Module

LFP_exists_fg_nonempty_iso_pullbackAlong.lean

Nyll.JacobianVarieties.Picard.LFP_exists_fg_nonempty_iso_pullbackAlong

118 declarations indexed from sourceView source ↗

Module documentation

Descent of a rigidified line bundle to a finitely generated stage

For a curve with a two-affine-chart cover, this file descends a rigidified line bundle to a finitely generated subalgebra of the coefficient algebra. The affine-chart modules and their overlap maps are constructed at that stage, and the pulled-back underlying sheaf is isomorphic to the original underlying sheaf.

Imports30

Nyll.AlgebraicCurves.Geometry.Picard.Functor
Nyll.SchemeTheory.Cohomology.Cech.TwoAffineOpenCover
Nyll.AlgebraicCurves.Cohomology.Cech.TwoChartCechSectionsOf
Nyll.AlgebraicCurves.Geometry.Schemes.AffineLimit
Nyll.SchemeTheory.Descent.FGSubalgebra
Nyll.AlgebraicCurves.Geometry.Schemes.FGSubalgebraTensorStage
Nyll.AlgebraicCurves.Geometry.Picard.StageHom
Nyll.AlgebraicCurves.Geometry.Schemes.RigidifiedLineBundleOfInvertible
Nyll.AlgebraicCurves.Foundations.Algebra.Module_IsDirectLimit_exists_invertible_linearEquiv_baseChange
Nyll.AlgebraicCurves.Foundations.Algebra.Module_IsDirectLimit_exists_stage_linearEquiv_of_finitePresentation_compat
Nyll.AlgebraicCurves.Foundations.Algebra.Module_Invertible_of_projective_of_forall_finrank_eq_one
Nyll.AlgebraicCurves.Geometry.Projective.Scheme_TwoAffineOpenCover_sectionsOf_isInvertible_projective_finite_finrank
Nyll.AlgebraicCurves.Geometry.Schemes.Scheme_TwoAffineOpenCover_exists_sectionsOf_overlap_linearEquiv_baseChange_of_isInvertible
Nyll.AlgebraicCurves.Geometry.Schemes.Scheme_TwoAffineOpenCover_exists_baseChangeIsos_structureSheaf
Nyll.AlgebraicCurves.Geometry.Schemes.Scheme_TwoAffineOpenCover_exists_stage_baseChangeIsos_structureSheaf
Nyll.AlgebraicCurves.Geometry.Schemes.Scheme_TwoAffineOpenCover_exists_stage_sectionsOf_baseChange_equiv_of_locallyTrivial
Nyll.AlgebraicCurves.Geometry.Projective.Scheme_TwoAffineOpenCover_exists_isInvertible_sectionsOf_equiv_of_projective
Nyll.AlgebraicCurves.Geometry.Schemes.Scheme_TwoAffineOpenCover_nonempty_iso_of_sectionsOf_linearEquiv_of_isInvertible
Nyll.JacobianVarieties.Picard.Designation
Nyll.JacobianVarieties.Picard.Representation
Nyll.SchemeTheory.CategoryTheory.ModuleCatTensorExtension
Mathlib.Algebra.Category.ModuleCat.ChangeOfRings
Mathlib.Algebra.Algebra.Basic
Mathlib.Algebra.Algebra.Hom
Mathlib.Algebra.Group.Action.Defs
Mathlib.Algebra.Module.Equiv.Defs
Mathlib.Algebra.Module.FinitePresentation
Mathlib.RingTheory.TensorProduct.Maps
Mathlib.LinearAlgebra.TensorProduct.Tower
Mathlib.RingTheory.PicardGroup

Imported by2

theorem

cancelBaseChange_one_tmul_rTensor

L76 ↗

ModuleCat.cancelBaseChange_one_tmul_rTensor

Documentation

Tensor cancellation on a unit outer tensor is the coefficient map induced by g.

Source context 5 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section CategoricalClutching
universe u v
open CategoryTheory TensorProduct
namespace ModuleCat
open scoped ChangeOfRings

Lean source · full declaration

theorem cancelBaseChange_one_tmul_rTensor
    {R S T : Type u} [CommRing R] [CommRing S] [CommRing T]
    [Algebra R S] [Algebra R T] (g : S →ₐ[R] T) (M : ModuleCat.{u} R)
    (x : S ⊗[R] M) :
    let _ : Algebra S T := g.toRingHom.toAlgebra
    let _ : IsScalarTower R S T :=
      IsScalarTower.of_algebraMap_eq fun r => (g.commutes r).symm
    TensorProduct.AlgebraTensorModule.cancelBaseChange R S T T M
        ((1 : T) ⊗ₜ[S] x) = g.toLinearMap.rTensor M x := by
  intro algebraST towerRST
  induction x using TensorProduct.inductionOn with
  | tmul s m =>
    rw [TensorProduct.AlgebraTensorModule.cancelBaseChange_tmul, LinearMap.rTensor_tmul,
      AlgHom.toLinearMap_apply]
    change (g s * 1) ⊗ₜ[R] m = g s ⊗ₜ[R] m
    rw [mul_one]
  | add x y hx hy => simp only [tmul_add, map_add, hx, hy]

theorem

exists_stage_iso_of_finitePresentation_compat

L97 ↗

ModuleCat.exists_stage_iso_of_finitePresentation_compat

Documentation

A finite-presentation isomorphism descends as an isomorphism of scalar-extension objects, with the canonical cocone comparison square proved in ModuleCat. The stage owner's actual algebra hom is identified with the prescribed cocone map.

Source context 5 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section CategoricalClutching
universe u v
open CategoryTheory TensorProduct
namespace ModuleCat
open scoped ChangeOfRings

Lean source · full declaration

theorem exists_stage_iso_of_finitePresentation_compat
    {ι : Type v} [Preorder ι] [IsDirectedOrder ι] [Nonempty ι]
    {K : Type u} [CommRing K]
    (B : ι → Type u) [∀ i, CommRing (B i)] [∀ i, Algebra K (B i)]
    (τ : ∀ i j, i ≤ j → B i →ₐ[K] B j)
    [DirectedSystem B (fun i j h => τ i j h)]
    (U : Type u) [CommRing U] [Algebra K U]
    (g : ∀ i, B i →ₐ[K] U)
    (hg : ∀ i j (h : i ≤ j), (g j).comp (τ i j h) = g i)
    (hsurj : ∀ b : U, ∃ i x, g i x = b)
    (hzero : ∀ i (x : B i), g i x = 0 → ∃ (j : ι) (h : i ≤ j), τ i j h x = 0)
    (i₀ : ι) (P Q : ModuleCat.{u} (B i₀))
    [Module.FinitePresentation (B i₀) P] [Module.FinitePresentation (B i₀) Q]
    (e : (extendScalars (g i₀).toRingHom).obj P ≅
      (extendScalars (g i₀).toRingHom).obj Q) :
    ∃ (j : ι) (hj : i₀ ≤ j)
      (ej : (extendScalars (τ i₀ j hj).toRingHom).obj P ≅
        (extendScalars (τ i₀ j hj).toRingHom).obj Q),
      let hcomp : (g j).toRingHom.comp (τ i₀ j hj).toRingHom = (g i₀).toRingHom :=
        congrArg (fun f : B i₀ →ₐ[K] U => f.toRingHom) (hg i₀ j hj)
      let c : extendScalars (g i₀).toRingHom ≅
          extendScalars (τ i₀ j hj).toRingHom ⋙ extendScalars (g j).toRingHom :=
        (eqToIso (congrArg (fun f : B i₀ →+* U => extendScalars.{u, u, u} f) hcomp)).symm ≪≫
          extendScalarsComp (τ i₀ j hj).toRingHom (g j).toRingHom
      (extendScalars (g j).toRingHom).map ej.hom ≫ c.inv.app Q =
        c.inv.app P ≫ e.hom := by
  let : Algebra (B i₀) U := (g i₀).toRingHom.toAlgebra
  obtain ⟨j, hj, gj, hgj, ej, hcompat⟩ :=
    Module.IsDirectLimit.exists_stage_linearEquiv_of_finitePresentation_compat
      B τ U g hg hsurj hzero i₀ (fun _ => rfl) P Q e.toLinearEquiv
  let : Algebra (B i₀) (B j) := (τ i₀ j hj).toRingHom.toAlgebra
  let γ : B j →ₐ[B i₀] U :=
    { (g j).toRingHom with
      commutes' := fun b => DFunLike.congr_fun (hg i₀ j hj) b }
  have hgjEq : gj = γ := AlgHom.ext hgj
  rw [hgjEq] at hcompat
  let stageIso : (extendScalars (τ i₀ j hj).toRingHom).obj P ≅
      (extendScalars (τ i₀ j hj).toRingHom).obj Q := ej.toModuleIso
  refine ⟨j, hj, stageIso, ?_⟩
  let hcomp : (g j).toRingHom.comp (τ i₀ j hj).toRingHom = (g i₀).toRingHom :=
    congrArg (fun f : B i₀ →ₐ[K] U => f.toRingHom) (hg i₀ j hj)
  let c : extendScalars (g i₀).toRingHom ≅
      extendScalars (τ i₀ j hj).toRingHom ⋙ extendScalars (g j).toRingHom :=
    (eqToIso (congrArg (fun f : B i₀ →+* U => extendScalars.{u, u, u} f) hcomp)).symm ≪≫
      extendScalarsComp (τ i₀ j hj).toRingHom (g j).toRingHom
  change (extendScalars (g j).toRingHom).map stageIso.hom ≫ c.inv.app Q =
    c.inv.app P ≫ e.hom
  let : Algebra (B j) U := γ.toRingHom.toAlgebra
  let : IsScalarTower (B i₀) (B j) U :=
    IsScalarTower.of_algebraMap_eq fun b => (γ.commutes b).symm
  have hcp : c.app P =
      (TensorProduct.AlgebraTensorModule.cancelBaseChange (B i₀) (B j) U U P).symm.toModuleIso :=
    extendScalars_comparison_eq_cancel (τ i₀ j hj).toRingHom (g j).toRingHom
      (g i₀).toRingHom hcomp P
  have hcq : c.app Q =
      (TensorProduct.AlgebraTensorModule.cancelBaseChange (B i₀) (B j) U U Q).symm.toModuleIso :=
    extendScalars_comparison_eq_cancel (τ i₀ j hj).toRingHom (g j).toRingHom
      (g i₀).toRingHom hcomp Q
  apply ExtendScalars.hom_ext (f := (g j).toRingHom)
  intro x
  change (c.app Q).inv
      ((extendScalars (g j).toRingHom).map stageIso.hom ((1 : U) ⊗ₜ[B j,(g j).toRingHom] x)) =
    e.hom ((c.app P).inv ((1 : U) ⊗ₜ[B j,(g j).toRingHom] x))
  rw [ExtendScalars.map_tmul, hcq, hcp]
  change TensorProduct.AlgebraTensorModule.cancelBaseChange (B i₀) (B j) U U Q
      ((1 : U) ⊗ₜ[B j] (ej x)) =
    e.hom (TensorProduct.AlgebraTensorModule.cancelBaseChange (B i₀) (B j) U U P
      ((1 : U) ⊗ₜ[B j] x))
  rw [cancelBaseChange_one_tmul_rTensor γ Q (ej x), cancelBaseChange_one_tmul_rTensor γ P x]
  exact (hcompat x).symm

theorem

extendScalars_one_chart_pasting

L172 ↗

ModuleCat.extendScalars_one_chart_pasting

Documentation

Raising both the chart stage and the overlap stage preserves the realization of a module. All four supplied squares are equalities of actual ring maps; the old realization square is derived from them.

Source context 5 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section CategoricalClutching
universe u v
open CategoryTheory TensorProduct
namespace ModuleCat
open scoped ChangeOfRings

Lean source · full declaration

theorem extendScalars_one_chart_pasting
    {B J C D V U : Type u}
    [CommRing B] [CommRing J] [CommRing C] [CommRing D] [CommRing V] [CommRing U]
    (a : B →+* J) (q : B →+* C) (r : J →+* D) (s : C →+* D)
    (p : B →+* V) (p' : J →+* V) (uC : C →+* U) (uD : D →+* U) (v : V →+* U)
    (hc : p'.comp a = p) (ho : uD.comp s = uC)
    (hd : r.comp a = s.comp q) (hn : uD.comp r = v.comp p')
    (X : ModuleCat.{u} B) (M : ModuleCat.{u} V)
    (e : (extendScalars p).obj X ≅ M) :
  let hu : uC.comp q = v.comp p := by
    calc
      uC.comp q = (uD.comp s).comp q :=
        congrArg (fun t : C →+* U => t.comp q) ho.symm
      _ = uD.comp (s.comp q) := rfl
      _ = uD.comp (r.comp a) := congrArg (fun t : B →+* D => uD.comp t) hd.symm
      _ = (uD.comp r).comp a := rfl
      _ = (v.comp p').comp a := congrArg (fun t : J →+* U => t.comp a) hn
      _ = v.comp p := congrArg (fun t : B →+* V => v.comp t) hc
  let L : ModuleCat.{u} J := (extendScalars a).obj X
  let Q : ModuleCat.{u} C := (extendScalars q).obj X
  let N : ModuleCat.{u} D := (extendScalars r).obj L
  let cChart : extendScalars p ≅ extendScalars a ⋙ extendScalars p' :=
    (eqToIso (congrArg (fun t : B →+* V => extendScalars.{u, u, u} t) hc)).symm ≪≫
      extendScalarsComp a p'
  let e' : (extendScalars p').obj L ≅ M := cChart.symm.app X ≪≫ e
  let cStage : extendScalars (r.comp a) ≅ extendScalars q ⋙ extendScalars s :=
    (eqToIso (congrArg (fun t : B →+* D => extendScalars.{u, u, u} t) hd.symm)).symm ≪≫
      extendScalarsComp q s
  let stage : N ≅ (extendScalars s).obj Q :=
    (extendScalarsComp a r).symm.app X ≪≫ cStage.app X
  let cOld : extendScalars (v.comp p) ≅ extendScalars q ⋙ extendScalars uC :=
    (eqToIso (congrArg (fun t : B →+* U => extendScalars.{u, u, u} t) hu)).symm ≪≫
      extendScalarsComp q uC
  let oldRealization : (extendScalars uC).obj Q ≅ (extendScalars v).obj M :=
    cOld.symm.app X ≪≫ (extendScalarsComp p v).app X ≪≫ (extendScalars v).mapIso e
  let cNew : extendScalars (v.comp p') ≅ extendScalars r ⋙ extendScalars uD :=
    (eqToIso (congrArg (fun t : J →+* U => extendScalars.{u, u, u} t) hn)).symm ≪≫
      extendScalarsComp r uD
  let newRealization : (extendScalars uD).obj N ≅ (extendScalars v).obj M :=
    cNew.symm.app L ≪≫ (extendScalarsComp p' v).app L ≪≫ (extendScalars v).mapIso e'
  let cOverlap : extendScalars uC ≅ extendScalars s ⋙ extendScalars uD :=
    (eqToIso (congrArg (fun t : C →+* U => extendScalars.{u, u, u} t) ho)).symm ≪≫
      extendScalarsComp s uD
  (extendScalars uD).mapIso stage ≪≫ cOverlap.symm.app Q ≪≫ oldRealization =
    newRealization := by
  let hu : uC.comp q = v.comp p := by
    calc
      uC.comp q = (uD.comp s).comp q :=
        congrArg (fun t : C →+* U => t.comp q) ho.symm
      _ = uD.comp (s.comp q) := rfl
      _ = uD.comp (r.comp a) := congrArg (fun t : B →+* D => uD.comp t) hd.symm
      _ = (uD.comp r).comp a := rfl
      _ = (v.comp p').comp a := congrArg (fun t : J →+* U => t.comp a) hn
      _ = v.comp p := congrArg (fun t : B →+* V => v.comp t) hc
  let L : ModuleCat.{u} J := (extendScalars a).obj X
  let Q : ModuleCat.{u} C := (extendScalars q).obj X
  let N : ModuleCat.{u} D := (extendScalars r).obj L
  let cChart : extendScalars p ≅ extendScalars a ⋙ extendScalars p' :=
    (eqToIso (congrArg (fun t : B →+* V => extendScalars.{u, u, u} t) hc)).symm ≪≫
      extendScalarsComp a p'
  let e' : (extendScalars p').obj L ≅ M := cChart.symm.app X ≪≫ e
  let cStage : extendScalars (r.comp a) ≅ extendScalars q ⋙ extendScalars s :=
    (eqToIso (congrArg (fun t : B →+* D => extendScalars.{u, u, u} t) hd.symm)).symm ≪≫
      extendScalarsComp q s
  let stage : N ≅ (extendScalars s).obj Q :=
    (extendScalarsComp a r).symm.app X ≪≫ cStage.app X
  let cOld : extendScalars (v.comp p) ≅ extendScalars q ⋙ extendScalars uC :=
    (eqToIso (congrArg (fun t : B →+* U => extendScalars.{u, u, u} t) hu)).symm ≪≫
      extendScalarsComp q uC
  let oldRealization : (extendScalars uC).obj Q ≅ (extendScalars v).obj M :=
    cOld.symm.app X ≪≫ (extendScalarsComp p v).app X ≪≫ (extendScalars v).mapIso e
  let cNew : extendScalars (v.comp p') ≅ extendScalars r ⋙ extendScalars uD :=
    (eqToIso (congrArg (fun t : J →+* U => extendScalars.{u, u, u} t) hn)).symm ≪≫
      extendScalarsComp r uD
  let newRealization : (extendScalars uD).obj N ≅ (extendScalars v).obj M :=
    cNew.symm.app L ≪≫ (extendScalarsComp p' v).app L ≪≫ (extendScalars v).mapIso e'
  let cOverlap : extendScalars uC ≅ extendScalars s ⋙ extendScalars uD :=
    (eqToIso (congrArg (fun t : C →+* U => extendScalars.{u, u, u} t) ho)).symm ≪≫
      extendScalarsComp s uD
  change (extendScalars uD).mapIso stage ≪≫ cOverlap.symm.app Q ≪≫ oldRealization =
    newRealization
  let k : (extendScalars (uD.comp (r.comp a))).obj X ≅ (extendScalars uD).obj N :=
    (extendScalarsComp (r.comp a) uD).app X ≪≫
      (extendScalars uD).mapIso ((extendScalarsComp a r).app X)
  let H : (extendScalars uD).obj N ⟶ (extendScalars v).obj M :=
    (extendScalars uD).map stage.hom ≫ cOverlap.inv.app Q ≫ oldRealization.hom
  apply CategoryTheory.Iso.ext
  change H = newRealization.hom
  apply (cancel_epi k.hom).mp
  apply ExtendScalars.hom_ext (f := uD.comp (r.comp a))
  intro x
  let xa : L := (extendRestrictScalarsAdj a).unit.app X x
  let xq : Q := (extendRestrictScalarsAdj q).unit.app X x
  let xr : N := (extendRestrictScalarsAdj r).unit.app L xa
  have hk : k.hom ((extendRestrictScalarsAdj (uD.comp (r.comp a))).unit.app X x) =
      (extendRestrictScalarsAdj uD).unit.app N xr := by
    change (extendScalars uD).map ((extendScalarsComp a r).hom.app X)
      ((extendScalarsComp (r.comp a) uD).hom.app X
        ((extendRestrictScalarsAdj (uD.comp (r.comp a))).unit.app X x)) = _
    have houter := extendScalars_comparison_hom_unit (r.comp a) uD
      (uD.comp (r.comp a)) rfl X x
    change (extendScalarsComp (r.comp a) uD).hom.app X
      ((extendRestrictScalarsAdj (uD.comp (r.comp a))).unit.app X x) = _ at houter
    rw [houter]
    have hinner := extendScalars_comparison_hom_unit a r (r.comp a) rfl X x
    change (extendScalarsComp a r).hom.app X
      ((extendRestrictScalarsAdj (r.comp a)).unit.app X x) = xr at hinner
    exact (extendScalars_map_unit uD ((extendScalarsComp a r).hom.app X)
      ((extendRestrictScalarsAdj (r.comp a)).unit.app X x)).trans
      (congrArg (fun z : N => (extendRestrictScalarsAdj uD).unit.app N z) hinner)
  have hstage : stage.hom xr = (extendRestrictScalarsAdj s).unit.app Q xq := by
    change cStage.hom.app X ((extendScalarsComp a r).inv.app X xr) = _
    have ha : (extendScalarsComp a r).inv.app X xr =
        (extendRestrictScalarsAdj (r.comp a)).unit.app X x :=
      extendScalars_comparison_inv_unit a r (r.comp a) rfl X x
    rw [ha]
    exact extendScalars_comparison_hom_unit q s (r.comp a) hd.symm X x
  have hold : oldRealization.hom ((extendRestrictScalarsAdj uC).unit.app Q xq) =
      (extendRestrictScalarsAdj v).unit.app M
        (e.hom ((extendRestrictScalarsAdj p).unit.app X x)) := by
    change (extendScalars v).map e.hom
      ((extendScalarsComp p v).hom.app X
        (cOld.inv.app X ((extendRestrictScalarsAdj uC).unit.app Q xq))) = _
    rw [extendScalars_comparison_inv_unit q uC (v.comp p) hu X x]
    have hp := extendScalars_comparison_hom_unit p v (v.comp p) rfl X x
    change (extendScalarsComp p v).hom.app X
      ((extendRestrictScalarsAdj (v.comp p)).unit.app X x) = _ at hp
    rw [hp]
    exact extendScalars_map_unit v e.hom ((extendRestrictScalarsAdj p).unit.app X x)
  have hnew : newRealization.hom ((extendRestrictScalarsAdj uD).unit.app N xr) =
      (extendRestrictScalarsAdj v).unit.app M
        (e.hom ((extendRestrictScalarsAdj p).unit.app X x)) := by
    change (extendScalars v).map e'.hom
      ((extendScalarsComp p' v).hom.app L
        (cNew.inv.app L ((extendRestrictScalarsAdj uD).unit.app N xr))) = _
    rw [extendScalars_comparison_inv_unit r uD (v.comp p') hn L xa]
    have hp := extendScalars_comparison_hom_unit p' v (v.comp p') rfl L xa
    change (extendScalarsComp p' v).hom.app L
      ((extendRestrictScalarsAdj (v.comp p')).unit.app L xa) = _ at hp
    rw [hp]
    exact (extendScalars_map_unit v e'.hom
      ((extendRestrictScalarsAdj p').unit.app L xa)).trans
      (congrArg
        (fun z : (extendScalars p).obj X =>
          (extendRestrictScalarsAdj v).unit.app M (e.hom z))
        (extendScalars_comparison_inv_unit a p' p hc X x))
  have hmiddle : cOverlap.inv.app Q
      ((extendScalars uD).map stage.hom ((extendRestrictScalarsAdj uD).unit.app N xr)) =
      (extendRestrictScalarsAdj uC).unit.app Q xq :=
    (congrArg
      (fun z : (extendScalars uD).obj ((extendScalars s).obj Q) => cOverlap.inv.app Q z)
      ((extendScalars_map_unit uD stage.hom xr).trans
        (congrArg
          (fun z : (extendScalars s).obj Q =>
            (extendRestrictScalarsAdj uD).unit.app ((extendScalars s).obj Q) z)
          hstage))).trans
      (extendScalars_comparison_inv_unit s uD uC ho Q xq)
  have hleft : H ((extendRestrictScalarsAdj uD).unit.app N xr) =
      newRealization.hom ((extendRestrictScalarsAdj uD).unit.app N xr) :=
    (congrArg
      (fun z : (extendScalars uC).obj Q => oldRealization.hom z) hmiddle).trans
      (hold.trans hnew.symm)
  clear_value H newRealization k
  have heval : H (k.hom ((extendRestrictScalarsAdj (uD.comp (r.comp a))).unit.app X x)) =
      newRealization.hom
        (k.hom ((extendRestrictScalarsAdj (uD.comp (r.comp a))).unit.app X x)) :=
    (congrArg (fun z : (extendScalars uD).obj N => H z) hk).trans
      (hleft.trans
        (congrArg (fun z : (extendScalars uD).obj N => newRealization.hom z) hk).symm)
  exact (ModuleCat.comp_apply k.hom H
    ((extendRestrictScalarsAdj (uD.comp (r.comp a))).unit.app X x)).trans
    (heval.trans (ModuleCat.comp_apply k.hom newRealization.hom
      ((extendRestrictScalarsAdj (uD.comp (r.comp a))).unit.app X x)).symm)

