MonoidHom.id G),
4484 zcToStdAugIdeal C G (MonoidHom.id G) x = 0 →
4485 zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
4486 C hC psi x = 0)
4487 (hcont :
4488 letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
4489 zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
4490 @Continuous
4491 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
4492 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)
4493 inferInstance
4494 (zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom)
4495 (stdAugIdealToZCSepDiffOfBoundaryKernel
4496 C hC psi hker))
4497 {x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G}
4498 (hx : x ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
4499 C hC hForm psi hpsi hfopen) :
4500 stdAugIdealToZCSepDiffOfBoundaryKernel
4501 C hC psi hker x = 0 := by
4502 letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
4503 zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
4504 let M :=
4505 stdAugIdealToZCSepDiffOfBoundaryKernel
4506 C hC psi hker
4507 have hxcl :
4508 x ∈ closure
4509 ((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :
4510 Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G))) := by
4511 rw [closure_zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_eq_closed
4512 C hC hForm psi hpsi hfopen]
4513 exact hx
4514 have hclosed_preimage :
4515 IsClosed
4516 (M ⁻¹'
4517 ({0} : Set (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom))) := by
4518 exact
4519 (isClosed_zero_zcSeparatedCompletedDifferentialModuleNaturalTopology
4520 C psi.toMonoidHom).preimage hcont
4521 have hsubset :
4522 ((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :
4523 Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G))) ⊆
4524 M ⁻¹' ({0} : Set (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)) := by
4525 intro y hy
4526 exact
4527 stdAugIdealToZCSepDiffOfBoundaryKernel_kills_kernelMulStandard
4528 C hC psi hker hy
4529 exact closure_minimal hsubset hclosed_preimage hxcl
4531/-- If the source-standard map is continuous for the separated quotient topology, then it kills
4532the closed finite-stage denominator. -/
4533theorem
4534 stdAugIdealToZCSepDiff_kills_kernelMulStandardClosed_of_continuous
4537 (hForm : ProCGroups.FiniteGroupClass.Formation C)
4538 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
4539 (hfopen : IsOpenMap psi)
4540 (hcont :
4541 letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
4542 zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
4543 @Continuous
4544 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
4545 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)
4546 inferInstance
4547 (zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom)
4548 (stdAugIdealToZCSepDiff
4549 C hC psi))
4550 {x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G}
4551 (hx : x ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
4552 C hC hForm psi hpsi hfopen) :
4553 stdAugIdealToZCSepDiff
4554 C hC psi x = 0 := by
4555 have hcont' :
4556 letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
4557 zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
4558 @Continuous
4559 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
4560 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)
4561 inferInstance
4562 (zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom)
4563 (stdAugIdealToZCSepDiffOfBoundaryKernel
4564 C hC psi
4565 (zcDiffModuleIdToZCSepDiffModule_eq_zero_of_zcToStandard_eq_zero
4566 C hC psi)) := by
4567 simpa [stdAugIdealToZCSepDiff] using hcont
4568 exact
4569 stdAugIdealToZCSepDiffOfBoundaryKernel_kills_kernelMulStandardClosed_of_continuous
4570 C hC hForm psi hpsi hfopen
4571 (zcDiffModuleIdToZCSepDiffModule_eq_zero_of_zcToStandard_eq_zero
4572 C hC psi) hcont' hx
4574/-- The source-standard map kills the closed finite-stage denominator. -/
4575theorem
4576 stdAugIdealToZCSepDiff_kills_kernelMulStandardClosed
4579 (hForm : ProCGroups.FiniteGroupClass.Formation C)
4580 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
4581 (hfopen : IsOpenMap psi)
4582 {x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G}
4583 (hx : x ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
4584 C hC hForm psi hpsi hfopen) :
4585 stdAugIdealToZCSepDiff
4586 C hC psi x = 0 :=
4587 stdAugIdealToZCSepDiff_kills_kernelMulStandardClosed_of_continuous
4588 C hC hForm psi hpsi hfopen
4589 (continuous_stdAugIdealToZCSepDiff
4590 C hC hForm psi) hx
4592/-- A conditional reverse map from the closed source augmentation quotient to the separated
4593module. The remaining closed-denominator input is isolated as `hclosed_kill`. -/
4594noncomputable def
4595 kerAugIdealQuotToZCSepDiffOfBoundaryKernelOfClosedKill
4598 (hForm : ProCGroups.FiniteGroupClass.Formation C)
4599 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
4600 (hfopen : IsOpenMap psi)
4601 (hker :
4602 ∀ x : ZCCompletedDifferentialModule C (MonoidHom.id G),
4603 zcToStdAugIdeal C G (MonoidHom.id G) x = 0 →
4604 zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
4605 C hC psi x = 0)
4606 (hclosed_kill :
4607 ∀ x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
4608 x ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
4609 C hC hForm psi hpsi hfopen →
4610 stdAugIdealToZCSepDiffOfBoundaryKernel
4611 C hC psi hker x = 0) :
4612 letI : Module (ZCCompletedGroupAlgebra C G)
4613 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
4614 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
4615 KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen
4616 →ₗ[ZCCompletedGroupAlgebra C G]
4617 ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom := by
4618 letI : Module (ZCCompletedGroupAlgebra C G)
4619 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
4620 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
4621 let M :=
4622 stdAugIdealToZCSepDiffOfBoundaryKernel
4623 C hC psi hker
4624 exact
4625 (zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
4626 C hC hForm psi hpsi hfopen).liftQ M
4627 (by
4628 intro x hx
4629 rw [LinearMap.mem_ker]
4630 exact hclosed_kill x hx)
4632/-- The conditional map from the closed augmentation quotient sends the class of a boundary to
4633the separated universal differential. -/
4634@[simp 900]
4635theorem
4636 kerAugIdealQuotToZCSepDiffOfBoundaryKernelOfClosedKill_boundary
4639 (hForm : ProCGroups.FiniteGroupClass.Formation C)
4640 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
4641 (hfopen : IsOpenMap psi)
4642 (hker :
4643 ∀ x : ZCCompletedDifferentialModule C (MonoidHom.id G),
4644 zcToStdAugIdeal C G (MonoidHom.id G) x = 0 →
4645 zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
4646 C hC psi x = 0)
4647 (hclosed_kill :
4648 ∀ x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
4649 x ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
4650 C hC hForm psi hpsi hfopen →
4651 stdAugIdealToZCSepDiffOfBoundaryKernel
4652 C hC psi hker x = 0)
4653 (g : G) :
4654 kerAugIdealQuotToZCSepDiffOfBoundaryKernelOfClosedKill
4655 C hC hForm psi hpsi hfopen hker hclosed_kill
4656 (Submodule.Quotient.mk
4657 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
4658 C hC hForm psi hpsi hfopen)
4659 ⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
4660 zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
4661 C G (MonoidHom.id G) g⟩) =
4662 zcSeparatedUniversalDifferential C psi.toMonoidHom g := by
4663 letI : Module (ZCCompletedGroupAlgebra C G)
4664 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
4665 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
4666 rw [kerAugIdealQuotToZCSepDiffOfBoundaryKernelOfClosedKill,
4667 Submodule.liftQ_apply]
4668 exact
4669 stdAugIdealToZCSepDiffOfBoundaryKernel_boundary
4670 C hC psi hker g
4672/-- Target-linear version of the conditional reverse map from the closed source augmentation
4673quotient to the separated module. -/
4674noncomputable def
4675 kerAugIdealQuotToZCSepDiffLinearOfBoundaryKernelOfClosedKill
4678 (hForm : ProCGroups.FiniteGroupClass.Formation C)
4679 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
4680 (hfopen : IsOpenMap psi)
4681 (hker :
4682 ∀ x : ZCCompletedDifferentialModule C (MonoidHom.id G),
4683 zcToStdAugIdeal C G (MonoidHom.id G) x = 0 →
4684 zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
4685 C hC psi x = 0)
4686 (hclosed_kill :
4687 ∀ x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
4688 x ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
4689 C hC hForm psi hpsi hfopen →
4690 stdAugIdealToZCSepDiffOfBoundaryKernel
4691 C hC psi hker x = 0) :
4692 letI : Module (ZCCompletedGroupAlgebra C H)
4693 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
4694 kerAugClosedQuotTargetCompletedModuleOfSurj
4695 C hC hForm psi hpsi hfopen
4696 KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen
4697 →ₗ[ZCCompletedGroupAlgebra C H]
4698 ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom := by
4699 letI : Module (ZCCompletedGroupAlgebra C H)
4700 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
4701 kerAugClosedQuotTargetCompletedModuleOfSurj
4702 C hC hForm psi hpsi hfopen
4703 letI : Module (ZCCompletedGroupAlgebra C G)
4704 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
4705 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
4706 let Q :=
4707 kerAugIdealQuotToZCSepDiffOfBoundaryKernelOfClosedKill
4708 C hC hForm psi hpsi hfopen hker hclosed_kill
4709 refine
4710 { toFun := Q
4711 map_add' := by
4712 intro x y
4713 exact map_add Q x y
4714 map_smul' := by
4715 intro a x
4716 change Q
4717 (zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi a • x) =
4718 a • Q x
4719 rw [map_smul]
4720 symm
4721 calc
4722 a • Q x =
4723 zcCompletedGroupAlgebraMap C hC psi
4724 (zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi a) •
4725 Q x := by
4726 rw [zcCompletedGroupAlgebraMap_targetLiftOfSurjective]
4727 _ =
4728 zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi a •
4729 Q x := by
4730 exact
4731 zcSeparatedCompletedDifferentialModule_source_map_smul
4732 C hC psi
4733 (zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi a)
4734 (Q x) }
4736/-- The target-linear conditional quotient map sends each boundary class to the separated
4737universal differential. -/
4738@[simp 900]
4739theorem
4740 kerAugIdealQuotToZCSepDiffLinearOfBoundaryKernelOfClosedKill_boundary
4743 (hForm : ProCGroups.FiniteGroupClass.Formation C)
4744 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
4745 (hfopen : IsOpenMap psi)
4746 (hker :
4747 ∀ x : ZCCompletedDifferentialModule C (MonoidHom.id G),
4748 zcToStdAugIdeal C G (MonoidHom.id G) x = 0 →
4749 zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
4750 C hC psi x = 0)
4751 (hclosed_kill :
4752 ∀ x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
4753 x ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
4754 C hC hForm psi hpsi hfopen →
4755 stdAugIdealToZCSepDiffOfBoundaryKernel
4756 C hC psi hker x = 0)
4757 (g : G) :
4758 kerAugIdealQuotToZCSepDiffLinearOfBoundaryKernelOfClosedKill
4759 C hC hForm psi hpsi hfopen hker hclosed_kill
4760 (Submodule.Quotient.mk
4761 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
4762 C hC hForm psi hpsi hfopen)
4763 ⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
4764 zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
4765 C G (MonoidHom.id G) g⟩) =
4766 zcSeparatedUniversalDifferential C psi.toMonoidHom g := by
4767 exact
4768 kerAugIdealQuotToZCSepDiffOfBoundaryKernelOfClosedKill_boundary
4769 C hC hForm psi hpsi hfopen hker hclosed_kill g
4771/-- Target-linear reverse map from the closed source augmentation quotient to the separated
4772module, reducing the closed-denominator condition to continuity of the source-standard map. -/
4773noncomputable def
4774 kerAugIdealQuotToZCSepDiffLinearOfBoundaryKernelOfContStdMap
4777 (hForm : ProCGroups.FiniteGroupClass.Formation C)
4778 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
4779 (hfopen : IsOpenMap psi)
4780 (hker :
4781 ∀ x : ZCCompletedDifferentialModule C (MonoidHom.id G),
4782 zcToStdAugIdeal C G (MonoidHom.id G) x = 0 →
4783 zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
4784 C hC psi x = 0)
4785 (hcont :
4786 letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
4787 zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
4788 @Continuous
4789 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
4790 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)
4791 inferInstance
4792 (zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom)
4793 (stdAugIdealToZCSepDiffOfBoundaryKernel
4794 C hC psi hker)) :
4795 letI : Module (ZCCompletedGroupAlgebra C H)
4796 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
4797 kerAugClosedQuotTargetCompletedModuleOfSurj
4798 C hC hForm psi hpsi hfopen
4799 KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen
4800 →ₗ[ZCCompletedGroupAlgebra C H]
4801 ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom :=
4802 kerAugIdealQuotToZCSepDiffLinearOfBoundaryKernelOfClosedKill
4803 C hC hForm psi hpsi hfopen hker
4804 (fun _ hx =>
4805 stdAugIdealToZCSepDiffOfBoundaryKernel_kills_kernelMulStandardClosed_of_continuous
4806 C hC hForm psi hpsi hfopen hker hcont hx)
4808/-- Under continuity of the conditional standard map, the target-linear quotient map sends a
4809boundary class to the separated universal differential. -/
4810@[simp 900]
4811theorem
4812 kerAugIdealQuotToZCSepDiffLinearOfBoundaryKernelOfContStdMap_boundary
4815 (hForm : ProCGroups.FiniteGroupClass.Formation C)
4816 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
4817 (hfopen : IsOpenMap psi)
4818 (hker :
4819 ∀ x : ZCCompletedDifferentialModule C (MonoidHom.id G),
4820 zcToStdAugIdeal C G (MonoidHom.id G) x = 0 →
4821 zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
4822 C hC psi x = 0)
4823 (hcont :
4824 letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
4825 zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
4826 @Continuous
4827 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
4828 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)
4829 inferInstance
4830 (zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom)
4831 (stdAugIdealToZCSepDiffOfBoundaryKernel
4832 C hC psi hker))
4833 (g : G) :
4834 kerAugIdealQuotToZCSepDiffLinearOfBoundaryKernelOfContStdMap
4835 C hC hForm psi hpsi hfopen hker hcont
4836 (Submodule.Quotient.mk