theorem

finitePresentation_extendScalars

L348 ↗

ModuleCat.finitePresentation_extendScalars

Documentation

Finite presentation is preserved by scalar extension along its prescribed ring map.

Source context 5 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section CategoricalClutching
universe u v
open CategoryTheory TensorProduct
namespace ModuleCat
open scoped ChangeOfRings

Lean source · statement

theorem finitePresentation_extendScalars
    {R S : Type u} [CommRing R] [CommRing S] (f : R →+* S)
    (M : ModuleCat.{u} R) [Module.FinitePresentation R M] :
    Module.FinitePresentation S ((extendScalars f).obj M)

theorem

exists_stage_iso_of_finitePresentation_ring_compat

L357 ↗

ModuleCat.exists_stage_iso_of_finitePresentation_ring_compat

Documentation

The finite-stage isomorphism descent owner applies to a directed system of ring maps, using only the canonical lifted-integer scalar actions internally.

Source context 5 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section CategoricalClutching
universe u v
open CategoryTheory TensorProduct
namespace ModuleCat
open scoped ChangeOfRings

Lean source · full declaration

theorem exists_stage_iso_of_finitePresentation_ring_compat
    {ι : Type v} [Preorder ι] [IsDirectedOrder ι] [Nonempty ι]
    (B : ι → Type u) [∀ i, CommRing (B i)]
    (τ : ∀ i j, i ≤ j → B i →+* B j)
    [DirectedSystem B (fun i j h => τ i j h)]
    (U : Type u) [CommRing U]
    (g : ∀ i, B i →+* U)
    (hg : ∀ i j (h : i ≤ j), (g j).comp (τ i j h) = g i)
    (hsurj : ∀ b : U, ∃ i x, g i x = b)
    (hzero : ∀ i (x : B i), g i x = 0 → ∃ (j : ι) (h : i ≤ j), τ i j h x = 0)
    (i₀ : ι) (P Q : ModuleCat.{u} (B i₀))
    [Module.FinitePresentation (B i₀) P] [Module.FinitePresentation (B i₀) Q]
    (e : (extendScalars (g i₀)).obj P ≅ (extendScalars (g i₀)).obj Q) :
    ∃ (j : ι) (hj : i₀ ≤ j)
      (ej : (extendScalars (τ i₀ j hj)).obj P ≅ (extendScalars (τ i₀ j hj)).obj Q),
      let c : extendScalars (g i₀) ≅ extendScalars (τ i₀ j hj) ⋙ extendScalars (g j) :=
        (eqToIso (congrArg (fun f : B i₀ →+* U => extendScalars.{u, u, u} f)
          (hg i₀ j hj))).symm ≪≫ extendScalarsComp (τ i₀ j hj) (g j)
      (extendScalars (g j)).map ej.hom ≫ c.inv.app Q = c.inv.app P ≫ e.hom := by
  let : ∀ i, Algebra (ULift.{u} ℤ) (B i) := fun i => ULift.algebra' ℤ (B i)
  let : Algebra (ULift.{u} ℤ) U := ULift.algebra' ℤ U
  let τA : ∀ i j, i ≤ j → B i →ₐ[ULift.{u} ℤ] B j := fun i j hij =>
    { τ i j hij with
      commutes' := fun n => by
        change τ i j hij (n.down : B i) = (n.down : B j)
        exact map_intCast (τ i j hij) n.down }
  let gA : ∀ i, B i →ₐ[ULift.{u} ℤ] U := fun i =>
    { g i with
      commutes' := fun n => by
        change g i (n.down : B i) = (n.down : U)
        exact map_intCast (g i) n.down }
  have hgA : ∀ i j (h : i ≤ j), (gA j).comp (τA i j h) = gA i := by
    intro i j h
    exact AlgHom.ext fun b => RingHom.congr_fun (hg i j h) b
  exact exists_stage_iso_of_finitePresentation_compat B τA U gA hgA hsurj hzero i₀ P Q e

theorem

exists_common_stage_clutching

L397 ↗

ModuleCat.exists_common_stage_clutching

Documentation

A clutching isomorphism between realizations of two finitely presented stage modules descends after raising both chart stages to one common index. The module pair and its realization maps are canonical scalar extensions, and the complete clutching realization equality is proved.

Source context 5 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section CategoricalClutching
universe u v
open CategoryTheory TensorProduct
namespace ModuleCat
open scoped ChangeOfRings

Lean source · full declaration

theorem exists_common_stage_clutching
    {ι : Type v} [Preorder ι] [IsDirectedOrder ι] [Nonempty ι]
    (B0 B1 C : ι → Type u)
    [∀ i, CommRing (B0 i)] [∀ i, CommRing (B1 i)] [∀ i, CommRing (C i)]
    (τ0 : ∀ i j, i ≤ j → B0 i →+* B0 j)
    (τ1 : ∀ i j, i ≤ j → B1 i →+* B1 j)
    (τC : ∀ i j, i ≤ j → C i →+* C j)
    [DirectedSystem B0 (fun i j h => τ0 i j h)]
    [DirectedSystem B1 (fun i j h => τ1 i j h)]
    [DirectedSystem C (fun i j h => τC i j h)]
    (U0 U1 U : Type u) [CommRing U0] [CommRing U1] [CommRing U]
    (g0 : ∀ i, B0 i →+* U0) (g1 : ∀ i, B1 i →+* U1) (gC : ∀ i, C i →+* U)
    (hg0 : ∀ i j (h : i ≤ j), (g0 j).comp (τ0 i j h) = g0 i)
    (hg1 : ∀ i j (h : i ≤ j), (g1 j).comp (τ1 i j h) = g1 i)
    (hgC : ∀ i j (h : i ≤ j), (gC j).comp (τC i j h) = gC i)
    (hsurj : ∀ b : U, ∃ i x, gC i x = b)
    (hzero : ∀ i (x : C i), gC i x = 0 → ∃ (j : ι) (h : i ≤ j), τC i j h x = 0)
    (ρ0 : ∀ i, B0 i →+* C i) (ρ1 : ∀ i, B1 i →+* C i)
    (hρ0 : ∀ i j (h : i ≤ j), (τC i j h).comp (ρ0 i) = (ρ0 j).comp (τ0 i j h))
    (hρ1 : ∀ i j (h : i ≤ j), (τC i j h).comp (ρ1 i) = (ρ1 j).comp (τ1 i j h))
    (v0 : U0 →+* U) (v1 : U1 →+* U)
    (hlim0 : ∀ i, (gC i).comp (ρ0 i) = v0.comp (g0 i))
    (hlim1 : ∀ i, (gC i).comp (ρ1 i) = v1.comp (g1 i))
    (i0 i1 : ι) (P0 : ModuleCat.{u} (B0 i0)) (P1 : ModuleCat.{u} (B1 i1))
    [Module.FinitePresentation (B0 i0) P0] [Module.FinitePresentation (B1 i1) P1]
    (σlim : (extendScalars v0).obj ((extendScalars (g0 i0)).obj P0) ≅
      (extendScalars v1).obj ((extendScalars (g1 i1)).obj P1)) :
    ∃ (j : ι) (h0 : i0 ≤ j) (h1 : i1 ≤ j),
      let L0 : ModuleCat.{u} (B0 j) := (extendScalars (τ0 i0 j h0)).obj P0
      let L1 : ModuleCat.{u} (B1 j) := (extendScalars (τ1 i1 j h1)).obj P1
      let M0 : ModuleCat.{u} U0 := (extendScalars (g0 i0)).obj P0
      let M1 : ModuleCat.{u} U1 := (extendScalars (g1 i1)).obj P1
      let Q0 : ModuleCat.{u} (C j) := (extendScalars (ρ0 j)).obj L0
      let Q1 : ModuleCat.{u} (C j) := (extendScalars (ρ1 j)).obj L1
      ∃ (E0 : (extendScalars (g0 j)).obj L0 ≅ M0)
        (E1 : (extendScalars (g1 j)).obj L1 ≅ M1)
        (F0 : (extendScalars (gC j)).obj Q0 ≅ (extendScalars v0).obj M0)
        (F1 : (extendScalars (gC j)).obj Q1 ≅ (extendScalars v1).obj M1)
        (σj : Q0 ≅ Q1),
        (∀ x : L0,
          F0.hom ((extendRestrictScalarsAdj (gC j)).unit.app Q0
            (show Q0 from (extendRestrictScalarsAdj (ρ0 j)).unit.app L0 x)) =
          (extendRestrictScalarsAdj v0).unit.app M0
            (E0.hom ((extendRestrictScalarsAdj (g0 j)).unit.app L0 x))) ∧
        (∀ x : L1,
          F1.hom ((extendRestrictScalarsAdj (gC j)).unit.app Q1
            (show Q1 from (extendRestrictScalarsAdj (ρ1 j)).unit.app L1 x)) =
          (extendRestrictScalarsAdj v1).unit.app M1
            (E1.hom ((extendRestrictScalarsAdj (g1 j)).unit.app L1 x))) ∧
        (extendScalars (gC j)).mapIso σj ≪≫ F1 = F0 ≪≫ σlim := by
  obtain ⟨i2, h02, h12⟩ := directed_of (· ≤ ·) i0 i1
  let q0 : B0 i0 →+* C i2 := (ρ0 i2).comp (τ0 i0 i2 h02)
  let Q0 : ModuleCat.{u} (C i2) := (extendScalars q0).obj P0
  have : Module.FinitePresentation (C i2) Q0 := finitePresentation_extendScalars q0 P0
  have hu0 : (gC i2).comp q0 = v0.comp (g0 i0) := by
    calc
      (gC i2).comp q0 = ((gC i2).comp (ρ0 i2)).comp (τ0 i0 i2 h02) := rfl
      _ = (v0.comp (g0 i2)).comp (τ0 i0 i2 h02) :=
        congrArg (fun t : B0 i2 →+* U => t.comp (τ0 i0 i2 h02)) (hlim0 i2)
      _ = v0.comp (g0 i0) :=
        congrArg (fun t : B0 i0 →+* U0 => v0.comp t) (hg0 i0 i2 h02)
  let cOld0 : extendScalars (v0.comp (g0 i0)) ≅
      extendScalars q0 ⋙ extendScalars (gC i2) :=
    (eqToIso (congrArg (fun t : B0 i0 →+* U => extendScalars.{u, u, u} t) hu0)).symm ≪≫
      extendScalarsComp q0 (gC i2)
  let oldF0 : (extendScalars (gC i2)).obj Q0 ≅
      (extendScalars v0).obj ((extendScalars (g0 i0)).obj P0) :=
    cOld0.symm.app P0 ≪≫ (extendScalarsComp (g0 i0) v0).app P0 ≪≫
      (extendScalars v0).mapIso (Iso.refl ((extendScalars (g0 i0)).obj P0))
  let q1 : B1 i1 →+* C i2 := (ρ1 i2).comp (τ1 i1 i2 h12)
  let Q1 : ModuleCat.{u} (C i2) := (extendScalars q1).obj P1
  have : Module.FinitePresentation (C i2) Q1 := finitePresentation_extendScalars q1 P1
  have hu1 : (gC i2).comp q1 = v1.comp (g1 i1) := by
    calc
      (gC i2).comp q1 = ((gC i2).comp (ρ1 i2)).comp (τ1 i1 i2 h12) := rfl
      _ = (v1.comp (g1 i2)).comp (τ1 i1 i2 h12) :=
        congrArg (fun t : B1 i2 →+* U => t.comp (τ1 i1 i2 h12)) (hlim1 i2)
      _ = v1.comp (g1 i1) :=
        congrArg (fun t : B1 i1 →+* U1 => v1.comp t) (hg1 i1 i2 h12)
  let cOld1 : extendScalars (v1.comp (g1 i1)) ≅
      extendScalars q1 ⋙ extendScalars (gC i2) :=
    (eqToIso (congrArg (fun t : B1 i1 →+* U => extendScalars.{u, u, u} t) hu1)).symm ≪≫
      extendScalarsComp q1 (gC i2)
  let oldF1 : (extendScalars (gC i2)).obj Q1 ≅
      (extendScalars v1).obj ((extendScalars (g1 i1)).obj P1) :=
    cOld1.symm.app P1 ≪≫ (extendScalarsComp (g1 i1) v1).app P1 ≪≫
      (extendScalars v1).mapIso (Iso.refl ((extendScalars (g1 i1)).obj P1))
  let θ : (extendScalars (gC i2)).obj Q0 ≅ (extendScalars (gC i2)).obj Q1 :=
    oldF0 ≪≫ σlim ≪≫ oldF1.symm
  obtain ⟨j, h2j, β, hβ⟩ := exists_stage_iso_of_finitePresentation_ring_compat
    C τC U gC hgC hsurj hzero i2 Q0 Q1 θ
  have h0 : i0 ≤ j := le_trans h02 h2j
  have h1 : i1 ≤ j := le_trans h12 h2j
  let s : C i2 →+* C j := τC i2 j h2j
  let cOverlap : extendScalars (gC i2) ≅ extendScalars s ⋙ extendScalars (gC j) :=
    (eqToIso (congrArg (fun t : C i2 →+* U => extendScalars.{u, u, u} t)
      (hgC i2 j h2j))).symm ≪≫ extendScalarsComp s (gC j)
  change (extendScalars (gC j)).map β.hom ≫ cOverlap.inv.app Q1 =
    cOverlap.inv.app Q0 ≫ θ.hom at hβ
  let L0 : ModuleCat.{u} (B0 j) := (extendScalars (τ0 i0 j h0)).obj P0
  let M0 : ModuleCat.{u} U0 := (extendScalars (g0 i0)).obj P0
  let cChart0 : extendScalars (g0 i0) ≅
      extendScalars (τ0 i0 j h0) ⋙ extendScalars (g0 j) :=
    (eqToIso (congrArg (fun t : B0 i0 →+* U0 => extendScalars.{u, u, u} t)
      (hg0 i0 j h0))).symm ≪≫ extendScalarsComp (τ0 i0 j h0) (g0 j)
  let e0 : (extendScalars (g0 j)).obj L0 ≅ M0 :=
    cChart0.symm.app P0 ≪≫ Iso.refl M0
  let cNew0 : extendScalars (v0.comp (g0 j)) ≅
      extendScalars (ρ0 j) ⋙ extendScalars (gC j) :=
    (eqToIso (congrArg (fun t : B0 j →+* U => extendScalars.{u, u, u} t)
      (hlim0 j))).symm ≪≫ extendScalarsComp (ρ0 j) (gC j)
  let F0 : (extendScalars (gC j)).obj ((extendScalars (ρ0 j)).obj L0) ≅
      (extendScalars v0).obj M0 :=
    cNew0.symm.app L0 ≪≫ (extendScalarsComp (g0 j) v0).app L0 ≪≫
      (extendScalars v0).mapIso e0
  let L1 : ModuleCat.{u} (B1 j) := (extendScalars (τ1 i1 j h1)).obj P1
  let M1 : ModuleCat.{u} U1 := (extendScalars (g1 i1)).obj P1
  let cChart1 : extendScalars (g1 i1) ≅
      extendScalars (τ1 i1 j h1) ⋙ extendScalars (g1 j) :=
    (eqToIso (congrArg (fun t : B1 i1 →+* U1 => extendScalars.{u, u, u} t)
      (hg1 i1 j h1))).symm ≪≫ extendScalarsComp (τ1 i1 j h1) (g1 j)
  let e1 : (extendScalars (g1 j)).obj L1 ≅ M1 :=
    cChart1.symm.app P1 ≪≫ Iso.refl M1
  let cNew1 : extendScalars (v1.comp (g1 j)) ≅
      extendScalars (ρ1 j) ⋙ extendScalars (gC j) :=
    (eqToIso (congrArg (fun t : B1 j →+* U => extendScalars.{u, u, u} t)
      (hlim1 j))).symm ≪≫ extendScalarsComp (ρ1 j) (gC j)
  let F1 : (extendScalars (gC j)).obj ((extendScalars (ρ1 j)).obj L1) ≅
      (extendScalars v1).obj M1 :=
    cNew1.symm.app L1 ≪≫ (extendScalarsComp (g1 j) v1).app L1 ≪≫
      (extendScalars v1).mapIso e1
  have hd0 : (ρ0 j).comp (τ0 i0 j h0) = s.comp q0 := by
    have ht : (τ0 i2 j h2j).comp (τ0 i0 i2 h02) = τ0 i0 j h0 :=
      RingHom.ext fun x => DirectedSystem.map_map' τ0 h02 h2j x
    calc
      (ρ0 j).comp (τ0 i0 j h0) =
          (ρ0 j).comp ((τ0 i2 j h2j).comp (τ0 i0 i2 h02)) :=
        congrArg (fun t : B0 i0 →+* B0 j => (ρ0 j).comp t) ht.symm
      _ = ((ρ0 j).comp (τ0 i2 j h2j)).comp (τ0 i0 i2 h02) := rfl
      _ = ((τC i2 j h2j).comp (ρ0 i2)).comp (τ0 i0 i2 h02) :=
        congrArg (fun t : B0 i2 →+* C j => t.comp (τ0 i0 i2 h02))
          (hρ0 i2 j h2j).symm
      _ = s.comp q0 := rfl
  let cStage0 : extendScalars ((ρ0 j).comp (τ0 i0 j h0)) ≅
      extendScalars q0 ⋙ extendScalars s :=
    (eqToIso (congrArg (fun t : B0 i0 →+* C j => extendScalars.{u, u, u} t) hd0.symm)).symm ≪≫
      extendScalarsComp q0 s
  let lam0 : (extendScalars (ρ0 j)).obj L0 ≅ (extendScalars s).obj Q0 :=
    (extendScalarsComp (τ0 i0 j h0) (ρ0 j)).symm.app P0 ≪≫ cStage0.app P0
  have hpaste0 := extendScalars_one_chart_pasting
    (τ0 i0 j h0) q0 (ρ0 j) s (g0 i0) (g0 j) (gC i2) (gC j) v0
    (hg0 i0 j h0) (hgC i2 j h2j) hd0 (hlim0 j)
    P0 M0 (Iso.refl M0)
  change (extendScalars (gC j)).mapIso lam0 ≪≫ cOverlap.symm.app Q0 ≪≫ oldF0 = F0
    at hpaste0
  have hd1 : (ρ1 j).comp (τ1 i1 j h1) = s.comp q1 := by
    have ht : (τ1 i2 j h2j).comp (τ1 i1 i2 h12) = τ1 i1 j h1 :=
      RingHom.ext fun x => DirectedSystem.map_map' τ1 h12 h2j x
    calc
      (ρ1 j).comp (τ1 i1 j h1) =
          (ρ1 j).comp ((τ1 i2 j h2j).comp (τ1 i1 i2 h12)) :=
        congrArg (fun t : B1 i1 →+* B1 j => (ρ1 j).comp t) ht.symm
      _ = ((ρ1 j).comp (τ1 i2 j h2j)).comp (τ1 i1 i2 h12) := rfl
      _ = ((τC i2 j h2j).comp (ρ1 i2)).comp (τ1 i1 i2 h12) :=
        congrArg (fun t : B1 i2 →+* C j => t.comp (τ1 i1 i2 h12))
          (hρ1 i2 j h2j).symm
      _ = s.comp q1 := rfl
  let cStage1 : extendScalars ((ρ1 j).comp (τ1 i1 j h1)) ≅
      extendScalars q1 ⋙ extendScalars s :=
    (eqToIso (congrArg (fun t : B1 i1 →+* C j => extendScalars.{u, u, u} t) hd1.symm)).symm ≪≫
      extendScalarsComp q1 s
  let lam1 : (extendScalars (ρ1 j)).obj L1 ≅ (extendScalars s).obj Q1 :=
    (extendScalarsComp (τ1 i1 j h1) (ρ1 j)).symm.app P1 ≪≫ cStage1.app P1
  have hpaste1 := extendScalars_one_chart_pasting
    (τ1 i1 j h1) q1 (ρ1 j) s (g1 i1) (g1 j) (gC i2) (gC j) v1
    (hg1 i1 j h1) (hgC i2 j h2j) hd1 (hlim1 j)
    P1 M1 (Iso.refl M1)
  change (extendScalars (gC j)).mapIso lam1 ≪≫ cOverlap.symm.app Q1 ≪≫ oldF1 = F1
    at hpaste1
  let σj : (extendScalars (ρ0 j)).obj L0 ≅ (extendScalars (ρ1 j)).obj L1 :=
    lam0 ≪≫ β ≪≫ lam1.symm
  refine ⟨j, h0, h1, e0, e1, F0, F1, σj, ?_, ?_, ?_⟩
  · intro x
    exact extendScalars_realization_unit (ρ0 j) (g0 j) (gC j) v0 (hlim0 j) L0 M0 e0 x
  · intro x
    exact extendScalars_realization_unit (ρ1 j) (g1 j) (gC j) v1 (hlim1 j) L1 M1 e1 x
  · change (extendScalars (gC j)).mapIso σj ≪≫ F1 = F0 ≪≫ σlim
    have hβIso : (extendScalars (gC j)).mapIso β ≪≫ cOverlap.symm.app Q1 =
        cOverlap.symm.app Q0 ≪≫ θ := by
      apply CategoryTheory.Iso.ext
      exact hβ
    have hβRealized : (extendScalars (gC j)).mapIso β ≪≫ cOverlap.symm.app Q1 ≪≫ oldF1 =
        cOverlap.symm.app Q0 ≪≫ oldF0 ≪≫ σlim := by
      calc
        (extendScalars (gC j)).mapIso β ≪≫ cOverlap.symm.app Q1 ≪≫ oldF1 =
            ((extendScalars (gC j)).mapIso β ≪≫ cOverlap.symm.app Q1) ≪≫ oldF1 :=
          (Iso.trans_assoc _ _ _).symm
        _ = (cOverlap.symm.app Q0 ≪≫ θ) ≪≫ oldF1 :=
          congrArg (fun z => z ≪≫ oldF1) hβIso
        _ = cOverlap.symm.app Q0 ≪≫ oldF0 ≪≫ σlim := by
          simp only [θ, Iso.trans_assoc, Iso.symm_self_id, Iso.trans_refl]
    clear_value F0 F1 oldF0 oldF1 lam0 lam1 cOverlap
    calc
      (extendScalars (gC j)).mapIso σj ≪≫ F1 =
          (extendScalars (gC j)).mapIso lam0 ≪≫ (extendScalars (gC j)).mapIso β ≪≫
            (extendScalars (gC j)).mapIso lam1.symm ≪≫ F1 := by
        simp only [σj, Functor.mapIso_trans, Iso.trans_assoc]
      _ = (extendScalars (gC j)).mapIso lam0 ≪≫ (extendScalars (gC j)).mapIso β ≪≫
          (extendScalars (gC j)).mapIso lam1.symm ≪≫
            (extendScalars (gC j)).mapIso lam1 ≪≫ cOverlap.symm.app Q1 ≪≫ oldF1 := by
        rw [← hpaste1]
      _ = (extendScalars (gC j)).mapIso lam0 ≪≫ (extendScalars (gC j)).mapIso β ≪≫
          cOverlap.symm.app Q1 ≪≫ oldF1 := by
        simp only [Functor.mapIso_symm, Iso.symm_self_id_assoc]
      _ = (extendScalars (gC j)).mapIso lam0 ≪≫ cOverlap.symm.app Q0 ≪≫ oldF0 ≪≫ σlim :=
        congrArg (fun z => (extendScalars (gC j)).mapIso lam0 ≪≫ z) hβRealized
      _ = ((extendScalars (gC j)).mapIso lam0 ≪≫ cOverlap.symm.app Q0 ≪≫ oldF0) ≪≫ σlim := by
        rw [Iso.trans_assoc, Iso.trans_assoc]
      _ = F0 ≪≫ σlim := congrArg (fun z => z ≪≫ σlim) hpaste0

theorem

extendScalars_square_unit_smul

L619 ↗

ModuleCat.extendScalars_square_unit_smul

Documentation

The canonical comparison for a commuting scalar-extension square transports a chart coefficient through the restriction ring map.

Source context 5 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section CategoricalClutching
universe u v
open CategoryTheory TensorProduct
namespace ModuleCat
open scoped ChangeOfRings

Lean source · full declaration

theorem extendScalars_square_unit_smul
    {B J C D : Type u} [CommRing B] [CommRing J] [CommRing C] [CommRing D]
    (a : B →+* J) (q : B →+* C) (r : J →+* D) (s : C →+* D)
    (hsquare : r.comp a = s.comp q) (X : ModuleCat.{u} B) (x : X) (j : J) :
    let c : extendScalars (r.comp a) ≅ extendScalars q ⋙ extendScalars s :=
      (eqToIso (congrArg (fun t : B →+* D => extendScalars.{u, u, u} t) hsquare.symm)).symm ≪≫
        extendScalarsComp q s
    let κ : (extendScalars r).obj ((extendScalars a).obj X) ≅
        (extendScalars s).obj ((extendScalars q).obj X) :=
      (extendScalarsComp a r).symm.app X ≪≫ c.app X
    let xa : (extendScalars a).obj X := (extendRestrictScalarsAdj a).unit.app X x
    κ.hom ((extendRestrictScalarsAdj r).unit.app ((extendScalars a).obj X) (j • xa)) =
      r j • (show (extendScalars s).obj ((extendScalars q).obj X) from
        (extendRestrictScalarsAdj s).unit.app ((extendScalars q).obj X)
          ((extendRestrictScalarsAdj q).unit.app X x)) := by
  let c : extendScalars (r.comp a) ≅ extendScalars q ⋙ extendScalars s :=
    (eqToIso (congrArg (fun t : B →+* D => extendScalars.{u, u, u} t) hsquare.symm)).symm ≪≫
      extendScalarsComp q s
  let κ : (extendScalars r).obj ((extendScalars a).obj X) ≅
      (extendScalars s).obj ((extendScalars q).obj X) :=
    (extendScalarsComp a r).symm.app X ≪≫ c.app X
  let xa : (extendScalars a).obj X := (extendRestrictScalarsAdj a).unit.app X x
  change κ.hom ((extendRestrictScalarsAdj r).unit.app ((extendScalars a).obj X) (j • xa)) = _
  have hunit : κ.hom ((extendRestrictScalarsAdj r).unit.app ((extendScalars a).obj X) xa) =
      (extendRestrictScalarsAdj s).unit.app ((extendScalars q).obj X)
        ((extendRestrictScalarsAdj q).unit.app X x) := by
    have hcontract := extendScalars_comparison_inv_unit a r (r.comp a) rfl X x
    change (extendScalarsComp a r).inv.app X
      ((extendRestrictScalarsAdj r).unit.app ((extendScalars a).obj X) xa) = _ at hcontract
    exact (congrArg
      (fun z : (extendScalars (r.comp a)).obj X => c.hom.app X z) hcontract).trans
      (extendScalars_comparison_hom_unit q s (r.comp a) hsquare.symm X x)
  clear_value κ
  have hbalance : (extendRestrictScalarsAdj r).unit.app ((extendScalars a).obj X) (j • xa) =
      r j • (show (extendScalars r).obj ((extendScalars a).obj X) from
        (extendRestrictScalarsAdj r).unit.app ((extendScalars a).obj X) xa) :=
    ((extendRestrictScalarsAdj r).unit.app ((extendScalars a).obj X)).hom.map_smul j xa
  exact (congrArg
    (fun z : (extendScalars r).obj ((extendScalars a).obj X) => κ.hom z) hbalance).trans
    ((κ.hom.hom.map_smul (r j)
      ((extendRestrictScalarsAdj r).unit.app ((extendScalars a).obj X) xa)).trans
      (congrArg
        (fun z : (extendScalars s).obj ((extendScalars q).obj X) => r j • z) hunit))

theorem

exists_scalarExtension_realization

L673 ↗

ModuleCat.exists_scalarExtension_realization

Documentation

Extend a chart realization through a supplied overlap realization. The resulting equivalence satisfies the restriction law on the canonical scalar-extension unit, using the original overlap law.

Source context 7 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section CategoricalClutching
universe u v
open CategoryTheory TensorProduct
namespace ModuleCat
open scoped ChangeOfRings
universe w
universe w'

Lean source · full declaration

theorem exists_scalarExtension_realization
    {V U : Type u} [CommRing V] [CommRing U] (v : V →+* U)
    (M : ModuleCat.{u} V)
    {N : Type w} [AddCommGroup N] [Module V N]
    {O : Type w'} [AddCommGroup O] [Module U O]
    (I : M ≃ₗ[V] N)
    (J : let _ : Algebra V U := v.toAlgebra; U ⊗[V] N ≃ₗ[U] O)
    (r : N → O)
    (hJ : let _ : Algebra V U := v.toAlgebra;
      ∀ n : N, J ((1 : U) ⊗ₜ[V] n) = r n) :
    ∃ E : (extendScalars v).obj M ≃ₗ[U] O,
      ∀ m : M, E ((extendRestrictScalarsAdj v).unit.app M m) = r (I m) := by
  let : Algebra V U := v.toAlgebra
  let E : (extendScalars v).obj M ≃ₗ[U] O :=
    (LinearEquiv.baseChange V U M N I) ≪≫ₗ J
  refine ⟨E, ?_⟩
  intro m
  change J (LinearEquiv.baseChange V U M N I ((1 : U) ⊗ₜ[V] m)) = r (I m)
  rw [LinearEquiv.baseChange_tmul, hJ]

theorem

invertible_extendScalars_of_invertible

L696 ↗

ModuleCat.invertible_extendScalars_of_invertible

Documentation

Scalar extension along the specified ring map preserves invertibility. The categorical object uses that same map's restricted scalar action, so the existing tensor-base-change invertibility instance applies in a fixed context.

Source context 7 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section CategoricalClutching
universe u v
open CategoryTheory TensorProduct
namespace ModuleCat
open scoped ChangeOfRings
universe w
universe w'

Lean source · statement

theorem invertible_extendScalars_of_invertible
    {R S : Type u} [CommRing R] [CommRing S]
    (f : R →+* S) (M : ModuleCat.{u} R) [Module.Invertible R M] :
    Module.Invertible S ((extendScalars f).obj M)

theorem

realization_baseChange_compat

L706 ↗

ModuleCat.realization_baseChange_compat

Documentation

A commuting scalar-extension diagram remains compatible with its specified realizations. The raw base-change map is interpreted with the same fixed ring-map action as the categorical scalar-extension functor.