4837 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
4838 C hC hForm psi hpsi hfopen)
4839 ⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
4840 zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
4841 C G (MonoidHom.id G) g⟩) =
4842 zcSeparatedUniversalDifferential C psi.toMonoidHom g := by
4843 exact
4844 kerAugIdealQuotToZCSepDiffLinearOfBoundaryKernelOfClosedKill_boundary
4845 C hC hForm psi hpsi hfopen hker
4846 (fun x hx =>
4847 stdAugIdealToZCSepDiffOfBoundaryKernel_kills_kernelMulStandardClosed_of_continuous
4848 C hC hForm psi hpsi hfopen hker hcont hx)
4849 g
4851/-- Reverse map from the closed source augmentation quotient to the separated module, assuming
4852only that the closed denominator is killed by the unconditional source-standard map. -/
4853noncomputable def
4854 kerAugIdealQuotToZCSepDiffOfClosedKill
4857 (hForm : ProCGroups.FiniteGroupClass.Formation C)
4858 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
4859 (hfopen : IsOpenMap psi)
4860 (hclosed_kill :
4861 ∀ x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
4862 x ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
4863 C hC hForm psi hpsi hfopen →
4864 stdAugIdealToZCSepDiff
4865 C hC psi x = 0) :
4866 letI : Module (ZCCompletedGroupAlgebra C G)
4867 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
4868 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
4869 KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen
4870 →ₗ[ZCCompletedGroupAlgebra C G]
4871 ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom :=
4872 kerAugIdealQuotToZCSepDiffOfBoundaryKernelOfClosedKill
4873 C hC hForm psi hpsi hfopen
4874 (zcDiffModuleIdToZCSepDiffModule_eq_zero_of_zcToStandard_eq_zero
4875 C hC psi)
4876 (fun x hx => by
4877 simpa [stdAugIdealToZCSepDiff]
4878 using hclosed_kill x hx)
4880/-- If the closed denominator is killed, the resulting quotient map sends a boundary class to
4881the separated universal differential. -/
4882@[simp 900]
4883theorem
4884 kerAugIdealQuotToZCSepDiffOfClosedKill_boundary
4887 (hForm : ProCGroups.FiniteGroupClass.Formation C)
4888 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
4889 (hfopen : IsOpenMap psi)
4890 (hclosed_kill :
4891 ∀ x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
4892 x ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
4893 C hC hForm psi hpsi hfopen →
4894 stdAugIdealToZCSepDiff
4895 C hC psi x = 0)
4896 (g : G) :
4897 kerAugIdealQuotToZCSepDiffOfClosedKill
4898 C hC hForm psi hpsi hfopen hclosed_kill
4899 (Submodule.Quotient.mk
4900 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
4901 C hC hForm psi hpsi hfopen)
4902 ⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
4903 zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
4904 C G (MonoidHom.id G) g⟩) =
4905 zcSeparatedUniversalDifferential C psi.toMonoidHom g := by
4906 exact
4907 kerAugIdealQuotToZCSepDiffOfBoundaryKernelOfClosedKill_boundary
4908 C hC hForm psi hpsi hfopen
4909 (zcDiffModuleIdToZCSepDiffModule_eq_zero_of_zcToStandard_eq_zero
4910 C hC psi)
4911 (fun x hx => by
4912 simpa [stdAugIdealToZCSepDiff]
4913 using hclosed_kill x hx) g
4915/-- The reverse map from the closed source augmentation quotient to the separated module is
4916continuous once the source-standard map is continuous. -/
4917theorem
4918 continuous_kerAugIdealQuotToZCSepDiffOfClosedKill
4921 (hForm : ProCGroups.FiniteGroupClass.Formation C)
4922 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
4923 (hfopen : IsOpenMap psi)
4924 (hclosed_kill :
4925 ∀ x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
4926 x ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
4927 C hC hForm psi hpsi hfopen →
4928 stdAugIdealToZCSepDiff
4929 C hC psi x = 0)
4930 (hcont :
4931 letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
4932 zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
4933 @Continuous
4934 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
4935 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)
4936 inferInstance
4937 (zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom)
4938 (stdAugIdealToZCSepDiff
4939 C hC psi)) :
4940 letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
4941 zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
4942 Continuous
4943 (kerAugIdealQuotToZCSepDiffOfClosedKill
4944 C hC hForm psi hpsi hfopen hclosed_kill) := by
4945 letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
4946 zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
4947 rw [continuous_kernelAugmentationIdealClosedQuotient_iff_comp_mkQ
4948 (C := C) (hC := hC) (hForm := hForm) (psi := psi)
4949 (hpsi := hpsi) (hfopen := hfopen)]
4950 have hcomp :
4951 (fun x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
4952 kerAugIdealQuotToZCSepDiffOfClosedKill
4953 C hC hForm psi hpsi hfopen hclosed_kill
4954 ((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
4955 C hC hForm psi hpsi hfopen).mkQ x)) =
4956 stdAugIdealToZCSepDiff
4957 C hC psi := by
4958 funext x
4959 rfl
4960 rw [hcomp]
4961 exact hcont
4963/-- Target-linear reverse map from the closed source augmentation quotient to the separated
4964module, assuming only that the closed denominator is killed by the unconditional source-standard
4965map. -/
4966noncomputable def
4967 kerAugIdealQuotToZCSepDiffLinearOfClosedKill
4970 (hForm : ProCGroups.FiniteGroupClass.Formation C)
4971 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
4972 (hfopen : IsOpenMap psi)
4973 (hclosed_kill :
4974 ∀ x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
4975 x ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
4976 C hC hForm psi hpsi hfopen →
4977 stdAugIdealToZCSepDiff
4978 C hC psi x = 0) :
4979 letI : Module (ZCCompletedGroupAlgebra C H)
4980 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
4981 kerAugClosedQuotTargetCompletedModuleOfSurj
4982 C hC hForm psi hpsi hfopen
4983 KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen
4984 →ₗ[ZCCompletedGroupAlgebra C H]
4985 ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom :=
4986 kerAugIdealQuotToZCSepDiffLinearOfBoundaryKernelOfClosedKill
4987 C hC hForm psi hpsi hfopen
4988 (zcDiffModuleIdToZCSepDiffModule_eq_zero_of_zcToStandard_eq_zero
4989 C hC psi)
4990 (fun x hx => by
4991 simpa [stdAugIdealToZCSepDiff]
4992 using hclosed_kill x hx)
4994/-- The target-linear map induced by annihilating the closed denominator has the expected value
4995on boundary classes. -/
4996@[simp 900]
4997theorem
4998 kerAugIdealQuotToZCSepDiffLinearOfClosedKill_boundary
5001 (hForm : ProCGroups.FiniteGroupClass.Formation C)
5002 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
5003 (hfopen : IsOpenMap psi)
5004 (hclosed_kill :
5005 ∀ x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
5006 x ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
5007 C hC hForm psi hpsi hfopen →
5008 stdAugIdealToZCSepDiff
5009 C hC psi x = 0)
5010 (g : G) :
5011 kerAugIdealQuotToZCSepDiffLinearOfClosedKill
5012 C hC hForm psi hpsi hfopen hclosed_kill
5013 (Submodule.Quotient.mk
5014 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
5015 C hC hForm psi hpsi hfopen)
5016 ⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
5017 zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
5018 C G (MonoidHom.id G) g⟩) =
5019 zcSeparatedUniversalDifferential C psi.toMonoidHom g :=
5020 kerAugIdealQuotToZCSepDiffOfClosedKill_boundary
5021 C hC hForm psi hpsi hfopen hclosed_kill g
5023/-- Target-linear reverse map from the closed source augmentation quotient to the separated
5024module, reducing the closed-denominator condition to continuity of the unconditional
5025source-standard map. -/
5026noncomputable def
5027 kerAugIdealQuotToZCSepDiffLinearOfContStdMap
5030 (hForm : ProCGroups.FiniteGroupClass.Formation C)
5031 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
5032 (hfopen : IsOpenMap psi)
5033 (hcont :
5034 letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
5035 zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
5036 @Continuous
5037 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
5038 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)
5039 inferInstance
5040 (zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom)
5041 (stdAugIdealToZCSepDiff
5042 C hC psi)) :
5043 letI : Module (ZCCompletedGroupAlgebra C H)
5044 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5045 kerAugClosedQuotTargetCompletedModuleOfSurj
5046 C hC hForm psi hpsi hfopen
5047 KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen
5048 →ₗ[ZCCompletedGroupAlgebra C H]
5049 ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom :=
5050 kerAugIdealQuotToZCSepDiffLinearOfClosedKill
5051 C hC hForm psi hpsi hfopen
5052 (fun _ hx =>
5053 stdAugIdealToZCSepDiff_kills_kernelMulStandardClosed_of_continuous
5054 C hC hForm psi hpsi hfopen hcont hx)
5056/-- Continuity of the standard map gives a target-linear quotient map with the expected value
5057on boundary classes. -/
5058@[simp 900]
5059theorem
5060 kerAugIdealQuotToZCSepDiffLinearOfContStdMap_boundary
5063 (hForm : ProCGroups.FiniteGroupClass.Formation C)
5064 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
5065 (hfopen : IsOpenMap psi)
5066 (hcont :
5067 letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
5068 zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
5069 @Continuous
5070 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
5071 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)
5072 inferInstance
5073 (zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom)
5074 (stdAugIdealToZCSepDiff
5075 C hC psi))
5076 (g : G) :
5077 kerAugIdealQuotToZCSepDiffLinearOfContStdMap
5078 C hC hForm psi hpsi hfopen hcont
5079 (Submodule.Quotient.mk
5080 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
5081 C hC hForm psi hpsi hfopen)
5082 ⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
5083 zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
5084 C G (MonoidHom.id G) g⟩) =
5085 zcSeparatedUniversalDifferential C psi.toMonoidHom g :=
5086 kerAugIdealQuotToZCSepDiffLinearOfClosedKill_boundary
5087 C hC hForm psi hpsi hfopen
5088 (fun _ hx =>
5089 stdAugIdealToZCSepDiff_kills_kernelMulStandardClosed_of_continuous
5090 C hC hForm psi hpsi hfopen hcont hx) g
5092/-- The target-linear reverse map obtained from a continuous source-standard map is continuous. -/
5093theorem
5094 continuous_kerAugIdealQuotToZCSepDiffLinearOfContStdMap
5097 (hForm : ProCGroups.FiniteGroupClass.Formation C)
5098 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
5099 (hfopen : IsOpenMap psi)
5100 (hcont :
5101 letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
5102 zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
5103 @Continuous
5104 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
5105 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)
5106 inferInstance
5107 (zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom)
5108 (stdAugIdealToZCSepDiff
5109 C hC psi)) :
5110 letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
5111 zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
5112 Continuous
5113 (kerAugIdealQuotToZCSepDiffLinearOfContStdMap
5114 C hC hForm psi hpsi hfopen hcont) := by
5115 letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
5116 zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
5117 change Continuous
5118 (kerAugIdealQuotToZCSepDiffOfClosedKill
5119 C hC hForm psi hpsi hfopen
5120 (fun _ hx =>
5121 stdAugIdealToZCSepDiff_kills_kernelMulStandardClosed_of_continuous
5122 C hC hForm psi hpsi hfopen hcont hx))
5123 exact
5124 continuous_kerAugIdealQuotToZCSepDiffOfClosedKill
5125 C hC hForm psi hpsi hfopen
5126 (fun _ hx =>
5127 stdAugIdealToZCSepDiff_kills_kernelMulStandardClosed_of_continuous
5128 C hC hForm psi hpsi hfopen hcont hx)
5129 hcont
5131/-- Target-linear reverse map from the closed source augmentation quotient to the separated
5132module. -/
5133noncomputable def
5134 kerAugIdealQuotToZCSepDiffLinear
5137 (hForm : ProCGroups.FiniteGroupClass.Formation C)
5138 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
5139 (hfopen : IsOpenMap psi) :
5140 letI : Module (ZCCompletedGroupAlgebra C H)
5141 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5142 kerAugClosedQuotTargetCompletedModuleOfSurj
5143 C hC hForm psi hpsi hfopen
5144 KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen
5145 →ₗ[ZCCompletedGroupAlgebra C H]
5146 ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom :=
5147 kerAugIdealQuotToZCSepDiffLinearOfContStdMap
5148 C hC hForm psi hpsi hfopen
5149 (continuous_stdAugIdealToZCSepDiff
5150 C hC hForm psi)
5152/-- The canonical target-linear reverse map sends each boundary class to the separated
5153universal differential. -/
5154@[simp 900]
5155theorem
5156 kerAugIdealQuotToZCSepDiffLinear_boundary
5159 (hForm : ProCGroups.FiniteGroupClass.Formation C)
5160 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
5161 (hfopen : IsOpenMap psi)
5162 (g : G) :
5163 kerAugIdealQuotToZCSepDiffLinear
5164 C hC hForm psi hpsi hfopen
5165 (Submodule.Quotient.mk
5166 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
5167 C hC hForm psi hpsi hfopen)
5168 ⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
5169 zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
5170 C G (MonoidHom.id G) g⟩) =
5171 zcSeparatedUniversalDifferential C psi.toMonoidHom g :=
5172 kerAugIdealQuotToZCSepDiffLinearOfContStdMap_boundary
5173 C hC hForm psi hpsi hfopen
5174 (continuous_stdAugIdealToZCSepDiff
5175 C hC hForm psi) g
5177/-- The target-linear reverse map from the closed source augmentation quotient to the separated
5178module is continuous. -/
5179theorem
5180 continuous_kerAugIdealQuotToZCSepDiffLinear
5183 (hForm : ProCGroups.FiniteGroupClass.Formation C)
5184 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
5185 (hfopen : IsOpenMap psi) :
5186 letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
5187 zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
5188 Continuous
5189 (kerAugIdealQuotToZCSepDiffLinear
5190 C hC hForm psi hpsi hfopen) :=
5191 continuous_kerAugIdealQuotToZCSepDiffLinearOfContStdMap
5192 C hC hForm psi hpsi hfopen
5193 (continuous_stdAugIdealToZCSepDiff
5194 C hC hForm psi)
5196/--
5197The completed universal differential module maps to the closed source augmentation quotient by
5198\(dg \mapsto\) \([g]-1\).