Source context 7 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section CategoricalClutching
universe u v
open CategoryTheory TensorProduct
namespace ModuleCat
open scoped ChangeOfRings
universe w
universe w'

Lean source · full declaration

theorem realization_baseChange_compat
    {R U : Type u} [CommRing R] [CommRing U] (f : R →+* U)
    (X0 X1 : ModuleCat.{u} R) (Y0 Y1 : ModuleCat.{u} U)
    {N : Type w} [AddCommGroup N] [Module U N]
    (σ : X0 ≅ X1) (F0 : (extendScalars f).obj X0 ≅ Y0)
    (F1 : (extendScalars f).obj X1 ≅ Y1) (τ : Y0 ≅ Y1)
    (I0 : (extendScalars f).obj X0 ≃ₗ[U] N) (I1 : (extendScalars f).obj X1 ≃ₗ[U] N)
    (J0 : Y0 ≃ₗ[U] N) (J1 : Y1 ≃ₗ[U] N)
    (h0 : ∀ z : (extendScalars f).obj X0, I0 z = J0 (F0.hom z))
    (h1 : ∀ z : (extendScalars f).obj X1, I1 z = J1 (F1.hom z))
    (hτ : ∀ y : Y0, J1 (τ.hom y) = J0 y)
    (hsquare : (extendScalars f).mapIso σ ≪≫ F1 = F0 ≪≫ τ) :
    let _ : Algebra R U := f.toAlgebra
    ∀ z : U ⊗[R] X0,
      I1 (LinearEquiv.baseChange R U X0 X1 σ.toLinearEquiv z) = I0 z := by
  intro algebraRU z
  have hleft : ((extendScalars f).mapIso σ ≪≫ F1).hom z =
      F1.hom (LinearEquiv.baseChange R U X0 X1 σ.toLinearEquiv z) :=
    ModuleCat.comp_apply ((extendScalars f).mapIso σ).hom F1.hom z
  have heval : ((extendScalars f).mapIso σ ≪≫ F1).hom z = (F0 ≪≫ τ).hom z :=
    congrArg (fun e : (extendScalars f).obj X0 ≅ Y1 => e.hom z) hsquare
  have hright : (F0 ≪≫ τ).hom z = τ.hom (F0.hom z) :=
    ModuleCat.comp_apply F0.hom τ.hom z
  exact (h1 (LinearEquiv.baseChange R U X0 X1 σ.toLinearEquiv z)).trans
    ((congrArg J1 (hleft.symm.trans (heval.trans hright))).trans
      ((hτ (F0.hom z)).trans (h0 z).symm))

theorem

presented_overlap_generator

L736 ↗

ModuleCat.presented_overlap_generator

Documentation

Evaluate an overlap presentation on a chart generator, keeping both coordinate modules in their native scalar-extension carriers. The three primitive generator laws determine the realized value.

Source context 7 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section CategoricalClutching
universe u v
open CategoryTheory TensorProduct
namespace ModuleCat
open scoped ChangeOfRings
universe w
universe w'

Lean source · full declaration

theorem presented_overlap_generator
    {J C D U : Type u} [CommRing J] [CommRing C] [CommRing D] [CommRing U]
    (r : J →+* D) (s : C →+* D) (t : C →+* U)
    (P : ModuleCat.{u} J) (Q : ModuleCat.{u} C)
    {X N : Type*} [AddCommGroup N] [Module U N]
    (g : J → X → P) (η : X → Q) (ψ : D → U) (b : X → N)
    (κ :
      let _ : Algebra J D := r.toAlgebra
      let _ : Algebra C D := s.toAlgebra
      D ⊗[J] P ≃ₗ[D] D ⊗[C] Q)
    (F :
      let _ : Algebra C D := s.toAlgebra
      let _ : Algebra C U := t.toAlgebra
      D ⊗[C] Q → U ⊗[C] Q)
    (I : (extendScalars t).obj Q ≃ₗ[U] N)
    (hκ :
      let _ : Algebra J D := r.toAlgebra
      let _ : Algebra C D := s.toAlgebra
      ∀ (j : J) (x : X), κ ((1 : D) ⊗ₜ[J] g j x) = r j ⊗ₜ[C] η x)
    (hF :
      let _ : Algebra C D := s.toAlgebra
      let _ : Algebra C U := t.toAlgebra
      ∀ (d : D) (q : Q), F (d ⊗ₜ[C] q) = ψ d ⊗ₜ[C] q)
    (hI : ∀ (a : U) (x : X), I (a ⊗ₜ[C,t] η x) = a • b x) :
    let _ : Algebra J D := r.toAlgebra
    let _ : Algebra C D := s.toAlgebra
    let _ : Algebra C U := t.toAlgebra
    ∀ (j : J) (x : X), I (F (κ ((1 : D) ⊗ₜ[J] g j x))) = ψ (r j) • b x := by
  intro algebraJD algebraCD algebraCU j x
  exact (congrArg (fun z : D ⊗[C] Q => I (F z)) (hκ j x)).trans
    ((congrArg I (hF (r j) (η x))).trans (hI (ψ (r j)) x))

theorem

exists_presentation_of_linearCoordinates

L775 ↗

TensorProduct.exists_presentation_of_linearCoordinates

Documentation

Extend the actual coefficient-linear coordinates of a module and compose with its tensor presentation. The source scalar action is preserved through its supplied tower, and the elementary-tensor law is constructed together.

Source context 4 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section CategoricalClutching
universe u v
open CategoryTheory TensorProduct
namespace TensorProduct

Lean source · statement

theorem exists_presentation_of_linearCoordinates
    (k : Type*) [CommRing k] (A : Type*) [CommRing A] [Algebra k A]
    (S : Type*) [CommRing S] [Algebra k S]
    (B : Type*) [CommRing B] [Algebra B S]
    (L : Type*) [AddCommGroup L] [Module B L] [SMulCommClass B k S]
    {M : Type*} [AddCommGroup M] [Module k M] [Module S M] [IsScalarTower k S M]
    {N : Type*} [AddCommGroup N] [Module A N]
    (e : M ≃ₗ[S] S ⊗[B] L) (F : A ⊗[k] (S ⊗[B] L) ≃ₗ[A] N) :
    ∃ E : A ⊗[k] M ≃ₗ[A] N,
      ∀ (a : A) (m : M), E (a ⊗ₜ[k] m) = F (a ⊗ₜ[k] e m)

theorem

exists_realized_presentation_of_linearCoordinates

L796 ↗

TensorProduct.exists_realized_presentation_of_linearCoordinates

Documentation

Extend coefficient-linear coordinates through a tensor presentation and its A-linear realization, constructing the final equivalence and its complete generator formula without rebuilding the coordinate restriction at the call site.

Source context 4 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section CategoricalClutching
universe u v
open CategoryTheory TensorProduct
namespace TensorProduct

Lean source · statement

theorem exists_realized_presentation_of_linearCoordinates
    (k : Type*) [CommRing k] (A : Type*) [CommRing A] [Algebra k A]
    (S : Type*) [CommRing S] [Algebra k S]
    (B : Type*) [CommRing B] [Algebra B S]
    (L : Type*) [AddCommGroup L] [Module B L] [SMulCommClass B k S]
    {M : Type*} [AddCommGroup M] [Module k M] [Module S M] [IsScalarTower k S M]
    {Q : Type*} [AddCommGroup Q] [Module A Q]
    {N : Type*} [AddCommGroup N] [Module A N]
    (e : M ≃ₗ[S] S ⊗[B] L) (F : A ⊗[k] (S ⊗[B] L) ≃ₗ[A] Q)
    (G : Q ≃ₗ[A] N) :
    ∃ Z : A ⊗[k] M ≃ₗ[A] N,
      ∀ (a : A) (m : M), Z (a ⊗ₜ[k] m) = G (F (a ⊗ₜ[k] e m))

theorem

realized_presentation_coordinate_transition

L814 ↗

TensorProduct.realized_presentation_coordinate_transition

Documentation

Compose the supplied coordinate transition, tensor-presentation naturality, and realization comparison on every elementary outer tensor.

Source context 4 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section CategoricalClutching
universe u v
open CategoryTheory TensorProduct
namespace TensorProduct

Lean source · statement

theorem realized_presentation_coordinate_transition
    (k A S C : Type*) [CommRing k] [CommRing A] [CommRing S] [CommRing C]
    [Algebra k A] [Algebra k S] [Algebra C S] [SMulCommClass C k S]
    (Q0 Q1 : Type*) [AddCommGroup Q0] [Module C Q0] [AddCommGroup Q1] [Module C Q1]
    (σ : Q0 ≃ₗ[C] Q1)
    {Y0 Y1 X0 X1 N : Type*}
    (K0 : Y0 → S ⊗[C] Q0) (K1 : Y1 → S ⊗[C] Q1) (s : Y0 → Y1)
    (F0 : A ⊗[k] (S ⊗[C] Q0) → X0) (F1 : A ⊗[k] (S ⊗[C] Q1) → X1)
    (u : X0 → X1) (I0 : X0 → N) (I1 : X1 → N)
    (hK : ∀ y : Y0, K1 (s y) = LinearEquiv.baseChange C S Q0 Q1 σ (K0 y))
    (hF : ∀ (a : A) (z : S ⊗[C] Q0),
      F1 (a ⊗ₜ[k] LinearEquiv.baseChange C S Q0 Q1 σ z) = u (F0 (a ⊗ₜ[k] z)))
    (hI : ∀ x : X0, I1 (u x) = I0 x) :
    ∀ (a : A) (y : Y0),
      I1 (F1 (a ⊗ₜ[k] K1 (s y))) = I0 (F0 (a ⊗ₜ[k] K0 y))

theorem

scalar_compatibility_of_unit_presentation

L835 ↗

TensorProduct.scalar_compatibility_of_unit_presentation

Documentation

Compose the coordinate, presentation and realization scalar laws at the outer unit coefficient. No tensor action is reconstructed.

Source context 4 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section CategoricalClutching
universe u v
open CategoryTheory TensorProduct
namespace TensorProduct

Lean source · statement

theorem scalar_compatibility_of_unit_presentation
    {A S U M P Q N : Type*} [Monoid A] [MulAction A U]
    [SMul S M] [SMul S P] [SMul U Q] [SMul U N]
    (φ : S → U) (e : M → P) (F : P → Q) (G : Q → N) (Z : M → N)
    (he : ∀ (s : S) (m : M), e (s • m) = s • e m)
    (hF : ∀ (s : S) (p : P), F (s • p) = ((1 : A) • φ s) • F p)
    (hG : ∀ (u : U) (q : Q), G (u • q) = u • G q)
    (hZ : ∀ m : M, Z m = G (F (e m))) :
    ∀ (s : S) (m : M), Z (s • m) = φ s • Z m

theorem

baseChange_square_of_coordinate_generators

L855 ↗

TensorProduct.baseChange_square_of_coordinate_generators

Documentation

A square of scalar-extended linear maps is determined by unit tensors in additive coordinates. The coefficient tensors span the coordinate module, and A-linearity then supplies every outer tensor coefficient.

Source context 4 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section CategoricalClutching
universe u v
open CategoryTheory TensorProduct
namespace TensorProduct

Lean source · statement

theorem baseChange_square_of_coordinate_generators
    {k A B : Type*} [CommRing k] [CommRing A] [Algebra k A] [CommRing B]
    {S L M N P Q : Type*}
    [AddCommGroup S] [Module B S] [AddCommGroup L] [Module B L]
    [AddCommGroup M] [Module k M] [AddCommGroup N] [Module k N]
    [AddCommGroup P] [Module A P] [AddCommGroup Q] [Module A Q]
    (e : M ≃+ S ⊗[B] L) (r : M →ₗ[k] N) (r' : P →ₗ[A] Q)
    (Z : A ⊗[k] M ≃ₗ[A] P) (W : A ⊗[k] N ≃ₗ[A] Q)
    (hgen : ∀ (s : S) (l : L),
      W ((1 : A) ⊗ₜ[k] r (e.symm (s ⊗ₜ[B] l))) =
        r' (Z ((1 : A) ⊗ₜ[k] e.symm (s ⊗ₜ[B] l)))) :
    ∀ x : A ⊗[k] M, W (r.baseChange A x) = r' (Z x)

theorem

baseChange_square_of_presented_coordinates

L890 ↗

TensorProduct.baseChange_square_of_presented_coordinates

Documentation

Transport a coordinate-generator square through supplied endpoint presentations and a source coordinate law. The overlap coordinate carrier requires no additional algebraic structure.

Source context 4 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section CategoricalClutching
universe u v
open CategoryTheory TensorProduct
namespace TensorProduct

Lean source · statement

theorem baseChange_square_of_presented_coordinates
    {k A B : Type*} [CommRing k] [CommRing A] [Algebra k A] [baseRing : CommRing B]
    {S L M N P Q Y : Type*}
    [sourceAdd : AddCommGroup S] [sourceModule : Module B S] [AddCommGroup L] [Module B L]
    [AddCommGroup M] [Module k M] [AddCommGroup N] [Module k N]
    [AddCommGroup P] [Module A P] [AddCommGroup Q] [Module A Q]
    (e : M ≃+ S ⊗[B] L) (r : M →ₗ[k] N) (r' : P →ₗ[A] Q)
    (Z : A ⊗[k] M ≃ₗ[A] P) (W : A ⊗[k] N ≃ₗ[A] Q)
    (d : N → Y) (u : S ⊗[B] L → Y) (F : Y → Q) (G : S ⊗[B] L → P)
    (hW : ∀ n : N, W ((1 : A) ⊗ₜ[k] n) = F (d n))
    (hZ : ∀ m : M, Z ((1 : A) ⊗ₜ[k] m) = G (e m))
    (hd : ∀ m : M, d (r m) = u (e m))
    (hgen : ∀ (s : S) (l : L), F (u (s ⊗ₜ[B] l)) = r' (G (s ⊗ₜ[B] l))) :
    ∀ x : A ⊗[k] M, W (r.baseChange A x) = r' (Z x)

theorem

baseChange_square_of_native_chart_coordinates

L910 ↗

TensorProduct.baseChange_square_of_native_chart_coordinates

Documentation

Prove the base-change restriction square from the actual chart and overlap coordinate presentations and their generated pure-tensor laws.

Source context 4 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section CategoricalClutching
universe u v
open CategoryTheory TensorProduct
namespace TensorProduct

Lean source · statement

theorem baseChange_square_of_native_chart_coordinates
    {k A B J C D U V : Type u}
    [CommRing k] [CommRing A] [CommRing B] [CommRing J]
    [CommRing C] [CommRing D] [CommRing U] [CommRing V]
    [Algebra k A]
    [algBJ : Algebra B J] [algBU : Algebra B U]
    [algJD : Algebra J D] [algCD : Algebra C D] [algCV : Algebra C V]
    [Algebra A U] [Algebra A V]
    {X Q : Type u} [AddCommGroup X] [Module B X]
    [AddCommGroup Q] [Module C Q]
    {M N T0 T1 : Type v}
    [AddCommGroup M] [Module k M] [AddCommGroup N] [Module k N]
    [AddCommGroup T0] [Module A T0] [Module U T0]
    [AddCommGroup T1] [Module A T1] [Module V T1]
    (e : M ≃+ J ⊗[B] X)
    (sourceRestrict : M →ₗ[k] N) (targetRestrict : T0 →ₗ[A] T1)
    (Z : A ⊗[k] M ≃ₗ[A] T0) (W : A ⊗[k] N ≃ₗ[A] T1)
    (d : N → D ⊗[J] (J ⊗[B] X))
    (kappa : D ⊗[J] (J ⊗[B] X) ≃ₗ[D] D ⊗[C] Q)
    (newI : U ⊗[B] X ≃ₗ[U] T0) (Xi : V ⊗[C] Q ≃ₗ[V] T1)
    (G : J ⊗[B] X → U ⊗[B] X) (H : D ⊗[C] Q → V ⊗[C] Q)
    (eta : X → Q) (rho : J →+* D) (rhoU : U →+* V)
    (phi : J →+* U) (psi : D →+* V)
    (hKappa : ∀ (j : J) (x : X),
      kappa ((1 : D) ⊗ₜ[J] (j ⊗ₜ[B] x)) = rho j ⊗ₜ[C] eta x)
    (hPresent : ∀ (d : D) (q : Q),
      H (d ⊗ₜ[C] q) = ((1 : A) • psi d) ⊗ₜ[C] q)
    (hXi : ∀ (a : V) (x : X),
      Xi (a ⊗ₜ[C] eta x) = a • targetRestrict (newI ((1 : U) ⊗ₜ[B] x)))
    (hChart : ∀ (j : J) (x : X),
      G (j ⊗ₜ[B] x) = ((1 : A) • phi j) ⊗ₜ[B] x)
    (hW : ∀ n : N, W ((1 : A) ⊗ₜ[k] n) = Xi (H (kappa (d n))))
    (hZ : ∀ m : M, Z ((1 : A) ⊗ₜ[k] m) = newI (G (e m)))
    (hC : ∀ m : M, d (sourceRestrict m) = (1 : D) ⊗ₜ[J] e m)
    (hRestrict : ∀ (a : U) (m : T0),
      targetRestrict (a • m) = rhoU a • targetRestrict m)
    (hPhiRho : ∀ j : J, psi (rho j) = rhoU (phi j)) :
    ∀ x : A ⊗[k] M,
      W (sourceRestrict.baseChange A x) = targetRestrict (Z x)

def

actR

L1029 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.actR

Documentation

The action of a chart scalar on the module after base extension.