5199-/
5200def zcCompletedDifferentialModuleToKernelAugmentationClosedQuotientOfSurjective
5203 (hForm : ProCGroups.FiniteGroupClass.Formation C)
5204 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
5205 (hfopen : IsOpenMap psi) :
5206 letI : Module (ZCCompletedGroupAlgebra C H)
5207 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5208 kerAugClosedQuotTargetCompletedModuleOfSurj
5209 C hC hForm psi hpsi hfopen
5210 ZCCompletedDifferentialModule C psi.toMonoidHom →ₗ[ZCCompletedGroupAlgebra C H]
5211 KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen := by
5212 letI : Module (ZCCompletedGroupAlgebra C H)
5213 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5214 kerAugClosedQuotTargetCompletedModuleOfSurj
5215 C hC hForm psi hpsi hfopen
5216 exact
5217 crossedHomModuleLift
5218 (A := KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
5219 (zcCompletedGroupAlgebraScalar C psi.toMonoidHom)
5220 (zcCompletedGASourceBoundaryToKerAugClosedQuotTargetCompletedCrossedHomOfSurjective
5221 C hC hForm psi hpsi hfopen)
5223/--
5224The closed augmentation-quotient map sends the universal differential to the class of the
5225corresponding augmentation generator.
5226-/
5227@[simp 900]
5228theorem zcDiffToKerAugClosedQuotOfSurj_universal
5231 (hForm : ProCGroups.FiniteGroupClass.Formation C)
5232 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
5233 (hfopen : IsOpenMap psi) (g : G) :
5234 zcCompletedDifferentialModuleToKernelAugmentationClosedQuotientOfSurjective
5235 C hC hForm psi hpsi hfopen
5236 (zcUniversalDifferential C psi.toMonoidHom g) =
5237 zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
5238 C hC hForm psi hpsi hfopen g := by
5239 letI : Module (ZCCompletedGroupAlgebra C H)
5240 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5241 kerAugClosedQuotTargetCompletedModuleOfSurj
5242 C hC hForm psi hpsi hfopen
5243 exact
5244 crossedHomModuleLift_universal
5245 (A := KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
5246 (zcCompletedGroupAlgebraScalar C psi.toMonoidHom)
5247 (zcCompletedGASourceBoundaryToKerAugClosedQuotTargetCompletedCrossedHomOfSurjective
5248 C hC hForm psi hpsi hfopen) g
5250/--
5251If the pre-quotient source-boundary lift to the closed augmentation quotient is continuous for
5252the finite-stage pre-module topology, then it kills the finite-stage closed relation
5253denominator. This is the descent criterion needed to factor the algebraic map through the
5254separated completed differential module.
5255-/
5256theorem zcDiffToKerAugClosedQuotOfSurj_kills_finiteClosedSubmodule_of_continuous_lift
5259 (hForm : ProCGroups.FiniteGroupClass.Formation C)
5260 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
5261 (hfopen : IsOpenMap psi)
5262 [Nonempty (ZCCompletedDifferentialModuleIndex C psi.toMonoidHom)]
5263 (hdir : Directed (· ≤ ·)
5264 (id : ZCCompletedDifferentialModuleIndex C psi.toMonoidHom →
5265 ZCCompletedDifferentialModuleIndex C psi.toMonoidHom))
5266 (hcont :
5267 letI : Module (ZCCompletedGroupAlgebra C H)
5268 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5269 kerAugClosedQuotTargetCompletedModuleOfSurj
5270 C hC hForm psi hpsi hfopen
5271 @Continuous
5272 (CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)
5273 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
5274 (zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom)
5275 inferInstance
5276 (crossedDifferentialModuleLiftLinear
5277 (R := ZCCompletedGroupAlgebra C H)
5278 (zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
5279 C hC hForm psi hpsi hfopen)))
5280 {x : CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G}
5281 (hx : x ∈ zcCompletedDifferentialRelationFiniteClosedSubmodule C psi.toMonoidHom) :
5282 letI : Module (ZCCompletedGroupAlgebra C H)
5283 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5284 kerAugClosedQuotTargetCompletedModuleOfSurj
5285 C hC hForm psi hpsi hfopen
5286 crossedDifferentialModuleLiftLinear
5287 (R := ZCCompletedGroupAlgebra C H)
5288 (zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
5289 C hC hForm psi hpsi hfopen) x = 0 := by
5290 letI : Module (ZCCompletedGroupAlgebra C H)
5291 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5292 kerAugClosedQuotTargetCompletedModuleOfSurj
5293 C hC hForm psi hpsi hfopen
5294 letI : T1Space
5295 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5296 t1Space_kernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen
5297 exact
5298 crossedDifferentialModuleLiftLinear_kills_finiteClosedSubmodule_of_continuous
5299 C psi.toMonoidHom hdir
5300 (zcCompletedGASourceBoundaryToKerAugClosedQuotTargetCompletedCrossedHomOfSurjective
5301 C hC hForm psi hpsi hfopen)
5302 hcont hx
5304/--
5305Version of zcDiffToKerAugClosedQuotOfSurj_kills_finiteClosedSubmodule_of_continuous_lift with
5306nonemptiness and directedness of finite stages supplied by the continuous source map.
5307-/
5308theorem zcDiffToKerAugClosedQuotOfSurj_kills_finiteClosedSubmodule_of_continuous_lift_of_surj
5311 (hForm : ProCGroups.FiniteGroupClass.Formation C)
5312 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
5313 (hfopen : IsOpenMap psi)
5314 (hcont :
5315 letI : Module (ZCCompletedGroupAlgebra C H)
5316 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5317 kerAugClosedQuotTargetCompletedModuleOfSurj
5318 C hC hForm psi hpsi hfopen
5319 @Continuous
5320 (CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)
5321 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
5322 (zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom)
5323 inferInstance
5324 (crossedDifferentialModuleLiftLinear
5325 (R := ZCCompletedGroupAlgebra C H)
5326 (zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
5327 C hC hForm psi hpsi hfopen)))
5328 {x : CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G}
5329 (hx : x ∈ zcCompletedDifferentialRelationFiniteClosedSubmodule C psi.toMonoidHom) :
5330 letI : Module (ZCCompletedGroupAlgebra C H)
5331 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5332 kerAugClosedQuotTargetCompletedModuleOfSurj
5333 C hC hForm psi hpsi hfopen
5334 crossedDifferentialModuleLiftLinear
5335 (R := ZCCompletedGroupAlgebra C H)
5336 (zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
5337 C hC hForm psi hpsi hfopen) x = 0 := by
5338 letI : Nonempty (ZCCompletedDifferentialModuleIndex C psi.toMonoidHom) :=
5339 nonempty_zcCompletedDifferentialModuleIndex C hC psi
5340 exact
5341 zcDiffToKerAugClosedQuotOfSurj_kills_finiteClosedSubmodule_of_continuous_lift
5342 C hC hForm psi hpsi hfopen
5343 (directed_zcCompletedDifferentialModuleIndex C hForm hC psi)
5344 hcont hx
5346/--
5347Under the explicit continuity hypothesis for the pre-quotient source-boundary lift, the closed
5348source augmentation quotient receives the separated completed universal differential module.
5349-/
5350def zcSepDiffToKerAugClosedQuotOfSurjective
5353 (hForm : ProCGroups.FiniteGroupClass.Formation C)
5354 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
5355 (hfopen : IsOpenMap psi)
5356 [Nonempty (ZCCompletedDifferentialModuleIndex C psi.toMonoidHom)]
5357 (hdir : Directed (· ≤ ·)
5358 (id : ZCCompletedDifferentialModuleIndex C psi.toMonoidHom →
5359 ZCCompletedDifferentialModuleIndex C psi.toMonoidHom))
5360 (hcont :
5361 letI : Module (ZCCompletedGroupAlgebra C H)
5362 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5363 kerAugClosedQuotTargetCompletedModuleOfSurj
5364 C hC hForm psi hpsi hfopen
5365 @Continuous
5366 (CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)
5367 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
5368 (zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom)
5369 inferInstance
5370 (crossedDifferentialModuleLiftLinear
5371 (R := ZCCompletedGroupAlgebra C H)
5372 (zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
5373 C hC hForm psi hpsi hfopen))) :
5374 letI : Module (ZCCompletedGroupAlgebra C H)
5375 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5376 kerAugClosedQuotTargetCompletedModuleOfSurj
5377 C hC hForm psi hpsi hfopen
5378 ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom →ₗ[ZCCompletedGroupAlgebra C H]
5379 KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen := by
5380 letI : Module (ZCCompletedGroupAlgebra C H)
5381 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5382 kerAugClosedQuotTargetCompletedModuleOfSurj
5383 C hC hForm psi hpsi hfopen
5384 letI : T1Space
5385 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5386 t1Space_kernelAugmentationIdealClosedQuotient
5387 C hC hForm psi hpsi hfopen
5388 exact
5389 zcSeparatedCompletedDifferentialModuleLiftOfContinuousPrelift
5390 C psi.toMonoidHom hdir
5391 (zcCompletedGASourceBoundaryToKerAugClosedQuotTargetCompletedCrossedHomOfSurjective
5392 C hC hForm psi hpsi hfopen)
5393 hcont
5395/--
5396Public version of the closed-augmentation descent map with finite-stage nonemptiness and
5397directedness supplied by the continuous source map.
5398-/
5399def zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift
5402 (hForm : ProCGroups.FiniteGroupClass.Formation C)
5403 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
5404 (hfopen : IsOpenMap psi)
5405 (hcont :
5406 letI : Module (ZCCompletedGroupAlgebra C H)
5407 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5408 kerAugClosedQuotTargetCompletedModuleOfSurj
5409 C hC hForm psi hpsi hfopen
5410 @Continuous
5411 (CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)
5412 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
5413 (zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom)
5414 inferInstance
5415 (crossedDifferentialModuleLiftLinear
5416 (R := ZCCompletedGroupAlgebra C H)
5417 (zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
5418 C hC hForm psi hpsi hfopen))) :
5419 letI : Module (ZCCompletedGroupAlgebra C H)
5420 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5421 kerAugClosedQuotTargetCompletedModuleOfSurj
5422 C hC hForm psi hpsi hfopen
5423 ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom →ₗ[ZCCompletedGroupAlgebra C H]
5424 KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen := by
5425 letI : Nonempty (ZCCompletedDifferentialModuleIndex C psi.toMonoidHom) :=
5426 nonempty_zcCompletedDifferentialModuleIndex C hC psi
5427 exact
5428 zcSepDiffToKerAugClosedQuotOfSurjective
5429 C hC hForm psi hpsi hfopen
5430 (directed_zcCompletedDifferentialModuleIndex C hForm hC psi)
5431 hcont
5433/--
5434The separated differential module maps universally to the closed kernel-augmentation quotient in
5435the surjective case.