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section ActR
variable (k : Type*) [CommRing k] {Γ : Type*} [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

def actR (s : Γ) : A' ⊗[k] P →ₗ[A'] A' ⊗[k] P

theorem

actR_tmul

L1032 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.actR_tmul

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section ActR
variable (k : Type*) [CommRing k] {Γ : Type*} [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

theorem actR_tmul (s : Γ) (a : A') (m : P) :
    actR k A' s (a ⊗ₜ[k] m) = a ⊗ₜ[k] (s • m)

theorem

actR_one

L1036 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.actR_one

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section ActR
variable (k : Type*) [CommRing k] {Γ : Type*} [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

theorem actR_one : actR k A' (1 : Γ) = (LinearMap.id : A' ⊗[k] P →ₗ[A'] A' ⊗[k] P)

theorem

actR_mul

L1041 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.actR_mul

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section ActR
variable (k : Type*) [CommRing k] {Γ : Type*} [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

theorem actR_mul (s t : Γ) : actR k A' (P := P) (s * t) = actR k A' s ∘ₗ actR k A' t

theorem

actR_add

L1047 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.actR_add

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section ActR
variable (k : Type*) [CommRing k] {Γ : Type*} [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

theorem actR_add (s t : Γ) : actR k A' (P := P) (s + t) = actR k A' s + actR k A' t

theorem

rTensor_actR

L1053 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.rTensor_actR

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section ActR
variable (k : Type*) [CommRing k] {Γ : Type*} [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

theorem rTensor_actR {A'' : Type*} [CommRing A''] [Algebra k A''] (h : A' →ₐ[k] A'') (s : Γ)
    (x : A' ⊗[k] P) :
    h.toLinearMap.rTensor P (actR k A' s x) = actR k A'' s (h.toLinearMap.rTensor P x)

theorem

rTensor_smul'

L1061 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.rTensor_smul'

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section ActR
variable (k : Type*) [CommRing k] {Γ : Type*} [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

theorem rTensor_smul' {A'' : Type*} [CommRing A''] [Algebra k A''] (h : A' →ₐ[k] A'')
    {Q : Type*} [AddCommGroup Q] [Module k Q] (a : A') (x : A' ⊗[k] Q) :
    h.toLinearMap.rTensor Q (a • x) = h a • h.toLinearMap.rTensor Q x

theorem

tmul_eq_smul_one_tmul

L1070 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.tmul_eq_smul_one_tmul

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section ActR
variable (k : Type*) [CommRing k] {Γ : Type*} [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

theorem tmul_eq_smul_one_tmul {Q : Type*} [AddCommGroup Q] [Module k Q] (a : A') (m : Q) :
    a ⊗ₜ[k] m = a • ((1 : A') ⊗ₜ[k] m)

def

Intertwines

L1087 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.Intertwines

Documentation

Compatibility of a linear map with the indicated actions of chart scalars.

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Intertwines
variable {k : Type*} [CommRing k] {Γ Γ' Γ'' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ']
    [Algebra k Γ']
  [CommRing Γ''] [Algebra k Γ'']
  {A' : Type*} [CommRing A'] [Algebra k A']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module Γ'' N] [Module k N] [IsScalarTower k Γ'' N]

Lean source · statement

def Intertwines (σ : Γ → Γ') (f : A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q) : Prop

theorem

comp

L1090 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.Intertwines.comp

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Intertwines
variable {k : Type*} [CommRing k] {Γ Γ' Γ'' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ']
    [Algebra k Γ']
  [CommRing Γ''] [Algebra k Γ'']
  {A' : Type*} [CommRing A'] [Algebra k A']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module Γ'' N] [Module k N] [IsScalarTower k Γ'' N]

Lean source · statement

theorem Intertwines.comp {σ : Γ → Γ'} {σ' : Γ' → Γ''} {f : A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q}
    {g : A' ⊗[k] Q →ₗ[A'] A' ⊗[k] N} (hf : Intertwines σ f) (hg : Intertwines σ' g) :
    Intertwines (σ' ∘ σ) (g ∘ₗ f)

theorem

id

L1096 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.Intertwines.id

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Intertwines
variable {k : Type*} [CommRing k] {Γ Γ' Γ'' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ']
    [Algebra k Γ']
  [CommRing Γ''] [Algebra k Γ'']
  {A' : Type*} [CommRing A'] [Algebra k A']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module Γ'' N] [Module k N] [IsScalarTower k Γ'' N]

Lean source · statement

theorem Intertwines.id : Intertwines (_root_.id : Γ → Γ) (LinearMap.id : A' ⊗[k] P →ₗ[A'] A' ⊗[k]
    P)

theorem

symm

L1100 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.Intertwines.symm

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Intertwines
variable {k : Type*} [CommRing k] {Γ Γ' Γ'' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ']
    [Algebra k Γ']
  [CommRing Γ''] [Algebra k Γ'']
  {A' : Type*} [CommRing A'] [Algebra k A']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module Γ'' N] [Module k N] [IsScalarTower k Γ'' N]

Lean source · statement

theorem Intertwines.symm {e : A' ⊗[k] P ≃ₗ[A'] A' ⊗[k] Q} {σ : Γ ≃ Γ'}
    (he : Intertwines σ (e : A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q)) :
    Intertwines σ.symm (e.symm : A' ⊗[k] Q →ₗ[A'] A' ⊗[k] P)

theorem

baseChange

L1110 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.Intertwines.baseChange

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Intertwines
variable {k : Type*} [CommRing k] {Γ Γ' Γ'' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ']
    [Algebra k Γ']
  [CommRing Γ''] [Algebra k Γ'']
  {A' : Type*} [CommRing A'] [Algebra k A']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module Γ'' N] [Module k N] [IsScalarTower k Γ'' N]

Lean source · statement

theorem Intertwines.baseChange (σ : Γ → Γ') (r : P →ₗ[k] Q) (hr : ∀ (s : Γ) (m : P), r (s • m) = σ
    s • r m) :
    Intertwines σ (r.baseChange A')

theorem

linearMap_ext_of_span

L1126 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.linearMap_ext_of_span

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Ext
variable {k : Type*} [CommRing k] {Γ : Type*} [CommRing Γ] [Algebra k Γ]
  {A' : Type*} [CommRing A'] [Algebra k A']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {N : Type*} [AddCommGroup N] [Module A' N]

Lean source · statement

theorem linearMap_ext_of_span (F G : A' ⊗[k] P →ₗ[A'] N) (T : Γ → N → N)
    (hF : ∀ (s : Γ) (x : A' ⊗[k] P), F (actR k A' s x) = T s (F x))
    (hG : ∀ (s : Γ) (x : A' ⊗[k] P), G (actR k A' s x) = T s (G x))
    (gens : Set P) (hgens : Submodule.span Γ gens = ⊤)
    (h : ∀ m ∈ gens, F ((1 : A') ⊗ₜ[k] m) = G ((1 : A') ⊗ₜ[k] m)) : F = G

def

push

L1162 ↗

AlgebraicGeometry.RelPicard.LFP.push

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Push
variable {k : Type*} [CommRing k] {Γ Γ' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ'] [Algebra
    k Γ']
  {A' A'' : Type*} [CommRing A'] [Algebra k A'] [CommRing A''] [Algebra k A'']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module k N]

Lean source · statement

def _root_.AlgebraicGeometry.RelPicard.LFP.push (h : A' →ₐ[k] A'') (f : A'
    ⊗[k] P →ₗ[A'] A' ⊗[k] Q) :
    A'' ⊗[k] P →ₗ[A''] A'' ⊗[k] Q

theorem

push_tmul

L1169 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.push_tmul

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Push
variable {k : Type*} [CommRing k] {Γ Γ' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ'] [Algebra
    k Γ']
  {A' A'' : Type*} [CommRing A'] [Algebra k A'] [CommRing A''] [Algebra k A'']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module k N]
open _root_.AlgebraicGeometry.RelPicard.LFP

Lean source · statement

theorem push_tmul (h : A' →ₐ[k] A'') (f : A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q) (a : A'') (m
    : P) :
    push h f (a ⊗ₜ[k] m) = a • h.toLinearMap.rTensor Q (f ((1 : A') ⊗ₜ[k] m))

theorem

push_rTensor

L1175 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.push_rTensor

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Push
variable {k : Type*} [CommRing k] {Γ Γ' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ'] [Algebra
    k Γ']
  {A' A'' : Type*} [CommRing A'] [Algebra k A'] [CommRing A''] [Algebra k A'']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module k N]
open _root_.AlgebraicGeometry.RelPicard.LFP

Lean source · statement

theorem push_rTensor (h : A' →ₐ[k] A'') (f : A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q) (x : A' ⊗[k] P) :
    push h f (h.toLinearMap.rTensor P x) = h.toLinearMap.rTensor Q (f x)

theorem

push_id

L1183 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.push_id

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Push
variable {k : Type*} [CommRing k] {Γ Γ' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ'] [Algebra
    k Γ']
  {A' A'' : Type*} [CommRing A'] [Algebra k A'] [CommRing A''] [Algebra k A'']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module k N]
open _root_.AlgebraicGeometry.RelPicard.LFP

Lean source · statement

theorem push_id (h : A' →ₐ[k] A'') :
    push h (LinearMap.id : A' ⊗[k] P →ₗ[A'] A' ⊗[k] P) = LinearMap.id

theorem

push_comp

L1191 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.push_comp

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Push
variable {k : Type*} [CommRing k] {Γ Γ' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ'] [Algebra
    k Γ']
  {A' A'' : Type*} [CommRing A'] [Algebra k A'] [CommRing A''] [Algebra k A'']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module k N]
open _root_.AlgebraicGeometry.RelPicard.LFP

Lean source · statement

theorem push_comp (h : A' →ₐ[k] A'') (f : A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q)
    (g : A' ⊗[k] Q →ₗ[A'] A' ⊗[k] N) :
    push h (g ∘ₗ f) = push h g ∘ₗ push h f

theorem

push

L1199 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.Intertwines.push

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Push
variable {k : Type*} [CommRing k] {Γ Γ' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ'] [Algebra
    k Γ']
  {A' A'' : Type*} [CommRing A'] [Algebra k A'] [CommRing A''] [Algebra k A'']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module k N]
open _root_.AlgebraicGeometry.RelPicard.LFP

Lean source · statement

theorem Intertwines.push {σ : Γ → Γ'} (h : A' →ₐ[k] A'') {f : A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q}
    (hf : Intertwines σ f) : Intertwines σ (push h f)

theorem

rTensor_baseChange

L1206 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.rTensor_baseChange

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Push
variable {k : Type*} [CommRing k] {Γ Γ' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ'] [Algebra
    k Γ']
  {A' A'' : Type*} [CommRing A'] [Algebra k A'] [CommRing A''] [Algebra k A'']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module k N]
open _root_.AlgebraicGeometry.RelPicard.LFP

Lean source · statement

theorem rTensor_baseChange (h : A' →ₐ[k] A'') {M M' : Type*} [AddCommGroup M] [Module k M]
    [AddCommGroup M']
    [Module k M'] (r : M →ₗ[k] M') (x : A' ⊗[k] M) :
    h.toLinearMap.rTensor M' (r.baseChange A' x) = r.baseChange A'' (h.toLinearMap.rTensor M x)

theorem

push_square

L1217 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.push_square

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Push
variable {k : Type*} [CommRing k] {Γ Γ' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ'] [Algebra
    k Γ']
  {A' A'' : Type*} [CommRing A'] [Algebra k A'] [CommRing A''] [Algebra k A'']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module k N]
open _root_.AlgebraicGeometry.RelPicard.LFP

Lean source · statement

theorem push_square (h : A' →ₐ[k] A'') {P' : Type*} [AddCommGroup P'] [Module k P'] {Q' : Type*}
    [AddCommGroup Q']
    [Module k Q'] (f : A' ⊗[k] P →ₗ[A'] A' ⊗[k] P') (g : A' ⊗[k] Q →ₗ[A'] A' ⊗[k] Q')
    (r : P →ₗ[k] Q) (r' : P' →ₗ[k] Q') (hsq : ∀ x, g (r.baseChange A' x) = r'.baseChange A' (f x))
    (y : A'' ⊗[k] P) : push h g (r.baseChange A'' y) = r'.baseChange A'' (push h f y)

theorem

mem_span_range_of_algEquiv

L1228 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.mem_span_range_of_algEquiv

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Push
variable {k : Type*} [CommRing k] {Γ Γ' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ'] [Algebra
    k Γ']
  {A' A'' : Type*} [CommRing A'] [Algebra k A'] [CommRing A''] [Algebra k A'']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module k N]
open _root_.AlgebraicGeometry.RelPicard.LFP

Lean source · statement

theorem mem_span_range_of_algEquiv {Γ'' : Type*} [CommRing Γ''] [Algebra A' Γ'']
    (e : A' ⊗[k] Γ ≃ₐ[A'] Γ'') (φ : Γ → Γ'') (he : ∀ s, e ((1 : A') ⊗ₜ[k] s) = φ s) (y : Γ'') :
    y ∈ Submodule.span A' (Set.range φ)

def

pushEquiv

L1239 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.pushEquiv

Documentation

A tensor product equivalence extended along a homomorphism of base algebras.

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Push
variable {k : Type*} [CommRing k] {Γ Γ' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ'] [Algebra
    k Γ']
  {A' A'' : Type*} [CommRing A'] [Algebra k A'] [CommRing A''] [Algebra k A'']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module k N]
open _root_.AlgebraicGeometry.RelPicard.LFP

Lean source · statement

def pushEquiv (h : A' →ₐ[k] A'') (e : A' ⊗[k] P ≃ₗ[A'] A' ⊗[k] Q) :
    A'' ⊗[k] P ≃ₗ[A''] A'' ⊗[k] Q

theorem

pushEquiv_apply

L1246 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.pushEquiv_apply

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Push
variable {k : Type*} [CommRing k] {Γ Γ' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ'] [Algebra
    k Γ']
  {A' A'' : Type*} [CommRing A'] [Algebra k A'] [CommRing A''] [Algebra k A'']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module k N]
open _root_.AlgebraicGeometry.RelPicard.LFP

Lean source · statement

theorem pushEquiv_apply (h : A' →ₐ[k] A'') (e : A' ⊗[k] P ≃ₗ[A'] A' ⊗[k] Q) (x :
    A'' ⊗[k] P) :
    pushEquiv h e x = push h (e : A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q) x

theorem

pushEquiv_symm_apply

L1250 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.pushEquiv_symm_apply

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Push
variable {k : Type*} [CommRing k] {Γ Γ' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ'] [Algebra
    k Γ']
  {A' A'' : Type*} [CommRing A'] [Algebra k A'] [CommRing A''] [Algebra k A'']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module k N]
open _root_.AlgebraicGeometry.RelPicard.LFP

Lean source · statement

theorem pushEquiv_symm_apply (h : A' →ₐ[k] A'') (e : A' ⊗[k] P ≃ₗ[A'] A' ⊗[k] Q)
    (x : A'' ⊗[k] Q) :
    (pushEquiv h e).symm x = push h (e.symm : A' ⊗[k] Q →ₗ[A'] A' ⊗[k] P) x

theorem

coe_pushEquiv

L1254 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.coe_pushEquiv

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Push
variable {k : Type*} [CommRing k] {Γ Γ' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ'] [Algebra
    k Γ']
  {A' A'' : Type*} [CommRing A'] [Algebra k A'] [CommRing A''] [Algebra k A'']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module k N]
open _root_.AlgebraicGeometry.RelPicard.LFP

Lean source · statement

theorem coe_pushEquiv (h : A' →ₐ[k] A'') (e : A' ⊗[k] P ≃ₗ[A'] A' ⊗[k] Q) :
    (pushEquiv h e : A'' ⊗[k] P →ₗ[A''] A'' ⊗[k] Q) = push h (e : A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q)

def

orient

L1266 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.orient

Documentation

The equivalence reordering the tensor factors of a base extended module.

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

def orient : (Γ ⊗[k] A') ⊗[Γ] P ≃ₗ[k] A' ⊗[k] P

theorem

orient_tmul

L1271 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.orient_tmul

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

theorem orient_tmul (γ : Γ) (a : A') (m : P) :
    orient k Γ A' P ((γ ⊗ₜ[k] a) ⊗ₜ[Γ] m) = a ⊗ₜ[k] (γ • m)

theorem

orient_symm_tmul

L1280 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.orient_symm_tmul

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

theorem orient_symm_tmul (a : A') (m : P) :
    (orient k Γ A' P).symm (a ⊗ₜ[k] m) = ((1 : Γ) ⊗ₜ[k] a) ⊗ₜ[Γ] m

theorem

orient_smul

L1284 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.orient_smul

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

theorem orient_smul (γ : Γ) (a : A') (x : (Γ ⊗[k] A') ⊗[Γ] P) :
    orient k Γ A' P ((γ ⊗ₜ[k] a) • x) = a • actR k A' γ (orient k Γ A' P x)

theorem

orient_smul_left

L1295 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.orient_smul_left

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

theorem orient_smul_left (a : A') (x : (Γ ⊗[k] A') ⊗[Γ] P) :
    orient k Γ A' P (((1 : Γ) ⊗ₜ[k] a) • x) = a • orient k Γ A' P x

theorem

orient_smul_right

L1299 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.orient_smul_right

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

theorem orient_smul_right (γ : Γ) (x : (Γ ⊗[k] A') ⊗[Γ] P) :
    orient k Γ A' P ((γ ⊗ₜ[k] (1 : A')) • x) = actR k A' γ (orient k Γ A' P x)

theorem

orient_symm_smul

L1303 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.orient_symm_smul

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

theorem orient_symm_smul (a : A') (y : A' ⊗[k] P) :
    (orient k Γ A' P).symm (a • y) = ((1 : Γ) ⊗ₜ[k] a) • (orient k Γ A' P).symm y

theorem

orient_symm_actR

L1307 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.orient_symm_actR

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

theorem orient_symm_actR (γ : Γ) (y : A' ⊗[k] P) :
    (orient k Γ A' P).symm (actR k A' γ y) = (γ ⊗ₜ[k] (1 : A')) • (orient k Γ A' P).symm y

theorem

orient_rTensor_map

L1311 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.orient_rTensor_map

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

theorem orient_rTensor_map {A'' : Type*} [CommRing A''] [Algebra k A''] (h : A' →ₐ[k] A'')
    (x : (Γ ⊗[k] A') ⊗[Γ] P) :
    orient k Γ A'' P ((Algebra.TensorProduct.map (AlgHom.id Γ Γ) h).toLinearMap.rTensor P x) =
      h.toLinearMap.rTensor P (orient k Γ A' P x)

def

fromOrient

L1328 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.fromOrient

Documentation

A linear map transported from the reordered tensor presentation.

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
variable {k Γ A' P}
variable {Q : Type*} [AddCommGroup Q] [Module Γ Q] [Module k Q] [IsScalarTower k Γ Q]

Lean source · full declaration

def fromOrient (f : (Γ ⊗[k] A') ⊗[Γ] P →ₗ[Γ ⊗[k] A'] (Γ ⊗[k] A') ⊗[Γ] Q) :
    A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q where
  toFun y := orient k Γ A' Q (f ((orient k Γ A' P).symm y))
  map_add' y y' := by simp only [map_add]
  map_smul' a y := by
    rw [orient_symm_smul, map_smul, orient_smul_left, RingHom.id_apply]

theorem

fromOrient_apply

L1335 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.fromOrient_apply

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
variable {k Γ A' P}
variable {Q : Type*} [AddCommGroup Q] [Module Γ Q] [Module k Q] [IsScalarTower k Γ Q]

Lean source · statement

theorem fromOrient_apply (f : (Γ ⊗[k] A') ⊗[Γ] P →ₗ[Γ ⊗[k] A'] (Γ ⊗[k] A') ⊗[Γ] Q) (y : A' ⊗[k] P) :
    fromOrient f y = orient k Γ A' Q (f ((orient k Γ A' P).symm y))

theorem

fromOrient_orient

L1338 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.fromOrient_orient

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
variable {k Γ A' P}
variable {Q : Type*} [AddCommGroup Q] [Module Γ Q] [Module k Q] [IsScalarTower k Γ Q]

Lean source · statement

theorem fromOrient_orient (f : (Γ ⊗[k] A') ⊗[Γ] P →ₗ[Γ ⊗[k] A'] (Γ ⊗[k] A') ⊗[Γ] Q)
    (x : (Γ ⊗[k] A') ⊗[Γ] P) :
    fromOrient f (orient k Γ A' P x) = orient k Γ A' Q (f x)

theorem

intertwines_fromOrient

L1343 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.intertwines_fromOrient

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
variable {k Γ A' P}
variable {Q : Type*} [AddCommGroup Q] [Module Γ Q] [Module k Q] [IsScalarTower k Γ Q]

Lean source · statement

theorem intertwines_fromOrient (f : (Γ ⊗[k] A') ⊗[Γ] P →ₗ[Γ ⊗[k] A'] (Γ ⊗[k] A') ⊗[Γ] Q) :
    Intertwines (_root_.id : Γ → Γ) (fromOrient f)

def

toOrient

L1348 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.toOrient

Documentation

A linear map compatible with chart scalars transported to the reordered tensor presentation.

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
variable {k Γ A' P}
variable {Q : Type*} [AddCommGroup Q] [Module Γ Q] [Module k Q] [IsScalarTower k Γ Q]

Lean source · full declaration

def toOrient (f : A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q) (hf : Intertwines (_root_.id : Γ → Γ)
    f) :
    (Γ ⊗[k] A') ⊗[Γ] P →ₗ[Γ ⊗[k] A'] (Γ ⊗[k] A') ⊗[Γ] Q where
  toFun x := (orient k Γ A' Q).symm (f (orient k Γ A' P x))
  map_add' x x' := by simp only [map_add]
  map_smul' b x := by
    rw [RingHom.id_apply]
    induction b using TensorProduct.inductionOn with
    | tmul γ a =>
      rw [orient_smul, map_smul, hf, LinearEquiv.symm_apply_eq, orient_smul,
          LinearEquiv.apply_symm_apply]
      rfl
    | add b b' hb hb' => simp only [add_smul, map_add, hb, hb']

theorem

toOrient_apply

L1362 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.toOrient_apply

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
variable {k Γ A' P}
variable {Q : Type*} [AddCommGroup Q] [Module Γ Q] [Module k Q] [IsScalarTower k Γ Q]

Lean source · statement

theorem toOrient_apply (f : A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q) (hf : Intertwines (_root_.id : Γ → Γ) f)
    (x : (Γ ⊗[k] A') ⊗[Γ] P) :
    toOrient f hf x = (orient k Γ A' Q).symm (f (orient k Γ A' P x))

theorem

fromOrient_toOrient

L1366 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.fromOrient_toOrient

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
variable {k Γ A' P}
variable {Q : Type*} [AddCommGroup Q] [Module Γ Q] [Module k Q] [IsScalarTower k Γ Q]

Lean source · statement

theorem fromOrient_toOrient (f : A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q) (hf : Intertwines (_root_.id : Γ → Γ)
    f) :
    fromOrient (toOrient f hf) = f

theorem

toOrient_fromOrient

L1372 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.toOrient_fromOrient

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
variable {k Γ A' P}
variable {Q : Type*} [AddCommGroup Q] [Module Γ Q] [Module k Q] [IsScalarTower k Γ Q]

Lean source · statement

theorem toOrient_fromOrient (f : (Γ ⊗[k] A') ⊗[Γ] P →ₗ[Γ ⊗[k] A'] (Γ ⊗[k] A') ⊗[Γ] Q) :
    toOrient (fromOrient f) (intertwines_fromOrient f) = f

def

fromOrientEquiv

L1378 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.fromOrientEquiv

Documentation

An equivalence transported from the reordered tensor presentation.

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
variable {k Γ A' P}
variable {Q : Type*} [AddCommGroup Q] [Module Γ Q] [Module k Q] [IsScalarTower k Γ Q]

Lean source · statement

def fromOrientEquiv (e : (Γ ⊗[k] A') ⊗[Γ] P ≃ₗ[Γ ⊗[k] A'] (Γ ⊗[k] A') ⊗[Γ] Q) :
    A' ⊗[k] P ≃ₗ[A'] A' ⊗[k] Q

theorem

coe_fromOrientEquiv

L1397 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.coe_fromOrientEquiv

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
variable {k Γ A' P}
variable {Q : Type*} [AddCommGroup Q] [Module Γ Q] [Module k Q] [IsScalarTower k Γ Q]

Lean source · statement

theorem coe_fromOrientEquiv (e : (Γ ⊗[k] A') ⊗[Γ] P ≃ₗ[Γ ⊗[k] A'] (Γ ⊗[k] A') ⊗[Γ]
    Q) :
    (fromOrientEquiv e : A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q) =
      fromOrient (e : (Γ ⊗[k] A') ⊗[Γ] P →ₗ[Γ ⊗[k] A'] (Γ ⊗[k] A') ⊗[Γ] Q)

theorem

fromOrientEquiv_apply

L1402 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.fromOrientEquiv_apply

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
variable {k Γ A' P}
variable {Q : Type*} [AddCommGroup Q] [Module Γ Q] [Module k Q] [IsScalarTower k Γ Q]

Lean source · statement

theorem fromOrientEquiv_apply (e : (Γ ⊗[k] A') ⊗[Γ] P ≃ₗ[Γ ⊗[k] A'] (Γ ⊗[k] A') ⊗[Γ] Q) (y : A'
    ⊗[k] P) :
    fromOrientEquiv e y = orient k Γ A' Q (e ((orient k Γ A' P).symm y))

def

toOrientEquiv

L1408 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.toOrientEquiv

Documentation

An equivalence compatible with chart scalars transported to the reordered tensor presentation.

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
variable {k Γ A' P}
variable {Q : Type*} [AddCommGroup Q] [Module Γ Q] [Module k Q] [IsScalarTower k Γ Q]

Lean source · statement

def toOrientEquiv (e : A' ⊗[k] P ≃ₗ[A'] A' ⊗[k] Q)
    (he : Intertwines (_root_.id : Γ → Γ) (e : A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q)) :
    (Γ ⊗[k] A') ⊗[Γ] P ≃ₗ[Γ ⊗[k] A'] (Γ ⊗[k] A') ⊗[Γ] Q

theorem

toOrientEquiv_apply

L1429 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.toOrientEquiv_apply

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
variable {k Γ A' P}
variable {Q : Type*} [AddCommGroup Q] [Module Γ Q] [Module k Q] [IsScalarTower k Γ Q]

Lean source · statement

theorem toOrientEquiv_apply (e : A' ⊗[k] P ≃ₗ[A'] A' ⊗[k] Q)
    (he : Intertwines (_root_.id : Γ → Γ) (e : A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q)) (x : (Γ ⊗[k] A') ⊗[Γ]
        P) :
    toOrientEquiv e he x = (orient k Γ A' Q).symm (e (orient k Γ A' P x))

theorem

smul_comm_of_tower

L1446 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.smul_comm_of_tower

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section OneChart
variable {k : Type*} [CommRing k] {A' : Type*} [CommRing A'] [Algebra k A']
  {Γ : Type*} [CommRing Γ] [Algebra k Γ] {Γ' : Type*} [CommRing Γ'] [Algebra A' Γ']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {P' : Type*} [AddCommGroup P'] [Module Γ P'] [Module k P'] [IsScalarTower k Γ P']
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module A' Q] [IsScalarTower A' Γ' Q]
  {Q' : Type*} [AddCommGroup Q'] [Module Γ' Q'] [Module A' Q'] [IsScalarTower A' Γ' Q']
omit [Algebra k A'] in

Lean source · statement

theorem smul_comm_of_tower (a : A') (t : Γ') (q : Q) : a • t • q = t • a • q

theorem

apply_actR_of_semilinear

L1449 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.apply_actR_of_semilinear

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section OneChart
variable {k : Type*} [CommRing k] {A' : Type*} [CommRing A'] [Algebra k A']
  {Γ : Type*} [CommRing Γ] [Algebra k Γ] {Γ' : Type*} [CommRing Γ'] [Algebra A' Γ']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {P' : Type*} [AddCommGroup P'] [Module Γ P'] [Module k P'] [IsScalarTower k Γ P']
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module A' Q] [IsScalarTower A' Γ' Q]
  {Q' : Type*} [AddCommGroup Q'] [Module Γ' Q'] [Module A' Q'] [IsScalarTower A' Γ' Q']

Lean source · statement

theorem apply_actR_of_semilinear (φ : Γ → Γ') (β : A' ⊗[k] P ≃ₗ[A'] Q)
    (hs : ∀ (s : Γ) (m : P), β ((1 : A') ⊗ₜ[k] (s • m)) = φ s • β ((1 : A') ⊗ₜ[k] m))
    (s : Γ) (x : A' ⊗[k] P) : β (actR k A' s x) = φ s • β x

theorem

symm_apply_smul_of_semilinear

L1459 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.symm_apply_smul_of_semilinear

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section OneChart
variable {k : Type*} [CommRing k] {A' : Type*} [CommRing A'] [Algebra k A']
  {Γ : Type*} [CommRing Γ] [Algebra k Γ] {Γ' : Type*} [CommRing Γ'] [Algebra A' Γ']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {P' : Type*} [AddCommGroup P'] [Module Γ P'] [Module k P'] [IsScalarTower k Γ P']
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module A' Q] [IsScalarTower A' Γ' Q]
  {Q' : Type*} [AddCommGroup Q'] [Module Γ' Q'] [Module A' Q'] [IsScalarTower A' Γ' Q']

Lean source · statement

theorem symm_apply_smul_of_semilinear (φ : Γ → Γ') (β : A' ⊗[k] P ≃ₗ[A'] Q)
    (hs : ∀ (s : Γ) (m : P), β ((1 : A') ⊗ₜ[k] (s • m)) = φ s • β ((1 : A') ⊗ₜ[k] m))
    (s : Γ) (q : Q) : β.symm (φ s • q) = actR k A' s (β.symm q)

def

conjIn

L1466 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.conjIn

Documentation

The equivalence between tensor presentations induced by the two chart comparison equivalences.

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section OneChart
variable {k : Type*} [CommRing k] {A' : Type*} [CommRing A'] [Algebra k A']
  {Γ : Type*} [CommRing Γ] [Algebra k Γ] {Γ' : Type*} [CommRing Γ'] [Algebra A' Γ']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {P' : Type*} [AddCommGroup P'] [Module Γ P'] [Module k P'] [IsScalarTower k Γ P']
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module A' Q] [IsScalarTower A' Γ' Q]
  {Q' : Type*} [AddCommGroup Q'] [Module Γ' Q'] [Module A' Q'] [IsScalarTower A' Γ' Q']

Lean source · statement

def conjIn (β : A' ⊗[k] P ≃ₗ[A'] Q) (β' : A' ⊗[k] P' ≃ₗ[A'] Q') (ψ : Q ≃ₗ[Γ'] Q') :
    A' ⊗[k] P ≃ₗ[A'] A' ⊗[k] P'

theorem

conjIn_apply

L1470 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.conjIn_apply

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section OneChart
variable {k : Type*} [CommRing k] {A' : Type*} [CommRing A'] [Algebra k A']
  {Γ : Type*} [CommRing Γ] [Algebra k Γ] {Γ' : Type*} [CommRing Γ'] [Algebra A' Γ']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {P' : Type*} [AddCommGroup P'] [Module Γ P'] [Module k P'] [IsScalarTower k Γ P']
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module A' Q] [IsScalarTower A' Γ' Q]
  {Q' : Type*} [AddCommGroup Q'] [Module Γ' Q'] [Module A' Q'] [IsScalarTower A' Γ' Q']

Lean source · statement

theorem conjIn_apply (β : A' ⊗[k] P ≃ₗ[A'] Q) (β' : A' ⊗[k] P' ≃ₗ[A'] Q') (ψ : Q ≃ₗ[Γ'] Q')
    (x : A' ⊗[k] P) : conjIn β β' ψ x = β'.symm (ψ (β x))

theorem

intertwines_conjIn

L1473 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.intertwines_conjIn

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section OneChart
variable {k : Type*} [CommRing k] {A' : Type*} [CommRing A'] [Algebra k A']
  {Γ : Type*} [CommRing Γ] [Algebra k Γ] {Γ' : Type*} [CommRing Γ'] [Algebra A' Γ']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {P' : Type*} [AddCommGroup P'] [Module Γ P'] [Module k P'] [IsScalarTower k Γ P']
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module A' Q] [IsScalarTower A' Γ' Q]
  {Q' : Type*} [AddCommGroup Q'] [Module Γ' Q'] [Module A' Q'] [IsScalarTower A' Γ' Q']

Lean source · statement

theorem intertwines_conjIn (φ : Γ → Γ') (β : A' ⊗[k] P ≃ₗ[A'] Q) (β' : A' ⊗[k] P' ≃ₗ[A'] Q')
    (hs : ∀ (s : Γ) (m : P), β ((1 : A') ⊗ₜ[k] (s • m)) = φ s • β ((1 : A') ⊗ₜ[k] m))
    (hs' : ∀ (s : Γ) (m : P'), β' ((1 : A') ⊗ₜ[k] (s • m)) = φ s • β' ((1 : A') ⊗ₜ[k] m))
    (ψ : Q ≃ₗ[Γ'] Q') :
    Intertwines (_root_.id : Γ → Γ) (conjIn β β' ψ : A' ⊗[k] P →ₗ[A'] A' ⊗[k] P')

def

conjOut

L1483 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.conjOut

Documentation

The chart scalar linear equivalence induced from a compatible equivalence of tensor presentations.

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section OneChart
variable {k : Type*} [CommRing k] {A' : Type*} [CommRing A'] [Algebra k A']
  {Γ : Type*} [CommRing Γ] [Algebra k Γ] {Γ' : Type*} [CommRing Γ'] [Algebra A' Γ']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {P' : Type*} [AddCommGroup P'] [Module Γ P'] [Module k P'] [IsScalarTower k Γ P']
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module A' Q] [IsScalarTower A' Γ' Q]
  {Q' : Type*} [AddCommGroup Q'] [Module Γ' Q'] [Module A' Q'] [IsScalarTower A' Γ' Q']

Lean source · statement

def conjOut (φ : Γ → Γ') (hgen : ∀ y : Γ', y ∈ Submodule.span A' (Set.range φ))
    (β : A' ⊗[k] P ≃ₗ[A'] Q) (β' : A' ⊗[k] P' ≃ₗ[A'] Q')
    (hs : ∀ (s : Γ) (m : P), β ((1 : A') ⊗ₜ[k] (s • m)) = φ s • β ((1 : A') ⊗ₜ[k] m))
    (hs' : ∀ (s : Γ) (m : P'), β' ((1 : A') ⊗ₜ[k] (s • m)) = φ s • β' ((1 : A') ⊗ₜ[k] m))
    (f : A' ⊗[k] P ≃ₗ[A'] A' ⊗[k] P') (hf : Intertwines (_root_.id : Γ → Γ) (f : A' ⊗[k] P →ₗ[A']
        A' ⊗[k] P')) :
    Q ≃ₗ[Γ'] Q'

theorem

conjOut_apply

L1513 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.conjOut_apply

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section OneChart
variable {k : Type*} [CommRing k] {A' : Type*} [CommRing A'] [Algebra k A']
  {Γ : Type*} [CommRing Γ] [Algebra k Γ] {Γ' : Type*} [CommRing Γ'] [Algebra A' Γ']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {P' : Type*} [AddCommGroup P'] [Module Γ P'] [Module k P'] [IsScalarTower k Γ P']
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module A' Q] [IsScalarTower A' Γ' Q]
  {Q' : Type*} [AddCommGroup Q'] [Module Γ' Q'] [Module A' Q'] [IsScalarTower A' Γ' Q']

Lean source · statement

theorem conjOut_apply (φ : Γ → Γ') (hgen : ∀ y : Γ', y ∈ Submodule.span A' (Set.range φ))
    (β : A' ⊗[k] P ≃ₗ[A'] Q) (β' : A' ⊗[k] P' ≃ₗ[A'] Q')
    (hs : ∀ (s : Γ) (m : P), β ((1 : A') ⊗ₜ[k] (s • m)) = φ s • β ((1 : A') ⊗ₜ[k] m))
    (hs' : ∀ (s : Γ) (m : P'), β' ((1 : A') ⊗ₜ[k] (s • m)) = φ s • β' ((1 : A') ⊗ₜ[k] m))
    (f : A' ⊗[k] P ≃ₗ[A'] A' ⊗[k] P') (hf : Intertwines (_root_.id : Γ → Γ) (f : A' ⊗[k] P →ₗ[A']
        A' ⊗[k] P'))
    (q : Q) : conjOut φ hgen β β' hs hs' f hf q = β' (f (β.symm q))

structure

ChartBC

L1531 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.ChartBC

Documentation

Chart section comparisons for base extension, compatible with restrictions and chart scalars.