5436-/
5437@[simp 900]
5438theorem zcSepDiffToKerAugClosedQuotOfSurjective_universal
5441 (hForm : ProCGroups.FiniteGroupClass.Formation C)
5442 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
5443 (hfopen : IsOpenMap psi)
5444 [Nonempty (ZCCompletedDifferentialModuleIndex C psi.toMonoidHom)]
5445 (hdir : Directed (· ≤ ·)
5446 (id : ZCCompletedDifferentialModuleIndex C psi.toMonoidHom →
5447 ZCCompletedDifferentialModuleIndex C psi.toMonoidHom))
5448 (hcont :
5449 letI : Module (ZCCompletedGroupAlgebra C H)
5450 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5451 kerAugClosedQuotTargetCompletedModuleOfSurj
5452 C hC hForm psi hpsi hfopen
5453 @Continuous
5454 (CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)
5455 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
5456 (zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom)
5457 inferInstance
5458 (crossedDifferentialModuleLiftLinear
5459 (R := ZCCompletedGroupAlgebra C H)
5460 (zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
5461 C hC hForm psi hpsi hfopen)))
5462 (g : G) :
5463 zcSepDiffToKerAugClosedQuotOfSurjective
5464 C hC hForm psi hpsi hfopen hdir hcont
5465 (zcSeparatedUniversalDifferential C psi.toMonoidHom g) =
5466 zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
5467 C hC hForm psi hpsi hfopen g := by
5468 letI : Module (ZCCompletedGroupAlgebra C H)
5469 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5470 kerAugClosedQuotTargetCompletedModuleOfSurj
5471 C hC hForm psi hpsi hfopen
5472 letI : T1Space
5473 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5474 t1Space_kernelAugmentationIdealClosedQuotient
5475 C hC hForm psi hpsi hfopen
5476 exact
5477 zcSeparatedCompletedDifferentialModuleLiftOfContinuousPrelift_universal
5478 C psi.toMonoidHom hdir
5479 (zcCompletedGASourceBoundaryToKerAugClosedQuotTargetCompletedCrossedHomOfSurjective
5480 C hC hForm psi hpsi hfopen)
5481 hcont g
5483/--
5484The lifted map from the separated completed differential module to the closed
5485augmentation-kernel quotient sends each separated universal differential to the corresponding
5486boundary quotient.
5487-/
5488@[simp 900]
5489theorem zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift_universal
5492 (hForm : ProCGroups.FiniteGroupClass.Formation C)
5493 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
5494 (hfopen : IsOpenMap psi)
5495 (hcont :
5496 letI : Module (ZCCompletedGroupAlgebra C H)
5497 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5498 kerAugClosedQuotTargetCompletedModuleOfSurj
5499 C hC hForm psi hpsi hfopen
5500 @Continuous
5501 (CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)
5502 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
5503 (zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom)
5504 inferInstance
5505 (crossedDifferentialModuleLiftLinear
5506 (R := ZCCompletedGroupAlgebra C H)
5507 (zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
5508 C hC hForm psi hpsi hfopen)))
5509 (g : G) :
5510 zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift
5511 C hC hForm psi hpsi hfopen hcont
5512 (zcSeparatedUniversalDifferential C psi.toMonoidHom g) =
5513 zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
5514 C hC hForm psi hpsi hfopen g := by
5515 letI : Nonempty (ZCCompletedDifferentialModuleIndex C psi.toMonoidHom) :=
5516 nonempty_zcCompletedDifferentialModuleIndex C hC psi
5517 exact
5518 zcSepDiffToKerAugClosedQuotOfSurjective_universal
5519 C hC hForm psi hpsi hfopen
5520 (directed_zcCompletedDifferentialModuleIndex C hForm hC psi)
5521 hcont g
5523/--
5524The separated closed-augmentation map is the factorization of the algebraic closed-augmentation
5525map through \(A_{\psi}(C) \to A_{\psi}(C)_{\mathrm{sep}}\).
5526-/
5527theorem zcSepDiffToKerAugClosedQuotOfSurjective_comp_toSep
5530 (hForm : ProCGroups.FiniteGroupClass.Formation C)
5531 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
5532 (hfopen : IsOpenMap psi)
5533 [Nonempty (ZCCompletedDifferentialModuleIndex C psi.toMonoidHom)]
5534 (hdir : Directed (· ≤ ·)
5535 (id : ZCCompletedDifferentialModuleIndex C psi.toMonoidHom →
5536 ZCCompletedDifferentialModuleIndex C psi.toMonoidHom))
5537 (hcont :
5538 letI : Module (ZCCompletedGroupAlgebra C H)
5539 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5540 kerAugClosedQuotTargetCompletedModuleOfSurj
5541 C hC hForm psi hpsi hfopen
5542 @Continuous
5543 (CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)
5544 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
5545 (zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom)
5546 inferInstance
5547 (crossedDifferentialModuleLiftLinear
5548 (R := ZCCompletedGroupAlgebra C H)
5549 (zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
5550 C hC hForm psi hpsi hfopen))) :
5551 letI : Module (ZCCompletedGroupAlgebra C H)
5552 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5553 kerAugClosedQuotTargetCompletedModuleOfSurj
5554 C hC hForm psi hpsi hfopen
5555 (zcSepDiffToKerAugClosedQuotOfSurjective
5556 C hC hForm psi hpsi hfopen hdir hcont).comp
5557 (zcCompletedDifferentialModuleToSeparated C psi.toMonoidHom) =
5558 zcCompletedDifferentialModuleToKernelAugmentationClosedQuotientOfSurjective
5559 C hC hForm psi hpsi hfopen := by
5560 letI : Module (ZCCompletedGroupAlgebra C H)
5561 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5562 kerAugClosedQuotTargetCompletedModuleOfSurj
5563 C hC hForm psi hpsi hfopen
5564 apply crossedDifferentialModuleHom_ext
5565 (zcCompletedGroupAlgebraScalar C psi.toMonoidHom)
5566 intro g
5567 change
5568 zcSepDiffToKerAugClosedQuotOfSurjective
5569 C hC hForm psi hpsi hfopen hdir hcont
5570 (zcCompletedDifferentialModuleToSeparated C psi.toMonoidHom
5571 (zcUniversalDifferential C psi.toMonoidHom g)) =
5572 zcCompletedDifferentialModuleToKernelAugmentationClosedQuotientOfSurjective
5573 C hC hForm psi hpsi hfopen
5574 (zcUniversalDifferential C psi.toMonoidHom g)
5575 rw [zcCompletedDifferentialModuleToSeparated_universal,
5576 zcSepDiffToKerAugClosedQuotOfSurjective_universal,
5577 zcDiffToKerAugClosedQuotOfSurj_universal]
5579/--
5580The continuous-lift map to the closed kernel-augmentation quotient composes with separation as
5581expected.
5582-/
5583theorem zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift_comp_toSep
5586 (hForm : ProCGroups.FiniteGroupClass.Formation C)
5587 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
5588 (hfopen : IsOpenMap psi)
5589 (hcont :
5590 letI : Module (ZCCompletedGroupAlgebra C H)
5591 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5592 kerAugClosedQuotTargetCompletedModuleOfSurj
5593 C hC hForm psi hpsi hfopen
5594 @Continuous
5595 (CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)
5596 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
5597 (zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom)
5598 inferInstance
5599 (crossedDifferentialModuleLiftLinear
5600 (R := ZCCompletedGroupAlgebra C H)
5601 (zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
5602 C hC hForm psi hpsi hfopen))) :
5603 letI : Module (ZCCompletedGroupAlgebra C H)
5604 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5605 kerAugClosedQuotTargetCompletedModuleOfSurj
5606 C hC hForm psi hpsi hfopen
5607 (zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift
5608 C hC hForm psi hpsi hfopen hcont).comp
5609 (zcCompletedDifferentialModuleToSeparated C psi.toMonoidHom) =
5610 zcCompletedDifferentialModuleToKernelAugmentationClosedQuotientOfSurjective
5611 C hC hForm psi hpsi hfopen := by
5612 letI : Nonempty (ZCCompletedDifferentialModuleIndex C psi.toMonoidHom) :=
5613 nonempty_zcCompletedDifferentialModuleIndex C hC psi
5614 exact
5615 zcSepDiffToKerAugClosedQuotOfSurjective_comp_toSep
5616 C hC hForm psi hpsi hfopen
5617 (directed_zcCompletedDifferentialModuleIndex C hForm hC psi)
5618 hcont
5620/--
5621The descended forward map to the closed augmentation quotient is continuous once the
5622pre-quotient source-boundary lift is continuous.
5623-/
5624theorem continuous_zcSepDiffToKerAugClosedQuotOfSurjective
5627 (hForm : ProCGroups.FiniteGroupClass.Formation C)
5628 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
5629 (hfopen : IsOpenMap psi)
5630 [Nonempty (ZCCompletedDifferentialModuleIndex C psi.toMonoidHom)]
5631 (hdir : Directed (· ≤ ·)
5632 (id : ZCCompletedDifferentialModuleIndex C psi.toMonoidHom →
5633 ZCCompletedDifferentialModuleIndex C psi.toMonoidHom))
5634 (hcont :
5635 letI : Module (ZCCompletedGroupAlgebra C H)
5636 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5637 kerAugClosedQuotTargetCompletedModuleOfSurj
5638 C hC hForm psi hpsi hfopen
5639 @Continuous
5640 (CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)
5641 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
5642 (zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom)
5643 inferInstance
5644 (crossedDifferentialModuleLiftLinear
5645 (R := ZCCompletedGroupAlgebra C H)
5646 (zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
5647 C hC hForm psi hpsi hfopen))) :
5648 letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
5649 zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
5650 letI : Module (ZCCompletedGroupAlgebra C H)
5651 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5652 kerAugClosedQuotTargetCompletedModuleOfSurj
5653 C hC hForm psi hpsi hfopen
5654 @Continuous
5655 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)
5656 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
5657 (zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom)
5658 inferInstance
5659 (zcSepDiffToKerAugClosedQuotOfSurjective
5660 C hC hForm psi hpsi hfopen hdir hcont) := by
5661 letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
5662 zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
5663 letI : Module (ZCCompletedGroupAlgebra C H)
5664 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5665 kerAugClosedQuotTargetCompletedModuleOfSurj
5666 C hC hForm psi hpsi hfopen
5667 rw [continuous_zcSeparatedCompletedDifferentialModule_iff_comp_mkQ
5668 (C := C) (G := G) (H := H) (ψ := psi.toMonoidHom)]
5669 have hcomp :
5670 (fun x : CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G =>
5671 zcSepDiffToKerAugClosedQuotOfSurjective
5672 C hC hForm psi hpsi hfopen hdir hcont
5673 ((zcCompletedDifferentialRelationFiniteClosedSubmodule C psi.toMonoidHom).mkQ x)) =
5674 crossedDifferentialModuleLiftLinear
5675 (R := ZCCompletedGroupAlgebra C H)
5676 (zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
5677 C hC hForm psi hpsi hfopen) := by
5678 funext x
5679 rw [zcSepDiffToKerAugClosedQuotOfSurjective,
5680 zcSeparatedCompletedDifferentialModuleLiftOfContinuousPrelift,
5681 Submodule.mkQ_apply, Submodule.liftQ_apply]
5682 rfl
5683 rw [hcomp]
5684 exact hcont
5686/--
5687Continuity of the forward map when the finite-stage index data are supplied by the continuous
5688source map.
5689-/
5690theorem continuous_zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift
5693 (hForm : ProCGroups.FiniteGroupClass.Formation C)
5694 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
5695 (hfopen : IsOpenMap psi)
5696 (hcont :
5697 letI : Module (ZCCompletedGroupAlgebra C H)
5698 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5699 kerAugClosedQuotTargetCompletedModuleOfSurj
5700 C hC hForm psi hpsi hfopen
5701 @Continuous
5702 (CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)
5703 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
5704 (zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom)
5705 inferInstance
5706 (crossedDifferentialModuleLiftLinear
5707 (R := ZCCompletedGroupAlgebra C H)
5708 (zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
5709 C hC hForm psi hpsi hfopen))) :
5710 letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
5711 zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
5712 letI : Module (ZCCompletedGroupAlgebra C H)
5713 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5714 kerAugClosedQuotTargetCompletedModuleOfSurj
5715 C hC hForm psi hpsi hfopen
5716 @Continuous
5717 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)
5718 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
5719 (zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom)
5720 inferInstance
5721 (zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift
5722 C hC hForm psi hpsi hfopen hcont) := by
5723 letI : Nonempty (ZCCompletedDifferentialModuleIndex C psi.toMonoidHom) :=
5724 nonempty_zcCompletedDifferentialModuleIndex C hC psi
5725 exact
5726 continuous_zcSepDiffToKerAugClosedQuotOfSurjective
5727 C hC hForm psi hpsi hfopen
5728 (directed_zcCompletedDifferentialModuleIndex C hForm hC psi)
5729 hcont
5731/--
5732The map from the completed differential module to the closed augmentation quotient is surjective
5733when finite-stage representatives can be lifted.