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section TwoChart
open TwoChartCech
variable {k : Type u} [CommRing k] {A' : Type u} [CommRing A'] [Algebra k A']
  {𝒰 : Cover.{u, u} k} {𝒰' : Cover.{u, u} A'}

Lean source · full declaration

structure ChartBC (φ0 : 𝒰.A0 → 𝒰'.A0) (φ1 : 𝒰.A1 → 𝒰'.A1) (φ01 : 𝒰.A01 → 𝒰'.A01)
    (S : Sections.{u, u, w} 𝒰) (T : Sections.{u, u, w} 𝒰') where
  /-- The first chart section equivalence after extending the base ring. -/
  β0 : A' ⊗[k] S.M0 ≃ₗ[A'] T.M0
  /-- The second chart section equivalence after extending the base ring. -/
  β1 : A' ⊗[k] S.M1 ≃ₗ[A'] T.M1
  /-- The overlap section equivalence after extending the base ring. -/
  β01 : A' ⊗[k] S.M01 ≃ₗ[A'] T.M01
  /-- Compatibility with restriction from the first chart to the overlap. -/
  hr0 : ∀ x, β01 (S.r0.baseChange A' x) = T.r0 (β0 x)
  /-- Compatibility with restriction from the second chart to the overlap. -/
  hr1 : ∀ x, β01 (S.r1.baseChange A' x) = T.r1 (β1 x)
  /-- Compatibility with the first chart scalar action on pure tensor generators. -/
  hs0 : ∀ (s : 𝒰.A0) (m : S.M0), β0 ((1 : A') ⊗ₜ[k] (s • m)) = φ0 s • β0 ((1 : A') ⊗ₜ[k] m)
  /-- Compatibility with the second chart scalar action on pure tensor generators. -/
  hs1 : ∀ (s : 𝒰.A1) (m : S.M1), β1 ((1 : A') ⊗ₜ[k] (s • m)) = φ1 s • β1 ((1 : A') ⊗ₜ[k] m)
  /-- Compatibility with the overlap scalar action on pure tensor generators. -/
  hs01 : ∀ (s : 𝒰.A01) (m : S.M01), β01 ((1 : A') ⊗ₜ[k] (s • m)) = φ01 s • β01 ((1 : A') ⊗ₜ[k] m)

theorem

exists_intertwines_of_chartIso

L1553 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.ChartBC.exists_intertwines_of_chartIso

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section TwoChart
open TwoChartCech
variable {k : Type u} [CommRing k] {A' : Type u} [CommRing A'] [Algebra k A']
  {𝒰 : Cover.{u, u} k} {𝒰' : Cover.{u, u} A'}
variable {φ0 : 𝒰.A0 → 𝒰'.A0} {φ1 : 𝒰.A1 → 𝒰'.A1} {φ01 : 𝒰.A01 → 𝒰'.A01}
  {S S' : Sections.{u, u, w} 𝒰} {T T' : Sections.{u, u, w} 𝒰'}

Lean source · statement

theorem ChartBC.exists_intertwines_of_chartIso (β : ChartBC φ0 φ1 φ01 S T) (β' : ChartBC φ0 φ1 φ01
    S' T')
    (ψ0 : T.M0 ≃ₗ[𝒰'.A0] T'.M0) (ψ1 : T.M1 ≃ₗ[𝒰'.A1] T'.M1) (ψ01 : T.M01 ≃ₗ[𝒰'.A01] T'.M01)
    (hψ0 : ∀ m, ψ01 (T.r0 m) = T'.r0 (ψ0 m)) (hψ1 : ∀ m, ψ01 (T.r1 m) = T'.r1 (ψ1 m)) :
    ∃ (e0 : A' ⊗[k] S.M0 ≃ₗ[A'] A' ⊗[k] S'.M0) (e1 : A' ⊗[k] S.M1 ≃ₗ[A'] A' ⊗[k] S'.M1)
      (e01 : A' ⊗[k] S.M01 ≃ₗ[A'] A' ⊗[k] S'.M01),
      Intertwines (_root_.id : 𝒰.A0 → 𝒰.A0) (e0 : A' ⊗[k] S.M0 →ₗ[A'] A' ⊗[k] S'.M0) ∧
      Intertwines (_root_.id : 𝒰.A1 → 𝒰.A1) (e1 : A' ⊗[k] S.M1 →ₗ[A'] A' ⊗[k] S'.M1) ∧
      Intertwines (_root_.id : 𝒰.A01 → 𝒰.A01) (e01 : A' ⊗[k] S.M01 →ₗ[A'] A' ⊗[k] S'.M01) ∧
      (∀ x, e01 (S.r0.baseChange A' x) = S'.r0.baseChange A' (e0 x)) ∧
      (∀ x, e01 (S.r1.baseChange A' x) = S'.r1.baseChange A' (e1 x))

theorem

exists_chartIso_of_intertwines

L1574 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.ChartBC.exists_chartIso_of_intertwines

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section TwoChart
open TwoChartCech
variable {k : Type u} [CommRing k] {A' : Type u} [CommRing A'] [Algebra k A']
  {𝒰 : Cover.{u, u} k} {𝒰' : Cover.{u, u} A'}
variable {φ0 : 𝒰.A0 → 𝒰'.A0} {φ1 : 𝒰.A1 → 𝒰'.A1} {φ01 : 𝒰.A01 → 𝒰'.A01}
  {S S' : Sections.{u, u, w} 𝒰} {T T' : Sections.{u, u, w} 𝒰'}

Lean source · statement

theorem ChartBC.exists_chartIso_of_intertwines (β : ChartBC φ0 φ1 φ01 S T) (β' : ChartBC φ0 φ1 φ01
    S' T')
    (hgen0 : ∀ y : 𝒰'.A0, y ∈ Submodule.span A' (Set.range φ0))
    (hgen1 : ∀ y : 𝒰'.A1, y ∈ Submodule.span A' (Set.range φ1))
    (hgen01 : ∀ y : 𝒰'.A01, y ∈ Submodule.span A' (Set.range φ01))
    (f0 : A' ⊗[k] S.M0 ≃ₗ[A'] A' ⊗[k] S'.M0) (f1 : A' ⊗[k] S.M1 ≃ₗ[A'] A' ⊗[k] S'.M1)
    (f01 : A' ⊗[k] S.M01 ≃ₗ[A'] A' ⊗[k] S'.M01)
    (hf0 : Intertwines (_root_.id : 𝒰.A0 → 𝒰.A0) (f0 : A' ⊗[k] S.M0 →ₗ[A'] A' ⊗[k] S'.M0))
    (hf1 : Intertwines (_root_.id : 𝒰.A1 → 𝒰.A1) (f1 : A' ⊗[k] S.M1 →ₗ[A'] A' ⊗[k] S'.M1))
    (hf01 : Intertwines (_root_.id : 𝒰.A01 → 𝒰.A01) (f01 : A' ⊗[k] S.M01 →ₗ[A'] A' ⊗[k] S'.M01))
    (hsq0 : ∀ x, f01 (S.r0.baseChange A' x) = S'.r0.baseChange A' (f0 x))
    (hsq1 : ∀ x, f01 (S.r1.baseChange A' x) = S'.r1.baseChange A' (f1 x)) :
    ∃ (E0 : T.M0 ≃ₗ[𝒰'.A0] T'.M0) (E1 : T.M1 ≃ₗ[𝒰'.A1] T'.M1) (E01 : T.M01 ≃ₗ[𝒰'.A01] T'.M01),
      (∀ m, E01 (T.r0 m) = T'.r0 (E0 m)) ∧ (∀ m, E01 (T.r1 m) = T'.r1 (E1 m))

theorem

exists_presentEquiv

L1610 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.exists_presentEquiv

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Present
variable (k : Type*) [CommRing k] (A' : Type*) [CommRing A'] [Algebra k A']
  (S : Type*) [CommRing S] [Algebra k S] (S' : Type*) [CommRing S'] [Algebra k S'] [Algebra A' S']
  [IsScalarTower k A' S']
  (B : Type*) [CommRing B] [Algebra B S] [Algebra B S']
  (L : Type*) [AddCommGroup L] [Module B L]
  [SMulCommClass B k S] [SMulCommClass B A' S']

Lean source · statement

theorem exists_presentEquiv (φ : S →+* S') (e : A' ⊗[k] S ≃ₐ[A'] S') (he : ∀ s, e ((1 : A') ⊗ₜ[k]
    s) = φ s)
    (hB : ∀ b, algebraMap B S' b = φ (algebraMap B S b)) :
    ∃ Z : A' ⊗[k] (S ⊗[B] L) ≃ₗ[A'] S' ⊗[B] L,
      (∀ (a : A') (s : S) (l : L), Z (a ⊗ₜ[k] (s ⊗ₜ[B] l)) = (a • φ s) ⊗ₜ[B] l) ∧
      (∀ (a : A') (s : S) (p : S ⊗[B] L), Z (a ⊗ₜ[k] (s • p)) = (a • φ s) • Z ((1 : A') ⊗ₜ[k] p))

theorem

tensorPresentation_naturality

L1684 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.tensorPresentation_naturality

Documentation

A tensor presentation commutes with a map of its coefficient module. The evaluation map on the left coefficient need not itself be linear.

Source context 20 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry

Lean source · statement

theorem tensorPresentation_naturality
    {B S U L M : Type*} [CommRing B]
    [AddCommGroup S] [Module B S] [AddCommGroup U] [Module B U]
    [AddCommGroup L] [Module B L] [AddCommGroup M] [Module B M]
    (χ : S → U)
    (F : S ⊗[B] L →+ U ⊗[B] L) (G : S ⊗[B] M →+ U ⊗[B] M)
    (hF : ∀ s l, F (s ⊗ₜ[B] l) = χ s ⊗ₜ[B] l)
    (hG : ∀ s m, G (s ⊗ₜ[B] m) = χ s ⊗ₜ[B] m)
    (f : L →ₗ[B] M) (x : S ⊗[B] L) :
    G (TensorProduct.map (LinearMap.id : S →ₗ[B] S) f x) =
      TensorProduct.map (LinearMap.id : U →ₗ[B] U) f (F x)

theorem

tensorPresentation_baseChange_naturality

L1704 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.tensorPresentation_baseChange_naturality

Documentation

Tensor presentations commute with the actual scalar-extension maps of an equivalence of coefficient modules. Only their pure-tensor presentation laws are required; no module comparison is assumed.

Source context 20 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry

Lean source · statement

theorem tensorPresentation_baseChange_naturality
    (k : Type*) [CommRing k] (A : Type*) [CommRing A] [Algebra k A]
    (S : Type*) [CommRing S] [Algebra k S]
    (U : Type*) [CommRing U] [Algebra A U]
    (B : Type*) [CommRing B] [Algebra B S] [Algebra B U]
    (L : Type*) [AddCommGroup L] [Module B L]
    (M : Type*) [AddCommGroup M] [Module B M]
    [SMulCommClass B k S] [SMulCommClass B A U]
    (φ : S → U)
    (F : A ⊗[k] (S ⊗[B] L) ≃ₗ[A] U ⊗[B] L)
    (G : A ⊗[k] (S ⊗[B] M) ≃ₗ[A] U ⊗[B] M)
    (hF : ∀ (a : A) (s : S) (l : L),
      F (a ⊗ₜ[k] (s ⊗ₜ[B] l)) = (a • φ s) ⊗ₜ[B] l)
    (hG : ∀ (a : A) (s : S) (m : M),
      G (a ⊗ₜ[k] (s ⊗ₜ[B] m)) = (a • φ s) ⊗ₜ[B] m)
    (e : L ≃ₗ[B] M) (a : A) (z : S ⊗[B] L) :
    G (a ⊗ₜ[k] LinearEquiv.baseChange B S L M e z) =
      LinearEquiv.baseChange B U L M e (F (a ⊗ₜ[k] z))

theorem

exists_presentEquivs_natural

L1730 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.exists_presentEquivs_natural

Documentation

Construct both coefficient-module presentations over one ring presentation, together with their scalar law and the actual base-change naturality law for the supplied coefficient equivalence.

Source context 20 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry

Lean source · statement

theorem exists_presentEquivs_natural
    (k : Type*) [CommRing k] (A : Type*) [CommRing A] [Algebra k A]
    (S : Type*) [CommRing S] [Algebra k S]
    (U : Type*) [CommRing U] [Algebra k U] [Algebra A U] [IsScalarTower k A U]
    (B : Type*) [CommRing B] [Algebra B S] [Algebra B U]
    (L : Type*) [AddCommGroup L] [Module B L]
    (M : Type*) [AddCommGroup M] [Module B M]
    [SMulCommClass B k S] [SMulCommClass B A U]
    (φ : S →+* U) (e : A ⊗[k] S ≃ₐ[A] U)
    (he : ∀ s, e ((1 : A) ⊗ₜ[k] s) = φ s)
    (hB : ∀ b, algebraMap B U b = φ (algebraMap B S b))
    (σ : L ≃ₗ[B] M) :
    ∃ (F : A ⊗[k] (S ⊗[B] L) ≃ₗ[A] U ⊗[B] L)
      (G : A ⊗[k] (S ⊗[B] M) ≃ₗ[A] U ⊗[B] M),
      (∀ (a : A) (s : S) (l : L),
        F (a ⊗ₜ[k] (s ⊗ₜ[B] l)) = (a • φ s) ⊗ₜ[B] l) ∧
      (∀ (a : A) (s : S) (p : S ⊗[B] L),
        F (a ⊗ₜ[k] (s • p)) = (a • φ s) • F ((1 : A) ⊗ₜ[k] p)) ∧
      (∀ (a : A) (s : S) (m : M),
        G (a ⊗ₜ[k] (s ⊗ₜ[B] m)) = (a • φ s) ⊗ₜ[B] m) ∧
      ∀ (a : A) (z : S ⊗[B] L),
        G (a ⊗ₜ[k] LinearEquiv.baseChange B S L M σ z) =
          LinearEquiv.baseChange B U L M σ (F (a ⊗ₜ[k] z))

theorem

vertex_comp

L1777 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.vertex_comp

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Vertex
variable {ι : Type v} {B₀ : Type u} [CommRing B₀] {B : ι → Type u} [∀ i, CommRing (B i)] [∀ i,
    Algebra B₀ (B i)]
    {Bω : Type u} [CommRing Bω] [Algebra B₀ Bω] {Bω' : Type u} [CommRing Bω'] [Algebra B₀ Bω']

Lean source · statement

theorem vertex_comp [Preorder ι] (τ : ∀ i j, i ≤ j → B i →ₐ[B₀] B j) (g : ∀ i, B i →ₐ[B₀] Bω)
    (hg : ∀ i j (h : i ≤ j), (g j).comp (τ i j h) = g i) (h : Bω ≃ₐ[B₀] Bω') (i j : ι) (hij : i ≤
        j) :
    ((h : Bω →ₐ[B₀] Bω').comp (g j)).comp (τ i j hij) = (h : Bω →ₐ[B₀] Bω').comp (g i)

theorem

vertex_surj

L1783 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.vertex_surj

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Vertex
variable {ι : Type v} {B₀ : Type u} [CommRing B₀] {B : ι → Type u} [∀ i, CommRing (B i)] [∀ i,
    Algebra B₀ (B i)]
    {Bω : Type u} [CommRing Bω] [Algebra B₀ Bω] {Bω' : Type u} [CommRing Bω'] [Algebra B₀ Bω']

Lean source · statement

theorem vertex_surj (g : ∀ i, B i →ₐ[B₀] Bω) (hsurj : ∀ b : Bω, ∃ i x, g i x = b) (h : Bω ≃ₐ[B₀]
    Bω')
    (b : Bω') : ∃ i x, ((h : Bω →ₐ[B₀] Bω').comp (g i)) x = b

theorem

vertex_zero

L1791 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.vertex_zero

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Vertex
variable {ι : Type v} {B₀ : Type u} [CommRing B₀] {B : ι → Type u} [∀ i, CommRing (B i)] [∀ i,
    Algebra B₀ (B i)]
    {Bω : Type u} [CommRing Bω] [Algebra B₀ Bω] {Bω' : Type u} [CommRing Bω'] [Algebra B₀ Bω']

Lean source · statement

theorem vertex_zero [Preorder ι] (τ : ∀ i j, i ≤ j → B i →ₐ[B₀] B j) (g : ∀ i, B i →ₐ[B₀] Bω)
    (hzero : ∀ i (x : B i), g i x = 0 → ∃ (j : ι) (h : i ≤ j), τ i j h x = 0) (h : Bω ≃ₐ[B₀] Bω')
    (i : ι) (x : B i) (hx : ((h : Bω →ₐ[B₀] Bω').comp (g i)) x = 0) : ∃ (j : ι) (hij : i ≤ j), τ i
        j hij x = 0

theorem

smulCommClass_of_algebras

L1799 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.smulCommClass_of_algebras

Source context 20 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry

Lean source · statement

theorem smulCommClass_of_algebras (R₁ R₂ T : Type*) [CommSemiring R₁] [CommSemiring R₂]
    [CommSemiring T]
    [Algebra R₁ T] [Algebra R₂ T] : SMulCommClass R₁ R₂ T

theorem

one_tmul_tmul_tmul

L1804 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.one_tmul_tmul_tmul

Source context 20 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry

Lean source · statement

theorem one_tmul_tmul_tmul {B0 B' B'' V : Type*} [CommRing B0] [CommRing B'] [CommRing B'']
    [CommRing V]
    [Algebra B0 B'] [Algebra B' B''] [Algebra B'' V] {L : Type*} [AddCommGroup L] [Module B0 L]
    (b : B'') (β : B') (l : L) :
    ((1 : V) ⊗ₜ[B''] (b ⊗ₜ[B'] (β ⊗ₜ[B0] l)) : V ⊗[B''] (B'' ⊗[B'] (B' ⊗[B0] L))) =
      (algebraMap B'' V (β • b)) ⊗ₜ[B''] ((1 : B'') ⊗ₜ[B'] ((1 : B') ⊗ₜ[B0] l))

theorem

finrank_baseChange_eq_one_of_invertible

L1814 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.finrank_baseChange_eq_one_of_invertible

Source context 20 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry

Lean source · statement

theorem finrank_baseChange_eq_one_of_invertible {A : Type*} [CommRing A] (P : Type*) [AddCommGroup
    P] [Module A P]
    [Module.Invertible A P] (K : Type*) [Field K] [Algebra A K] : Module.finrank K (K ⊗[A] P) = 1

theorem

appLE_app_apply_of_comp_eq

L1832 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.Scheme.Hom.appLE_app_apply_of_comp_eq

Source context 16 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
open CategoryTheory Opposite
open scoped _root_.AlgebraicGeometry.Scheme _root_.AlgebraicGeometry.Scheme.Hom in

Lean source · full declaration

theorem Scheme.Hom.appLE_app_apply_of_comp_eq {X Y Z : Scheme.{u}} (f : X ⟶ Y) (g : Y ⟶ Z) (k : X
    ⟶ Z)
    (h : f ≫ g = k) (U : Z.Opens) (V : X.Opens) (e : V ≤ f ⁻¹ᵁ (g ⁻¹ᵁ U))
    (s : Γ(Z, U)) : (f.appLE (g ⁻¹ᵁ U) V e).hom ((g.app U).hom s) =
      (k.appLE U V (by
        subst h
        exact e)).hom s := by
  subst h
  rw [← CategoryTheory.ConcreteCategory.comp_apply, Scheme.Hom.app_eq_appLE,
      Scheme.Hom.appLE_comp_appLE]

theorem

appLE_apply_map

L1844 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.Scheme.Hom.appLE_apply_map

Source context 16 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
open CategoryTheory Opposite
open scoped _root_.AlgebraicGeometry.Scheme _root_.AlgebraicGeometry.Scheme.Hom in

Lean source · statement

theorem Scheme.Hom.appLE_apply_map {X Y : Scheme.{u}} (f : X ⟶ Y) {U U' : Y.Opens} {V V' : X.Opens}
    (e : V ≤ f ⁻¹ᵁ U) (e' : V' ≤ f ⁻¹ᵁ U') (hU : U' ≤ U) (hV : V' ≤ V) (s : Γ(Y, U)) :
    (f.appLE U' V' e').hom ((Y.presheaf.map (homOfLE hU).op).hom s) =
      (X.presheaf.map (homOfLE hV).op).hom ((f.appLE U V e).hom s)

theorem

app_apply_map

L1852 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.Scheme.Hom.app_apply_map

Source context 16 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
open CategoryTheory Opposite
open scoped _root_.AlgebraicGeometry.Scheme _root_.AlgebraicGeometry.Scheme.Hom in

Lean source · full declaration

theorem Scheme.Hom.app_apply_map {X Y : Scheme.{u}} (f : X ⟶ Y) {U U' : Y.Opens} (hU : U' ≤ U) (s
    : Γ(Y, U)) :
    (f.app U').hom ((Y.presheaf.map (homOfLE hU).op).hom s) =
      (X.presheaf.map (homOfLE (show f ⁻¹ᵁ U' ≤ f ⁻¹ᵁ U from fun _ hx => hU hx)).op).hom ((f.app
          U).hom s) := by
  rw [← CategoryTheory.ConcreteCategory.comp_apply, Scheme.Hom.naturality]
  rfl

def

algebraOfPresent

L1879 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.algebraOfPresent

Documentation

The chart algebra structure transported through the tensor presentation.