5734-/
5735theorem zcDiffToKerAugClosedQuotOfSurj_surj
5738 (hForm : ProCGroups.FiniteGroupClass.Formation C)
5739 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
5740 (hfopen : IsOpenMap psi) :
5741 Function.Surjective
5742 (zcCompletedDifferentialModuleToKernelAugmentationClosedQuotientOfSurjective
5743 C hC hForm psi hpsi hfopen) := by
5744 letI : Module (ZCCompletedGroupAlgebra C H)
5745 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5746 kerAugClosedQuotTargetCompletedModuleOfSurj
5747 C hC hForm psi hpsi hfopen
5748 intro x
5749 refine Submodule.Quotient.induction_on
5750 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
5751 C hC hForm psi hpsi hfopen) x ?_
5752 intro y
5753 let L :=
5754 zcCompletedDifferentialModuleToKernelAugmentationClosedQuotientOfSurjective
5755 C hC hForm psi hpsi hfopen
5756 let P : zcCompletedGroupAlgebraStandardAugmentationIdeal C G → Prop := fun y =>
5757 ∃ m : ZCCompletedDifferentialModule C psi.toMonoidHom,
5758 L m = Submodule.Quotient.mk
5759 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
5760 C hC hForm psi hpsi hfopen) y
5761 have hy : P y := by
5762 have hyspan : (y : ZCCompletedGroupAlgebra C G) ∈
5763 Submodule.span (ZCCompletedGroupAlgebra C G)
5764 (Set.range fun h : G => zcGroupLike C G h - 1) := by
5765 change (y : ZCCompletedGroupAlgebra C G) ∈
5766 zcCompletedGroupAlgebraStandardAugmentationIdeal C G
5767 exact y.2
5768 refine Submodule.span_induction
5769 (p := fun z hz =>
5770 P
5771 ⟨z, by
5772 simpa [zcCompletedGroupAlgebraStandardAugmentationIdeal_eq_span] using hz⟩)
5773 ?hgen ?hzero ?hadd ?hsmul hyspan
5774 · rintro _ ⟨g, rfl⟩
5775 refine ⟨zcUniversalDifferential C psi.toMonoidHom g, ?_⟩
5776 rw [zcDiffToKerAugClosedQuotOfSurj_universal]
5777 rfl
5778 · refine ⟨0, ?_⟩
5779 change (0 : KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) =
5780 Submodule.Quotient.mk
5781 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
5782 C hC hForm psi hpsi hfopen)
5783 (0 : zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
5784 rw [Submodule.Quotient.mk_zero]
5785 · intro a b ha hb hpa hpb
5786 rcases hpa with ⟨ma, hma⟩
5787 rcases hpb with ⟨mb, hmb⟩
5788 refine ⟨ma + mb, ?_⟩
5789 rw [map_add, hma, hmb, ← Submodule.Quotient.mk_add]
5790 apply congrArg (fun t : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
5791 Submodule.Quotient.mk
5792 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
5793 C hC hForm psi hpsi hfopen) t)
5794 exact Subtype.ext rfl
5795 · intro a b hb hpb
5796 rcases hpb with ⟨m, hm⟩
5797 refine ⟨zcCompletedGroupAlgebraMap C hC psi a • m, ?_⟩
5798 rw [map_smul, hm]
5799 calc
5800 zcCompletedGroupAlgebraMap C hC psi a •
5801 Submodule.Quotient.mk
5802 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
5803 C hC hForm psi hpsi hfopen)
5804 ⟨b, by
5805 simpa [zcCompletedGroupAlgebraStandardAugmentationIdeal_eq_span] using hb⟩ =
5806 a •
5807 Submodule.Quotient.mk
5808 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
5809 C hC hForm psi hpsi hfopen)
5810 ⟨b, by
5811 simpa [zcCompletedGroupAlgebraStandardAugmentationIdeal_eq_span] using hb⟩ := by
5812 exact
5813 kerAugClosedQuotTargetCompletedModuleOfSurj_map_smul
5814 C hC hForm psi hpsi hfopen a
5815 (Submodule.Quotient.mk
5816 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
5817 C hC hForm psi hpsi hfopen)
5818 ⟨b, by
5819 simpa [zcCompletedGroupAlgebraStandardAugmentationIdeal_eq_span] using hb⟩)
5820 _ = Submodule.Quotient.mk
5821 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
5822 C hC hForm psi hpsi hfopen)
5823 ⟨a • b, by
5824 rw [zcCompletedGroupAlgebraStandardAugmentationIdeal_eq_span]
5825 exact Submodule.smul_mem
5826 (Submodule.span (ZCCompletedGroupAlgebra C G)
5827 (Set.range fun h : G => zcGroupLike C G h - 1)) a hb⟩ := by
5828 rw [← Submodule.Quotient.mk_smul]
5829 apply congrArg (fun t : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
5830 Submodule.Quotient.mk
5831 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
5832 C hC hForm psi hpsi hfopen) t)
5833 exact Subtype.ext rfl
5834 exact hy
5836/--
5837The separated closed-augmentation map is surjective under the same pre-quotient continuity
5838hypothesis needed for the descent.
5839-/
5840theorem zcSepDiffToKerAugClosedQuotOfSurjective_surj
5843 (hForm : ProCGroups.FiniteGroupClass.Formation C)
5844 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
5845 (hfopen : IsOpenMap psi)
5846 [Nonempty (ZCCompletedDifferentialModuleIndex C psi.toMonoidHom)]
5847 (hdir : Directed (· ≤ ·)
5848 (id : ZCCompletedDifferentialModuleIndex C psi.toMonoidHom →
5849 ZCCompletedDifferentialModuleIndex C psi.toMonoidHom))
5850 (hcont :
5851 letI : Module (ZCCompletedGroupAlgebra C H)
5852 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5853 kerAugClosedQuotTargetCompletedModuleOfSurj
5854 C hC hForm psi hpsi hfopen
5855 @Continuous
5856 (CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)
5857 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
5858 (zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom)
5859 inferInstance
5860 (crossedDifferentialModuleLiftLinear
5861 (R := ZCCompletedGroupAlgebra C H)
5862 (zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
5863 C hC hForm psi hpsi hfopen))) :
5864 Function.Surjective
5865 (zcSepDiffToKerAugClosedQuotOfSurjective
5866 C hC hForm psi hpsi hfopen hdir hcont) := by
5867 letI : Module (ZCCompletedGroupAlgebra C H)
5868 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5869 kerAugClosedQuotTargetCompletedModuleOfSurj
5870 C hC hForm psi hpsi hfopen
5871 intro y
5872 rcases
5873 zcDiffToKerAugClosedQuotOfSurj_surj
5874 C hC hForm psi hpsi hfopen y with
5875 ⟨m, hm⟩
5876 refine ⟨zcCompletedDifferentialModuleToSeparated C psi.toMonoidHom m, ?_⟩
5877 have hfactor :=
5878 congrArg (fun L : ZCCompletedDifferentialModule C psi.toMonoidHom →ₗ[ZCCompletedGroupAlgebra
5879 C H]
5880 KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen => L m)
5881 (zcSepDiffToKerAugClosedQuotOfSurjective_comp_toSep
5882 C hC hForm psi hpsi hfopen hdir hcont)
5883 change
5884 zcSepDiffToKerAugClosedQuotOfSurjective
5885 C hC hForm psi hpsi hfopen hdir hcont
5886 (zcCompletedDifferentialModuleToSeparated C psi.toMonoidHom m) =
5887 zcCompletedDifferentialModuleToKernelAugmentationClosedQuotientOfSurjective
5888 C hC hForm psi hpsi hfopen m at hfactor
5889 exact hfactor.trans hm
5891/-- The continuous-lift map to the closed kernel-augmentation quotient is surjective. -/
5892theorem zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift_surj
5895 (hForm : ProCGroups.FiniteGroupClass.Formation C)
5896 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
5897 (hfopen : IsOpenMap psi)
5898 (hcont :
5899 letI : Module (ZCCompletedGroupAlgebra C H)
5900 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5901 kerAugClosedQuotTargetCompletedModuleOfSurj
5902 C hC hForm psi hpsi hfopen
5903 @Continuous
5904 (CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)
5905 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
5906 (zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom)
5907 inferInstance
5908 (crossedDifferentialModuleLiftLinear
5909 (R := ZCCompletedGroupAlgebra C H)
5910 (zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
5911 C hC hForm psi hpsi hfopen))) :
5912 Function.Surjective
5913 (zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift
5914 C hC hForm psi hpsi hfopen hcont) := by
5915 letI : Nonempty (ZCCompletedDifferentialModuleIndex C psi.toMonoidHom) :=
5916 nonempty_zcCompletedDifferentialModuleIndex C hC psi
5917 exact
5918 zcSepDiffToKerAugClosedQuotOfSurjective_surj
5919 C hC hForm psi hpsi hfopen
5920 (directed_zcCompletedDifferentialModuleIndex C hForm hC psi)
5921 hcont
5923/--
5924The separated completed universal differential module maps to the closed source augmentation
5925quotient by \(dg \mapsto\) \([g]-1\). The pre-quotient lift continuity is supplied by the
5926finite-stage factorization theorem.
5927-/
5928def zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient
5931 (hForm : ProCGroups.FiniteGroupClass.Formation C)
5932 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
5933 (hfopen : IsOpenMap psi) :
5934 letI : Module (ZCCompletedGroupAlgebra C H)
5935 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5936 kerAugClosedQuotTargetCompletedModuleOfSurj
5937 C hC hForm psi hpsi hfopen
5938 ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom →ₗ[ZCCompletedGroupAlgebra C H]
5939 KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen := by
5940 let hcont :=
5941 continuous_crossedDiffModuleLiftLinear_sourceBoundaryToKerAugClosedQuot
5942 C hC hForm psi hpsi hfopen
5943 exact
5944 zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift
5945 C hC hForm psi hpsi hfopen hcont
5947/--
5948The separated closed augmentation-quotient map sends the separated universal differential to the
5949closed augmentation quotient class.
5950-/
5951@[simp]
5952theorem zcSepDiffToKerAugClosedQuot_universal
5955 (hForm : ProCGroups.FiniteGroupClass.Formation C)
5956 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
5957 (hfopen : IsOpenMap psi) (g : G) :
5958 zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient
5959 C hC hForm psi hpsi hfopen
5960 (zcSeparatedUniversalDifferential C psi.toMonoidHom g) =
5961 zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
5962 C hC hForm psi hpsi hfopen g := by
5963 let hcont :=
5964 continuous_crossedDiffModuleLiftLinear_sourceBoundaryToKerAugClosedQuot
5965 C hC hForm psi hpsi hfopen
5966 simpa [zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient, hcont] using
5967 zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift_universal
5968 C hC hForm psi hpsi hfopen hcont g
5970/--
5971The unconditional separated closed-augmentation map factors the algebraic map through
5972\(A_{\psi}(C) \to A_{\psi}(C)_{\mathrm{sep}}\).
5973-/
5974theorem zcSepDiffToKerAugClosedQuot_comp_toSep
5977 (hForm : ProCGroups.FiniteGroupClass.Formation C)
5978 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
5979 (hfopen : IsOpenMap psi) :
5980 letI : Module (ZCCompletedGroupAlgebra C H)
5981 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
5982 kerAugClosedQuotTargetCompletedModuleOfSurj
5983 C hC hForm psi hpsi hfopen
5984 (zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient
5985 C hC hForm psi hpsi hfopen).comp
5986 (zcCompletedDifferentialModuleToSeparated C psi.toMonoidHom) =
5987 zcCompletedDifferentialModuleToKernelAugmentationClosedQuotientOfSurjective
5988 C hC hForm psi hpsi hfopen := by
5989 let hcont :=
5990 continuous_crossedDiffModuleLiftLinear_sourceBoundaryToKerAugClosedQuot
5991 C hC hForm psi hpsi hfopen
5992 simpa [zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient, hcont] using
5993 zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift_comp_toSep
5994 C hC hForm psi hpsi hfopen hcont
5996/--
5997The separated closed-augmentation map is continuous, with the pre-quotient lift continuity
5998provided by the finite-stage factorization theorem.
5999-/
6000theorem continuous_zcSepDiffToKerAugClosedQuot
6003 (hForm : ProCGroups.FiniteGroupClass.Formation C)
6004 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
6005 (hfopen : IsOpenMap psi) :
6006 letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
6007 zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
6008 letI : Module (ZCCompletedGroupAlgebra C H)
6009 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
6010 kerAugClosedQuotTargetCompletedModuleOfSurj
6011 C hC hForm psi hpsi hfopen
6012 @Continuous
6013 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)
6014 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
6015 (zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom)
6016 inferInstance
6017 (zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient
6018 C hC hForm psi hpsi hfopen) := by
6019 let hcont :=
6020 continuous_crossedDiffModuleLiftLinear_sourceBoundaryToKerAugClosedQuot
6021 C hC hForm psi hpsi hfopen
6022 simpa [zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient, hcont] using
6023 continuous_zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift
6024 C hC hForm psi hpsi hfopen hcont
6026/-- The separated map to the closed kernel-augmentation quotient is surjective. -/
6027theorem zcSepDiffToKerAugClosedQuot_surj
6030 (hForm : ProCGroups.FiniteGroupClass.Formation C)
6031 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
6032 (hfopen : IsOpenMap psi) :
6033 Function.Surjective
6034 (zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient
6035 C hC hForm psi hpsi hfopen) := by
6036 let hcont :=
6037 continuous_crossedDiffModuleLiftLinear_sourceBoundaryToKerAugClosedQuot
6038 C hC hForm psi hpsi hfopen
6039 simpa [zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient, hcont] using
6040 zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift_surj
6041 C hC hForm psi hpsi hfopen hcont
6043/--
6044The reverse map from the closed augmentation quotient to the separated differential module is
6045evaluated on quotient classes by the constructed linear representative.
6046-/
6047@[simp]
6048theorem kerAugIdealQuotToZCSepDiffLinear_mk
6051 (hForm : ProCGroups.FiniteGroupClass.Formation C)
6052 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
6053 (hfopen : IsOpenMap psi)
6054 (x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
6055 kerAugIdealQuotToZCSepDiffLinear
6056 C hC hForm psi hpsi hfopen
6057 (Submodule.Quotient.mk
6058 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
6059 C hC hForm psi hpsi hfopen) x) =
6060 stdAugIdealToZCSepDiff
6061 C hC psi x := by
6062 rfl
6064/--
6065Composing the reverse closed-augmentation map with the forward separated map gives the identity.
6066-/
6067theorem kerAugIdealQuotToZCSepDiffLinear_comp_zcSepDiffToKerAugClosedQuot
6070 (hForm : ProCGroups.FiniteGroupClass.Formation C)
6071 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
6072 (hfopen : IsOpenMap psi) :
6073 letI : Module (ZCCompletedGroupAlgebra C H)
6074 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
6075 kerAugClosedQuotTargetCompletedModuleOfSurj
6076 C hC hForm psi hpsi hfopen
6077 (kerAugIdealQuotToZCSepDiffLinear
6078 C hC hForm psi hpsi hfopen).comp
6079 (zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient
6080 C hC hForm psi hpsi hfopen) =
6081 LinearMap.id := by
6082 letI : Module (ZCCompletedGroupAlgebra C H)
6083 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
6084 kerAugClosedQuotTargetCompletedModuleOfSurj
6085 C hC hForm psi hpsi hfopen
6086 apply zcSeparatedCompletedDifferentialModuleHom_ext C psi.toMonoidHom
6087 intro g
6088 rw [LinearMap.comp_apply,
6089 zcSepDiffToKerAugClosedQuot_universal]
6090 rw [LinearMap.id_apply]
6091 change
6092 kerAugIdealQuotToZCSepDiffLinear C hC hForm psi hpsi hfopen
6093 (Submodule.Quotient.mk
6094 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
6095 C hC hForm psi hpsi hfopen)
6096 ⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
6097 zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
6098 C G (MonoidHom.id G) g⟩) =
6099 zcSeparatedUniversalDifferential C psi.toMonoidHom g
6100 exact
6101 kerAugIdealQuotToZCSepDiffLinear_boundary
6102 C hC hForm psi hpsi hfopen g
6104/--
6105Composing the forward separated map with the reverse closed-augmentation map gives the identity.
6106-/
6107theorem zcSepDiffToKerAugClosedQuot_comp_kerAugIdealQuotToZCSepDiffLinear
6110 (hForm : ProCGroups.FiniteGroupClass.Formation C)
6111 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
6112 (hfopen : IsOpenMap psi) :
6113 letI : Module (ZCCompletedGroupAlgebra C H)
6114 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
6115 kerAugClosedQuotTargetCompletedModuleOfSurj
6116 C hC hForm psi hpsi hfopen
6117 (zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient
6118 C hC hForm psi hpsi hfopen).comp
6119 (kerAugIdealQuotToZCSepDiffLinear
6120 C hC hForm psi hpsi hfopen) =
6121 LinearMap.id := by
6122 letI : Module (ZCCompletedGroupAlgebra C H)
6123 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
6124 kerAugClosedQuotTargetCompletedModuleOfSurj
6125 C hC hForm psi hpsi hfopen
6126 letI : Module (ZCCompletedGroupAlgebra C G)
6127 (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
6128 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
6129 apply LinearMap.ext
6130 intro x
6131 refine Submodule.Quotient.induction_on
6132 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
6133 C hC hForm psi hpsi hfopen) x ?_
6134 intro y
6135 let F :=
6136 zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient
6137 C hC hForm psi hpsi hfopen
6138 let S :=
6139 stdAugIdealToZCSepDiff
6140 C hC psi
6141 let Q : zcCompletedGroupAlgebraStandardAugmentationIdeal C G →
6142 KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen :=
6143 fun y => Submodule.Quotient.mk
6144 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
6145 C hC hForm psi hpsi hfopen) y
6146 have hmk :
6147 kerAugIdealQuotToZCSepDiffLinear
6148 C hC hForm psi hpsi hfopen (Q y) =
6149 S y := by
6150 exact
6151 kerAugIdealQuotToZCSepDiffLinear_mk
6152 C hC hForm psi hpsi hfopen y
6153 change F
6154 (kerAugIdealQuotToZCSepDiffLinear
6155 C hC hForm psi hpsi hfopen (Q y)) = Q y
6156 rw [hmk]
6157 let P : zcCompletedGroupAlgebraStandardAugmentationIdeal C G → Prop := fun y =>
6158 F (S y) = Q y
6159 have hyspan : (y : ZCCompletedGroupAlgebra C G) ∈
6160 Submodule.span (ZCCompletedGroupAlgebra C G)
6161 (Set.range fun h : G => zcGroupLike C G h - 1) := by
6162 change (y : ZCCompletedGroupAlgebra C G) ∈
6163 zcCompletedGroupAlgebraStandardAugmentationIdeal C G
6164 exact y.2
6165 exact
6166 (Submodule.span_induction
6167 (p := fun z hz =>
6168 ∀ y' : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
6169 (y' : ZCCompletedGroupAlgebra C G) = z → P y')
6170 (by
6171 rintro _ ⟨g, rfl⟩ y' hy'
6172 have hy'' :
6173 y' =
6174 ⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
6175 zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
6176 C G (MonoidHom.id G) g⟩ := by
6177 apply Subtype.ext
6178 change
6179 (y' : ZCCompletedGroupAlgebra C G) =
6180 zcGroupLike C G g - 1
6181 exact hy'
6182 rw [hy'']
6183 let yg : zcCompletedGroupAlgebraStandardAugmentationIdeal C G :=
6184 ⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
6185 zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
6186 C G (MonoidHom.id G) g⟩
6187 change P yg
6188 have hSg :
6189 S yg = zcSeparatedUniversalDifferential C psi.toMonoidHom g := by
6190 exact
6191 stdAugIdealToZCSepDiff_boundary
6192 C hC psi g
6193 dsimp [P, Q]
6194 calc
6195 F (S yg) = F (zcSeparatedUniversalDifferential C psi.toMonoidHom g) := by
6196 exact congrArg F hSg
6197 _ =
6198 zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
6199 C hC hForm psi hpsi hfopen g := by
6200 rw [zcSepDiffToKerAugClosedQuot_universal]
6201 _ = Submodule.Quotient.mk yg := rfl)
6202 (by
6203 intro y' hy'
6204 have hy'' : y' = 0 := by
6205 apply Subtype.ext
6206 simpa using hy'
6207 rw [hy'']
6208 change P (0 : zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
6209 change F (S 0) = 0
6210 rw [map_zero, map_zero])
6211 (by
6212 intro a b ha hb hpa hpb y' hy'
6213 let ya : zcCompletedGroupAlgebraStandardAugmentationIdeal C G :=
6214 ⟨a, by
6215 simpa [zcCompletedGroupAlgebraStandardAugmentationIdeal_eq_span] using ha⟩
6216 let yb : zcCompletedGroupAlgebraStandardAugmentationIdeal C G :=
6217 ⟨b, by
6218 simpa [zcCompletedGroupAlgebraStandardAugmentationIdeal_eq_span] using hb⟩
6219 have hpa' : P ya := hpa ya rfl
6220 have hpb' : P yb := hpb yb rfl
6221 change F (S ya) = Q ya at hpa'
6222 change F (S yb) = Q yb at hpb'
6223 have hy'' : y' = ya + yb := by
6224 apply Subtype.ext
6225 simpa [ya, yb] using hy'
6226 rw [hy'']
6227 change P (ya + yb)
6228 change F (S (ya + yb)) = Q (ya + yb)
6229 rw [map_add, map_add, hpa', hpb']
6230 simp only [Submodule.Quotient.mk_add, Q])
6231 (by
6232 intro a b hb hpb y' hy'
6233 let yb : zcCompletedGroupAlgebraStandardAugmentationIdeal C G :=
6234 ⟨b, by
6235 simpa [zcCompletedGroupAlgebraStandardAugmentationIdeal_eq_span] using hb⟩
6236 have hpb' : P yb := hpb yb rfl
6237 change F (S yb) = Q yb at hpb'
6238 have hy'' : y' = a • yb := by
6239 apply Subtype.ext
6240 simpa [yb] using hy'
6241 rw [hy'']
6242 change P (a • yb)
6243 change F (S (a • yb)) = Q (a • yb)
6244 have hsource :
6245 a • S yb =
6246 zcCompletedGroupAlgebraMap C hC psi a • S yb := by
6247 exact
6248 (zcSeparatedCompletedDifferentialModule_source_map_smul
6249 C hC psi a (S yb)).symm
6250 calc
6251 F (S (a • yb)) = F (a • S yb) := by
6252 rw [map_smul]
6253 _ = F (zcCompletedGroupAlgebraMap C hC psi a • S yb) := by
6254 rw [hsource]
6255 _ = zcCompletedGroupAlgebraMap C hC psi a • F (S yb) := by
6256 rw [map_smul]
6257 _ = zcCompletedGroupAlgebraMap C hC psi a • Q yb := by
6258 rw [hpb']
6259 _ = a • Q yb := by
6260 exact
6261 kerAugClosedQuotTargetCompletedModuleOfSurj_map_smul
6262 C hC hForm psi hpsi hfopen a (Q yb)
6263 _ = Q (a • yb) := by
6264 dsimp [Q])
6265 hyspan) y rfl
6267/--
6268The separated completed differential module is the closed source augmentation quotient for a
6269surjective open continuous homomorphism.
6270-/
6271def zcSepDiffEquivKerAugClosedQuot_of_surj
6274 (hForm : ProCGroups.FiniteGroupClass.Formation C)
6275 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
6276 (hfopen : IsOpenMap psi) :
6277 letI : Module (ZCCompletedGroupAlgebra C H)
6278 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
6279 kerAugClosedQuotTargetCompletedModuleOfSurj
6280 C hC hForm psi hpsi hfopen
6281 ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom
6282 ≃ₗ[ZCCompletedGroupAlgebra C H]
6283 KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen := by
6284 letI : Module (ZCCompletedGroupAlgebra C H)
6285 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
6286 kerAugClosedQuotTargetCompletedModuleOfSurj
6287 C hC hForm psi hpsi hfopen
6288 exact
6289 LinearEquiv.ofLinear
6290 (zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient
6291 C hC hForm psi hpsi hfopen)
6292 (kerAugIdealQuotToZCSepDiffLinear
6293 C hC hForm psi hpsi hfopen)
6294 (zcSepDiffToKerAugClosedQuot_comp_kerAugIdealQuotToZCSepDiffLinear
6295 C hC hForm psi hpsi hfopen)
6296 (kerAugIdealQuotToZCSepDiffLinear_comp_zcSepDiffToKerAugClosedQuot
6297 C hC hForm psi hpsi hfopen)
6299/--
6300The Fox-coordinate equivalence is evaluated by the finite-stage coordinate formula in the
6301completed differential complex.
6302-/
6303@[simp]
6304theorem zcSepDiffEquivKerAugClosedQuot_of_surj_apply
6307 (hForm : ProCGroups.FiniteGroupClass.Formation C)
6308 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
6309 (hfopen : IsOpenMap psi)
6310 (x : ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :
6311 zcSepDiffEquivKerAugClosedQuot_of_surj
6312 C hC hForm psi hpsi hfopen x =
6313 zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient
6314 C hC hForm psi hpsi hfopen x := rfl
6316/--
6317The inverse Fox-coordinate equivalence is evaluated by reconstructing the class from its
6318completed coordinate data.
6319-/
6320@[simp]
6321theorem zcSepDiffEquivKerAugClosedQuot_of_surj_symm_apply
6324 (hForm : ProCGroups.FiniteGroupClass.Formation C)
6325 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
6326 (hfopen : IsOpenMap psi)
6327 (x : KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :
6328 (zcSepDiffEquivKerAugClosedQuot_of_surj
6329 C hC hForm psi hpsi hfopen).symm x =
6330 kerAugIdealQuotToZCSepDiffLinear
6331 C hC hForm psi hpsi hfopen x := rfl
6333/--
6334The closed source augmentation quotient equivalence identifies \(A_{\psi}(C)\) over
6335\(\mathbb{Z}_C\) with the separated completed differential module, rather than with the raw
6336algebraic quotient.
6337-/
6338def zcApsiEquivKerAugClosedQuot_of_surj
6341 (hForm : ProCGroups.FiniteGroupClass.Formation C)
6342 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
6343 (hfopen : IsOpenMap psi) :
6344 letI : Module (ZCCompletedGroupAlgebra C H)
6345 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
6346 kerAugClosedQuotTargetCompletedModuleOfSurj
6347 C hC hForm psi hpsi hfopen
6348 ZCApsi C psi.toMonoidHom ≃ₗ[ZCCompletedGroupAlgebra C H]
6349 KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen :=
6350 zcSepDiffEquivKerAugClosedQuot_of_surj
6351 C hC hForm psi hpsi hfopen
6353/--
6354The Fox-coordinate equivalence is evaluated by the finite-stage coordinate formula in the
6355completed differential complex.
6356-/
6357@[simp]
6358theorem zcApsiEquivKerAugClosedQuot_of_surj_apply
6361 (hForm : ProCGroups.FiniteGroupClass.Formation C)
6362 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
6363 (hfopen : IsOpenMap psi)
6364 (x : ZCApsi C psi.toMonoidHom) :
6365 zcApsiEquivKerAugClosedQuot_of_surj
6366 C hC hForm psi hpsi hfopen x =
6367 zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient
6368 C hC hForm psi hpsi hfopen x := rfl
6370/--
6371The inverse Fox-coordinate equivalence is evaluated by reconstructing the class from its
6372completed coordinate data.
6373-/
6374@[simp]
6375theorem zcApsiEquivKerAugClosedQuot_of_surj_symm_apply
6378 (hForm : ProCGroups.FiniteGroupClass.Formation C)
6379 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
6380 (hfopen : IsOpenMap psi)
6381 (x : KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :
6382 (zcApsiEquivKerAugClosedQuot_of_surj
6383 C hC hForm psi hpsi hfopen).symm x =
6384 kerAugIdealQuotToZCSepDiffLinear
6385 C hC hForm psi hpsi hfopen x := rfl
6387/--
6388For a surjective target map, the completed target scalar action on the kernel-augmentation
6389quotient is compatible with applying the induced algebra map.
6390-/
6391theorem zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_map_smul
6394 (hForm : ProCGroups.FiniteGroupClass.Formation C)
6395 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
6396 (hker_mul :
6397 ∀ k : ZCCompletedGroupAlgebra C G,
6398 k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) →
6399 ∀ y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
6400 (⟨k * (y : ZCCompletedGroupAlgebra C G),
6401 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
6402 zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
6403 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi)
6404 (a : ZCCompletedGroupAlgebra C G) (x : KernelAugmentationIdealQuotient C psi) :
6405 letI : Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi) :=
6406 zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
6407 C hC hForm psi hpsi hker_mul
6408 zcCompletedGroupAlgebraMap C hC psi a • x = a • x := by
6409 letI : Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi) :=
6410 zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
6411 C hC hForm psi hpsi hker_mul
6412 change zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi
6413 (zcCompletedGroupAlgebraMap C hC psi a) • x =
6414 a • x
6415 have hdiff :
6416 zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi
6417 (zcCompletedGroupAlgebraMap C hC psi a) - a ∈
6418 RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) := by
6419 change zcCompletedGroupAlgebraMap C hC psi
6420 (zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi
6421 (zcCompletedGroupAlgebraMap C hC psi a) - a) = 0
6422 rw [map_sub, zcCompletedGroupAlgebraMap_targetLiftOfSurjective, sub_self]
6423 have hzero :=
6424 zcCompletedGAKerAugQuot_ker_map_smul_eq_zero_of_kernelMulStandard_le
6425 C hC psi hker_mul
6426 (zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi
6427 (zcCompletedGroupAlgebraMap C hC psi a) - a) hdiff x
6428 rw [sub_smul] at hzero
6429 exact sub_eq_zero.mp hzero
6431/--
6432In the descended target module structure, the group-like element of \(\psi(g)\) acts in the same
6433way as the source group-like element of \(g\).
6434-/
6435theorem zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_groupLike_smul
6438 (hForm : ProCGroups.FiniteGroupClass.Formation C)
6439 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
6440 (hker_mul :
6441 ∀ k : ZCCompletedGroupAlgebra C G,
6442 k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) →
6443 ∀ y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
6444 (⟨k * (y : ZCCompletedGroupAlgebra C G),
6445 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
6446 zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
6447 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi)
6448 (g : G) (x : KernelAugmentationIdealQuotient C psi) :
6449 letI : Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi) :=
6450 zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
6451 C hC hForm psi hpsi hker_mul
6452 zcGroupLike C H (psi g) • x = zcGroupLike C G g • x := by
6453 letI : Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi) :=
6454 zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
6455 C hC hForm psi hpsi hker_mul
6456 rw [← zcCompletedGroupAlgebraMap_groupLike (C := C) (hC := hC) psi g]
6457 exact
6458 zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_map_smul
6459 C hC hForm psi hpsi hker_mul (zcGroupLike C G g) x
6461/--
6462Under the explicit kernel-product hypothesis, the source boundary is a crossed differential for
6463the descended completed target scalars.
6464-/
6465def sourceBoundaryToKerAugTargetCrossedHomOfSurjectiveOfKernelMulStandardLe
6468 (hForm : ProCGroups.FiniteGroupClass.Formation C)
6469 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
6470 (hker_mul :
6471 ∀ k : ZCCompletedGroupAlgebra C G,
6472 k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) →
6473 ∀ y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
6474 (⟨k * (y : ZCCompletedGroupAlgebra C G),
6475 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
6476 zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
6477 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi) :
6478 letI : Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi) :=
6479 zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
6480 C hC hForm psi hpsi hker_mul
6481 ScalarCrossedHom
6482 (zcCompletedGroupAlgebraScalar C psi.toMonoidHom)
6483 (KernelAugmentationIdealQuotient C psi) := by
6484 letI : Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi) :=
6485 zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
6486 C hC hForm psi hpsi hker_mul
6487 exact
6488 { toFun := zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient C psi
6489 map_mul' := by
6490 intro g h
6491 rw [(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient
6492 C psi).map_mul]
6493 congr 1
6494 change zcGroupLike C G g •
6495 zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient C psi h =
6496 zcGroupLike C H (psi g) •
6497 zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient C psi h
6498 exact
6499 (zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_groupLike_smul
6500 C hC hForm psi hpsi hker_mul g
6501 (zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient
6502 C psi h)).symm }
6504/--
6505Under the explicit kernel-product hypothesis, the completed universal differential module maps
6506to the algebraic source augmentation quotient by \(dg \mapsto [g]-1\). The hypothesis is the
6507condition needed for the algebraic quotient \(I(G) / I(\ker \psi)I(G)\) to carry the completed
6508target \(\mathbb{Z}_C\llbracket H\rrbracket\)-module structure.
6509-/
6510def zcDiffToKerAugQuotOfSurj_of_kernelMulStandard_le
6513 (hForm : ProCGroups.FiniteGroupClass.Formation C)
6514 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
6515 (hker_mul :
6516 ∀ k : ZCCompletedGroupAlgebra C G,
6517 k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) →
6518 ∀ y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
6519 (⟨k * (y : ZCCompletedGroupAlgebra C G),
6520 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
6521 zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
6522 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi) :
6523 letI : Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi) :=
6524 zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
6525 C hC hForm psi hpsi hker_mul
6526 ZCCompletedDifferentialModule C psi.toMonoidHom →ₗ[ZCCompletedGroupAlgebra C H]
6527 KernelAugmentationIdealQuotient C psi := by
6528 letI : Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi) :=
6529 zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
6530 C hC hForm psi hpsi hker_mul
6531 exact
6532 crossedHomModuleLift
6533 (A := KernelAugmentationIdealQuotient C psi)
6534 (zcCompletedGroupAlgebraScalar C psi.toMonoidHom)
6535 (sourceBoundaryToKerAugTargetCrossedHomOfSurjectiveOfKernelMulStandardLe
6536 C hC hForm psi hpsi hker_mul)
6538/--
6539Under the kernel-product hypothesis, the algebraic source augmentation quotient map sends the
6540universal differential to the class of \([g]-1\).
6541-/
6542@[simp 900]
6543theorem zcDiffToKerAugQuotOfSurj_of_kernelMulStandard_le_universal
6546 (hForm : ProCGroups.FiniteGroupClass.Formation C)
6547 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
6548 (hker_mul :
6549 ∀ k : ZCCompletedGroupAlgebra C G,
6550 k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) →
6551 ∀ y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
6552 (⟨k * (y : ZCCompletedGroupAlgebra C G),
6553 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
6554 zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
6555 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi)
6556 (g : G) :
6557 zcDiffToKerAugQuotOfSurj_of_kernelMulStandard_le
6558 C hC hForm psi hpsi hker_mul
6559 (zcUniversalDifferential C psi.toMonoidHom g) =
6560 zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient C psi g := by
6561 letI : Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi) :=
6562 zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
6563 C hC hForm psi hpsi hker_mul
6564 exact
6565 crossedHomModuleLift_universal
6566 (A := KernelAugmentationIdealQuotient C psi)
6567 (zcCompletedGroupAlgebraScalar C psi.toMonoidHom)
6568 (sourceBoundaryToKerAugTargetCrossedHomOfSurjectiveOfKernelMulStandardLe
6569 C hC hForm psi hpsi hker_mul) g
6571/--
6572The algebraic map \(A_{\psi}(C) \to I(G) / I(\ker \psi)I(G)\) is onto once the algebraic
6573quotient has the completed target scalar action supplied by the explicit kernel-product
6574hypothesis.
6575-/
6576theorem zcDiffToKerAugQuotOfSurj_of_kernelMulStandard_le_surj
6579 (hForm : ProCGroups.FiniteGroupClass.Formation C)
6580 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
6581 (hker_mul :
6582 ∀ k : ZCCompletedGroupAlgebra C G,
6583 k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) →
6584 ∀ y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
6585 (⟨k * (y : ZCCompletedGroupAlgebra C G),
6586 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
6587 zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
6588 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi) :
6589 Function.Surjective
6590 (zcDiffToKerAugQuotOfSurj_of_kernelMulStandard_le
6591 C hC hForm psi hpsi hker_mul) := by
6592 letI : Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi) :=
6593 zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
6594 C hC hForm psi hpsi hker_mul
6595 intro x
6596 refine Submodule.Quotient.induction_on
6597 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi) x ?_
6598 intro y
6599 let L :=
6600 zcDiffToKerAugQuotOfSurj_of_kernelMulStandard_le
6601 C hC hForm psi hpsi hker_mul
6602 let P : zcCompletedGroupAlgebraStandardAugmentationIdeal C G → Prop := fun y =>
6603 ∃ m : ZCCompletedDifferentialModule C psi.toMonoidHom,
6604 L m = Submodule.Quotient.mk
6605 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi) y
6606 have hy : P y := by
6607 have hyspan : (y : ZCCompletedGroupAlgebra C G) ∈
6608 Submodule.span (ZCCompletedGroupAlgebra C G)
6609 (Set.range fun h : G => zcGroupLike C G h - 1) := by
6610 change (y : ZCCompletedGroupAlgebra C G) ∈
6611 zcCompletedGroupAlgebraStandardAugmentationIdeal C G
6612 exact y.2
6613 refine Submodule.span_induction
6614 (p := fun z hz =>
6615 P
6616 ⟨z, by
6617 simpa [zcCompletedGroupAlgebraStandardAugmentationIdeal_eq_span] using hz⟩)
6618 ?hgen ?hzero ?hadd ?hsmul hyspan
6619 · rintro _ ⟨g, rfl⟩
6620 refine ⟨zcUniversalDifferential C psi.toMonoidHom g, ?_⟩
6621 rw [zcDiffToKerAugQuotOfSurj_of_kernelMulStandard_le_universal]
6622 rfl
6623 · refine ⟨0, ?_⟩
6624 change (0 : KernelAugmentationIdealQuotient C psi) =
6625 Submodule.Quotient.mk
6626 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi)
6627 (0 : zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
6628 rw [Submodule.Quotient.mk_zero]
6629 · intro a b ha hb hpa hpb
6630 rcases hpa with ⟨ma, hma⟩
6631 rcases hpb with ⟨mb, hmb⟩
6632 refine ⟨ma + mb, ?_⟩
6633 rw [map_add, hma, hmb, ← Submodule.Quotient.mk_add]
6634 apply congrArg (fun t : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
6635 Submodule.Quotient.mk
6636 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi) t)
6637 exact Subtype.ext rfl
6638 · intro a b hb hpb
6639 rcases hpb with ⟨m, hm⟩
6640 refine ⟨zcCompletedGroupAlgebraMap C hC psi a • m, ?_⟩
6641 rw [map_smul, hm]
6642 calc
6643 zcCompletedGroupAlgebraMap C hC psi a •
6644 Submodule.Quotient.mk
6645 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi)
6646 ⟨b, by
6647 simpa [zcCompletedGroupAlgebraStandardAugmentationIdeal_eq_span] using hb⟩ =
6648 a •
6649 Submodule.Quotient.mk
6650 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi)
6651 ⟨b, by
6652 simpa [zcCompletedGroupAlgebraStandardAugmentationIdeal_eq_span] using hb⟩ := by
6653 exact
6654 zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_map_smul
6655 C hC hForm psi hpsi hker_mul a
6656 (Submodule.Quotient.mk
6657 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi)
6658 ⟨b, by
6659 simpa [zcCompletedGroupAlgebraStandardAugmentationIdeal_eq_span] using hb⟩)
6660 _ = Submodule.Quotient.mk
6661 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi)
6662 ⟨a • b, by
6663 rw [zcCompletedGroupAlgebraStandardAugmentationIdeal_eq_span]
6664 exact Submodule.smul_mem
6665 (Submodule.span (ZCCompletedGroupAlgebra C G)
6666 (Set.range fun h : G => zcGroupLike C G h - 1)) a hb⟩ := by
6667 rw [← Submodule.Quotient.mk_smul]
6668 apply congrArg (fun t : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
6669 Submodule.Quotient.mk
6670 (p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi) t)
6671 exact Subtype.ext rfl
6672 exact hy
6674/--
6675Under the explicit kernel-product hypothesis, the natural map from the algebraic source
6676augmentation quotient to the closed quotient is \(\mathbb{Z}_C\llbracket H\rrbracket\)-linear.
6677-/
6678def zcCompletedGAKerAugQuotToClosedQuotientTargetLinear_of_kernelMulStandard_le
6681 (hForm : ProCGroups.FiniteGroupClass.Formation C)
6682 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
6683 (hfopen : IsOpenMap psi)
6684 (hker_mul :
6685 ∀ k : ZCCompletedGroupAlgebra C G,
6686 k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) →
6687 ∀ y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
6688 (⟨k * (y : ZCCompletedGroupAlgebra C G),
6689 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
6690 zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
6691 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi) :
6692 letI : Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi) :=
6693 zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
6694 C hC hForm psi hpsi hker_mul
6695 letI : Module (ZCCompletedGroupAlgebra C H)
6696 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
6697 kerAugClosedQuotTargetCompletedModuleOfSurj
6698 C hC hForm psi hpsi hfopen
6699 KernelAugmentationIdealQuotient C psi →ₗ[ZCCompletedGroupAlgebra C H]
6700 KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen := by
6701 letI : Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi) :=
6702 zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
6703 C hC hForm psi hpsi hker_mul
6704 letI : Module (ZCCompletedGroupAlgebra C H)
6705 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
6706 kerAugClosedQuotTargetCompletedModuleOfSurj
6707 C hC hForm psi hpsi hfopen
6708 let Q :=
6709 zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient
6710 C hC hForm psi hpsi hfopen
6711 refine
6712 { toFun := Q
6713 map_add' := by
6714 intro x y
6715 exact map_add Q x y
6716 map_smul' := by
6717 intro a x
6718 change Q
6719 (zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi a • x) =
6720 a • Q x
6721 rw [map_smul]
6722 symm
6723 calc
6724 a • Q x =
6725 zcCompletedGroupAlgebraMap C hC psi
6726 (zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi a) •
6727 Q x := by
6728 rw [zcCompletedGroupAlgebraMap_targetLiftOfSurjective]
6729 _ =
6730 zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi a • Q x := by
6731 exact
6732 kerAugClosedQuotTargetCompletedModuleOfSurj_map_smul
6733 C hC hForm psi hpsi hfopen
6734 (zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi a)
6735 (Q x) }
6737/--
6738The natural map from the algebraic source augmentation quotient to the closed quotient sends
6739each algebraic quotient class to its closed quotient class.
6740-/
6741@[simp 900]
6742theorem zcCompletedGAKerAugQuotToClosedQuotientTargetLinear_of_kernelMulStandard_le_mk
6745 (hForm : ProCGroups.FiniteGroupClass.Formation C)
6746 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
6747 (hfopen : IsOpenMap psi)
6748 (hker_mul :
6749 ∀ k : ZCCompletedGroupAlgebra C G,
6750 k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) →
6751 ∀ y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
6752 (⟨k * (y : ZCCompletedGroupAlgebra C G),
6753 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
6754 zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
6755 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi)
6756 (x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
6757 zcCompletedGAKerAugQuotToClosedQuotientTargetLinear_of_kernelMulStandard_le
6758 C hC hForm psi hpsi hfopen hker_mul (Submodule.Quotient.mk x) =
6759 (Submodule.Quotient.mk x :
6760 KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) := by
6761 letI : Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi) :=
6762 zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
6763 C hC hForm psi hpsi hker_mul
6764 letI : Module (ZCCompletedGroupAlgebra C H)
6765 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
6766 kerAugClosedQuotTargetCompletedModuleOfSurj
6767 C hC hForm psi hpsi hfopen
6768 rw [zcCompletedGAKerAugQuotToClosedQuotientTargetLinear_of_kernelMulStandard_le]
6769 exact
6770 zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient_mk
6771 C hC hForm psi hpsi hfopen x
6773/--
6774The algebraic quotient map followed by the natural closed-quotient map is the closed quotient
6775map already constructed directly from \(A_{\psi}(C)\).
6776-/
6777theorem zcDiffToKerAugClosedQuotOfSurj_eq_toClosed_comp_quotient_of_kernelMulStandard_le
6780 (hForm : ProCGroups.FiniteGroupClass.Formation C)
6781 (psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
6782 (hfopen : IsOpenMap psi)
6783 (hker_mul :
6784 ∀ k : ZCCompletedGroupAlgebra C G,
6785 k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) →
6786 ∀ y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
6787 (⟨k * (y : ZCCompletedGroupAlgebra C G),
6788 (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
6789 zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
6790 zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi) :
6791 (zcCompletedGAKerAugQuotToClosedQuotientTargetLinear_of_kernelMulStandard_le
6792 C hC hForm psi hpsi hfopen hker_mul).comp
6793 (zcDiffToKerAugQuotOfSurj_of_kernelMulStandard_le
6794 C hC hForm psi hpsi hker_mul) =
6795 zcCompletedDifferentialModuleToKernelAugmentationClosedQuotientOfSurjective
6796 C hC hForm psi hpsi hfopen := by
6797 letI : Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi) :=
6798 zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
6799 C hC hForm psi hpsi hker_mul
6800 letI : Module (ZCCompletedGroupAlgebra C H)
6801 (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
6802 kerAugClosedQuotTargetCompletedModuleOfSurj
6803 C hC hForm psi hpsi hfopen
6804 apply crossedDifferentialModuleHom_ext
6805 (zcCompletedGroupAlgebraScalar C psi.toMonoidHom)
6806 intro g
6807 calc
6808 ((zcCompletedGAKerAugQuotToClosedQuotientTargetLinear_of_kernelMulStandard_le
6809 C hC hForm psi hpsi hfopen hker_mul).comp
6810 (zcDiffToKerAugQuotOfSurj_of_kernelMulStandard_le
6811 C hC hForm psi hpsi hker_mul))
6812 (zcUniversalDifferential C psi.toMonoidHom g)
6813 =
6814 zcCompletedGAKerAugQuotToClosedQuotientTargetLinear_of_kernelMulStandard_le
6815 C hC hForm psi hpsi hfopen hker_mul
6816 (zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient C psi g) := by
6817 rw [LinearMap.comp_apply,
6818 zcDiffToKerAugQuotOfSurj_of_kernelMulStandard_le_universal]
6819 _ =
6820 zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
6821 C hC hForm psi hpsi hfopen g := by
6822 rw [zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient]
6823 exact
6824 zcCompletedGAKerAugQuotToClosedQuotientTargetLinear_of_kernelMulStandard_le_mk
6825 C hC hForm psi hpsi hfopen hker_mul
6826 ⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
6827 zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
6828 C G (MonoidHom.id G) g⟩
6829 _ =
6830 zcCompletedDifferentialModuleToKernelAugmentationClosedQuotientOfSurjective
6831 C hC hForm psi hpsi hfopen
6832 (zcUniversalDifferential C psi.toMonoidHom g) := by
6833 rw [zcDiffToKerAugClosedQuotOfSurj_universal]
6835/--
6836Source kernel group-like differences act trivially on \(A_{\psi}(C)\) after restricting scalars
6837along \(\mathbb{Z}_C\llbracket G\rrbracket \to \mathbb{Z}_C\llbracket H\rrbracket\).
6838-/
6839theorem zcCompletedDifferentialModule_sourceKernelGroupLikeSubOne_smul_eq_zero
6840 (hC : ProCGroups.FiniteGroupClass.Hereditary C) (psi : ContinuousMonoidHom G H)
6841 (n : ProfiniteKernelSubgroup psi) (x : ZCCompletedDifferentialModule C psi.toMonoidHom) :
6842 letI : Module (ZCCompletedGroupAlgebra C G)
6843 (ZCCompletedDifferentialModule C psi.toMonoidHom) :=
6844 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
6845 (zcGroupLike C G n.1 - 1) • x = 0 := by
6846 letI : Module (ZCCompletedGroupAlgebra C G)
6847 (ZCCompletedDifferentialModule C psi.toMonoidHom) :=
6848 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
6849 have hmap :
6850 zcCompletedGroupAlgebraMap C hC psi (zcGroupLike C G n.1 - 1) = 0 := by
6851 rw [map_sub, zcCompletedGroupAlgebraMap_groupLike, map_one]
6852 rw [show psi (n : G) = 1 from n.2]
6853 rw [map_one, sub_self]
6854 change zcCompletedGroupAlgebraMap C hC psi (zcGroupLike C G n.1 - 1) • x = 0
6855 rw [hmap, zero_smul]
6857/--
6858The algebraic product \(I(\ker \psi)I(G)\) acts trivially on \(A_{\psi}(C)\) after restricting
6859scalars along \(\mathbb{Z}_C\llbracket G\rrbracket \to \mathbb{Z}_C\llbracket H\rrbracket\).
6860-/
6861theorem zcCompletedDifferentialModule_kernelAugmentationIdealMulStandard_smul_eq_zero
6862 (hC : ProCGroups.FiniteGroupClass.Hereditary C) (psi : ContinuousMonoidHom G H)
6863 (y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
6864 (hy : y ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi)
6865 (x : ZCCompletedDifferentialModule C psi.toMonoidHom) :
6866 letI : Module (ZCCompletedGroupAlgebra C G)
6867 (ZCCompletedDifferentialModule C psi.toMonoidHom) :=
6868 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
6869 (y : ZCCompletedGroupAlgebra C G) • x = 0 := by
6870 letI : Module (ZCCompletedGroupAlgebra C G)
6871 (ZCCompletedDifferentialModule C psi.toMonoidHom) :=
6872 Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
6873 change (y : ZCCompletedGroupAlgebra C G) • x = 0
6874 refine Submodule.span_induction
6875 (p := fun y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
6876 fun _ => (y : ZCCompletedGroupAlgebra C G) • x = 0) ?_ ?_ ?_ ?_ hy
6877 · rintro _ ⟨⟨n, s⟩, rfl⟩
6878 change ((zcGroupLike C G n.1 - 1) * (s : ZCCompletedGroupAlgebra C G)) • x = 0
6879 rw [mul_smul]
6880 exact zcCompletedDifferentialModule_sourceKernelGroupLikeSubOne_smul_eq_zero
6881 C hC psi n ((s : ZCCompletedGroupAlgebra C G) • x)
6882 · change (0 : ZCCompletedGroupAlgebra C G) • x = 0
6883 rw [zero_smul]
6884 · intro y z _ _ hy hz
6885 change ((y : ZCCompletedGroupAlgebra C G) +
6886 (z : ZCCompletedGroupAlgebra C G)) • x = 0
6887 rw [add_smul, hy, hz, zero_add]
6888 · intro a y _ hy
6889 change (a * (y : ZCCompletedGroupAlgebra C G)) • x = 0
6890 rw [mul_smul, hy, smul_zero]
6892end KernelAugmentationQuotient
6894end
6896end FoxDifferential