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section VertexGeo
open _root_.AlgebraicGeometry.FGSubalgebra
variable (R : Type u) [CommRing R] (A : Type u) [CommRing A] [Algebra R A]
  (Γ : Type u) [CommRing Γ] [Algebra R Γ]

Lean source · statement

def algebraOfPresent (U : Type u) [CommRing U] [Algebra A U] (eU : A
    ⊗[R] Γ ≃ₐ[A] U) :
    Algebra Γ U

def

vtxEquiv

L1886 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.vtxEquiv

Documentation

The reversed tensor presentation of the chart algebra.

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section VertexGeo
open _root_.AlgebraicGeometry.FGSubalgebra
variable (R : Type u) [CommRing R] (A : Type u) [CommRing A] [Algebra R A]
  (Γ : Type u) [CommRing Γ] [Algebra R Γ]

Lean source · full declaration

def vtxEquiv (U : Type u) [CommRing U] [Algebra A U] (eU : A ⊗[R] Γ ≃ₐ[A] U) :
    let _ : Algebra Γ U := algebraOfPresent R A Γ U eU
    Γ ⊗[R] A ≃ₐ[Γ] U := by
  intro chartAlgebra
  exact AlgEquiv.ofRingEquiv
    (f := (Algebra.TensorProduct.comm R Γ A).toRingEquiv.trans (eU : A ⊗[R] Γ ≃+* U))
    (fun _ => rfl)

def

vtxHom

L1895 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.vtxHom

Documentation

The chart algebra homomorphism from a finitely generated tensor stage to the limit.

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section VertexGeo
open _root_.AlgebraicGeometry.FGSubalgebra
variable (R : Type u) [CommRing R] (A : Type u) [CommRing A] [Algebra R A]
  (Γ : Type u) [CommRing Γ] [Algebra R Γ]

Lean source · full declaration

def vtxHom (U : Type u) [CommRing U] [Algebra A U] (eU : A ⊗[R] Γ ≃ₐ[A] U) (D :
    FGSubalgebra R A) :
    let _ : Algebra Γ U := algebraOfPresent R A Γ U eU
    tensorStage R A Γ D →ₐ[Γ] U := by
  intro chartAlgebra
  exact (vtxEquiv R A Γ U eU : Γ ⊗[R] A →ₐ[Γ] U).comp (tensorStageToColimit R A Γ D)

theorem

vtxHom_tmul

L1904 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.vtxHom_tmul

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section VertexGeo
open _root_.AlgebraicGeometry.FGSubalgebra
variable (R : Type u) [CommRing R] (A : Type u) [CommRing A] [Algebra R A]
  (Γ : Type u) [CommRing Γ] [Algebra R Γ]
variable (U : Type u) [CommRing U] [Algebra A U] (eU : A ⊗[R] Γ ≃ₐ[A] U)

Lean source · statement

theorem vtxHom_tmul (D : FGSubalgebra R A) (γ : Γ) (d : ↥D.1) :
    vtxHom R A Γ U eU D (γ ⊗ₜ[R] d) = eU ((d : A) ⊗ₜ[R] γ)

theorem

vtxHom_comp

L1907 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.vtxHom_comp

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section VertexGeo
open _root_.AlgebraicGeometry.FGSubalgebra
variable (R : Type u) [CommRing R] (A : Type u) [CommRing A] [Algebra R A]
  (Γ : Type u) [CommRing Γ] [Algebra R Γ]
variable (U : Type u) [CommRing U] [Algebra A U] (eU : A ⊗[R] Γ ≃ₐ[A] U)

Lean source · full declaration

theorem vtxHom_comp (i j : FGSubalgebra R A) (h : i ≤ j) :
    let _ : Algebra Γ U := algebraOfPresent R A Γ U eU
    (vtxHom R A Γ U eU j).comp (tensorStageMap R A Γ i j h) = vtxHom R A Γ U eU i := by
  intro chartAlgebra
  exact vertex_comp (fun i j h => tensorStageMap R A Γ i j h) (tensorStageToColimit R A Γ)
    (tensorStageToColimit_comp R A Γ) (vtxEquiv R A Γ U eU) i j h

theorem

vtxHom_surj

L1914 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.vtxHom_surj

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section VertexGeo
open _root_.AlgebraicGeometry.FGSubalgebra
variable (R : Type u) [CommRing R] (A : Type u) [CommRing A] [Algebra R A]
  (Γ : Type u) [CommRing Γ] [Algebra R Γ]
variable (U : Type u) [CommRing U] [Algebra A U] (eU : A ⊗[R] Γ ≃ₐ[A] U)

Lean source · statement

theorem vtxHom_surj (b : U) : ∃ (i : FGSubalgebra R A) (x : tensorStage R A Γ i), vtxHom R A Γ U
    eU i x = b

theorem

vtxHom_zero

L1919 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.vtxHom_zero

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section VertexGeo
open _root_.AlgebraicGeometry.FGSubalgebra
variable (R : Type u) [CommRing R] (A : Type u) [CommRing A] [Algebra R A]
  (Γ : Type u) [CommRing Γ] [Algebra R Γ]
variable (U : Type u) [CommRing U] [Algebra A U] (eU : A ⊗[R] Γ ≃ₐ[A] U)

Lean source · statement

theorem vtxHom_zero (i : FGSubalgebra R A) (x : tensorStage R A Γ i) (hx : vtxHom R A Γ U eU i x =
    0) :
    ∃ (j : FGSubalgebra R A) (h : i ≤ j), tensorStageMap R A Γ i j h x = 0

def

stgHom

L1928 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.stgHom

Documentation

The ring homomorphism from a tensor stage to its chart presentation.

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section VertexGeo
open _root_.AlgebraicGeometry.FGSubalgebra
variable (R : Type u) [CommRing R] (A : Type u) [CommRing A] [Algebra R A]
  (Γ : Type u) [CommRing Γ] [Algebra R Γ]
variable (U : Type u) [CommRing U] [Algebra A U] (eU : A ⊗[R] Γ ≃ₐ[A] U)

Lean source · statement

def stgHom (D : FGSubalgebra R A) (W : Type u) [CommRing W] [Algebra (↥D.1) W]
    (eW : ↥D.1 ⊗[R] Γ ≃ₐ[↥D.1] W) : tensorStage R A Γ D →+* W

theorem

stgHom_tmul

L1933 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.stgHom_tmul

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section VertexGeo
open _root_.AlgebraicGeometry.FGSubalgebra
variable (R : Type u) [CommRing R] (A : Type u) [CommRing A] [Algebra R A]
  (Γ : Type u) [CommRing Γ] [Algebra R Γ]
variable (U : Type u) [CommRing U] [Algebra A U] (eU : A ⊗[R] Γ ≃ₐ[A] U)

Lean source · statement

theorem stgHom_tmul (D : FGSubalgebra R A) (W : Type u) [CommRing W] [Algebra (↥D.1) W]
    (eW : ↥D.1 ⊗[R] Γ ≃ₐ[↥D.1] W) (γ : Γ) (d : ↥D.1) : stgHom R A Γ D W eW (γ ⊗ₜ[R] d) = eW (d
        ⊗ₜ[R] γ)

theorem

stgHom_restriction

L1939 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.stgHom_restriction

Documentation

Compatible stage presentations identify the chart restriction with the restriction of the tensor-stage rings.

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section VertexGeo
open _root_.AlgebraicGeometry.FGSubalgebra
variable (R : Type u) [CommRing R] (A : Type u) [CommRing A] [Algebra R A]
  (Γ : Type u) [CommRing Γ] [Algebra R Γ]
variable (U : Type u) [CommRing U] [Algebra A U] (eU : A ⊗[R] Γ ≃ₐ[A] U)

Lean source · statement

theorem stgHom_restriction
    (Δ : Type u) [CommRing Δ] [Algebra R Δ] (ρ : Γ →ₐ[R] Δ)
    (D : FGSubalgebra R A) (W Z : Type u) [CommRing W] [CommRing Z]
    [Algebra (↥D.1) W] [Algebra (↥D.1) Z]
    (eW : ↥D.1 ⊗[R] Γ ≃ₐ[↥D.1] W) (eZ : ↥D.1 ⊗[R] Δ ≃ₐ[↥D.1] Z)
    (v : W →ₐ[↥D.1] Z)
    (hρ : ∀ (d : ↥D.1) (γ : Γ), v (eW (d ⊗ₜ[R] γ)) = eZ (d ⊗ₜ[R] ρ γ)) :
    v.toRingHom.comp (stgHom R A Γ D W eW) =
      (stgHom R A Δ D Z eZ).comp
        (Algebra.TensorProduct.map ρ (AlgHom.id R ↥D.1)).toRingHom

theorem

vtxHom_eq_comp_stgHom

L1957 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.vtxHom_eq_comp_stgHom

Documentation

Compatibility of two tensor presentations determines the factorization of the vertex map through the stage map. Only the two tensor-ring generators are needed, so the statement applies equally to a chart and to its overlap.

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section VertexGeo
open _root_.AlgebraicGeometry.FGSubalgebra
variable (R : Type u) [CommRing R] (A : Type u) [CommRing A] [Algebra R A]
  (Γ : Type u) [CommRing Γ] [Algebra R Γ]
variable (U : Type u) [CommRing U] [Algebra A U] (eU : A ⊗[R] Γ ≃ₐ[A] U)

Lean source · full declaration

theorem vtxHom_eq_comp_stgHom
    (D : FGSubalgebra R A) (W : Type u) [CommRing W] [Algebra (↥D.1) W]
    (eW : ↥D.1 ⊗[R] Γ ≃ₐ[↥D.1] W) (φ : W →+* U)
    (hφe : ∀ (d : ↥D.1) (γ : Γ), φ (eW (d ⊗ₜ[R] γ)) = eU ((d : A) ⊗ₜ[R] γ)) :
    let _ : Algebra Γ U := algebraOfPresent R A Γ U eU
    (vtxHom R A Γ U eU D).toRingHom = φ.comp (stgHom R A Γ D W eW) := by
  intro chartAlgebra
  refine Algebra.TensorProduct.ringHom_ext
    (R := R) (A := Γ) (B := ↥D.1) (C := U) ?_ ?_
  · exact RingHom.ext fun γ => (hφe (1 : ↥D.1) γ).symm
  · exact RingHom.ext fun d => (hφe d (1 : Γ)).symm

theorem

tensorStage_restriction_natural

L1982 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.tensorStage_restriction_natural

Documentation

Restricting a tensor stage commutes with enlarging the finitely generated base subalgebra.

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section AbstractSurj
open _root_.TwoChartCech
open _root_.AlgebraicGeometry.FGSubalgebra
section NativeChartGeneratorNotation
open scoped ChangeOfRings
open _root_.ModuleCat

Lean source · statement

theorem tensorStage_restriction_natural
    (R A Γ Δ : Type u) [CommRing R] [CommRing A] [CommRing Γ] [CommRing Δ]
    [Algebra R A] [Algebra R Γ] [Algebra R Δ]
    (ρ : Γ →ₐ[R] Δ) (i j : FGSubalgebra R A) (h : i ≤ j) :
    (tensorStageMap R A Δ i j h).toRingHom.comp
        (Algebra.TensorProduct.map ρ (AlgHom.id R ↥i.1)).toRingHom =
      (Algebra.TensorProduct.map ρ (AlgHom.id R ↥j.1)).toRingHom.comp
        (tensorStageMap R A Γ i j h).toRingHom

theorem

tensorStage_presentation_restriction

L1997 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.tensorStage_presentation_restriction

Documentation

A tensor presentation compatible with a chart restriction gives the corresponding commutative ring diagram from every finite stage to the limit.

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section AbstractSurj
open _root_.TwoChartCech
open _root_.AlgebraicGeometry.FGSubalgebra
section NativeChartGeneratorNotation
open scoped ChangeOfRings
open _root_.ModuleCat

Lean source · full declaration

theorem tensorStage_presentation_restriction
    (R A Γ Δ U V : Type u) [CommRing R] [CommRing A] [CommRing Γ] [CommRing Δ]
    [CommRing U] [CommRing V] [Algebra R A] [Algebra R Γ] [Algebra R Δ]
    [Algebra A U] [Algebra A V]
    (ρ : Γ →ₐ[R] Δ) (eU : A ⊗[R] Γ ≃ₐ[A] U) (eV : A ⊗[R] Δ ≃ₐ[A] V)
    (v : U →ₐ[A] V)
    (hρ : ∀ (a : A) (γ : Γ), v (eU (a ⊗ₜ[R] γ)) = eV (a ⊗ₜ[R] ρ γ))
    (D : FGSubalgebra R A) :
    let _ : Algebra Γ U := algebraOfPresent R A Γ U eU
    let _ : Algebra Δ V := algebraOfPresent R A Δ V eV
    (vtxHom R A Δ V eV D).toRingHom.comp
        (Algebra.TensorProduct.map ρ (AlgHom.id R ↥D.1)).toRingHom =
      v.toRingHom.comp (vtxHom R A Γ U eU D).toRingHom := by
  intro chartAlgebra overlapAlgebra
  refine Algebra.TensorProduct.ringHom_ext
    (R := R) (A := Γ) (B := ↥D.1) (C := V) ?_ ?_
  · exact RingHom.ext fun γ => (hρ (1 : A) γ).symm
  · exact RingHom.ext fun d => (hρ (d : A) (1 : Γ)).symm

theorem

exists_invertible_chart_and_overlap

L2018 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.exists_invertible_chart_and_overlap

Documentation

Descend an invertible chart and construct its overlap realization from its original tensor presentation and restriction law.

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section AbstractSurj
open _root_.TwoChartCech
open _root_.AlgebraicGeometry.FGSubalgebra
section NativeChartGeneratorNotation
open scoped ChangeOfRings
open _root_.ModuleCat

Lean source · full declaration

theorem exists_invertible_chart_and_overlap
    (R A Γ U V : Type u) [CommRing R] [CommRing A] [CommRing Γ]
    [CommRing U] [CommRing V] [Algebra R A] [Algebra R Γ] [Algebra A U]
    (eU : A ⊗[R] Γ ≃ₐ[A] U) (v : U →+* V)
    {N O : Type w} [AddCommGroup N] [Module U N] [Module.Invertible U N]
    [AddCommGroup O] [Module V O]
    (rbc : let _ : Algebra U V := v.toAlgebra; V ⊗[U] N ≃ₗ[V] O) (r : N → O)
    (hrbc : let _ : Algebra U V := v.toAlgebra;
      ∀ n, rbc ((1 : V) ⊗ₜ[U] n) = r n) :
    let _ : Algebra Γ U := algebraOfPresent R A Γ U eU
    ∃ (i : FGSubalgebra R A) (P : ModuleCat.{u} (tensorStage R A Γ i)),
      Module.Invertible (tensorStage R A Γ i) P ∧
      Module.FinitePresentation (tensorStage R A Γ i) P ∧
      ∃ (I : (extendScalars (vtxHom R A Γ U eU i).toRingHom).obj P ≃ₗ[U] N)
        (J : (extendScalars v).obj
          ((extendScalars (vtxHom R A Γ U eU i).toRingHom).obj P) ≃ₗ[V] O),
        ∀ m : (extendScalars (vtxHom R A Γ U eU i).toRingHom).obj P,
          J ((extendRestrictScalarsAdj v).unit.app _ m) = r (I m) := by
  intro chartAlgebra
  classical
  have : IsDirectedOrder (FGSubalgebra R A) := ⟨IsDirected.directed⟩
  obtain ⟨i, K, _, _, hK, ⟨Iraw⟩⟩ :=
    Module.IsDirectLimit.exists_invertible_linearEquiv_baseChange
      (tensorStage R A Γ) (fun i j h => tensorStageMap R A Γ i j h)
      U (vtxHom R A Γ U eU) (vtxHom_comp R A Γ U eU)
      (vtxHom_surj R A Γ U eU) (vtxHom_zero R A Γ U eU) N
  have hfp : Module.FinitePresentation (tensorStage R A Γ i) K :=
    Module.finitePresentation_of_projective (tensorStage R A Γ i) K
  let P : ModuleCat.{u} (tensorStage R A Γ i) := ModuleCat.of _ K
  let M : ModuleCat.{u} U :=
    (extendScalars (vtxHom R A Γ U eU i).toRingHom).obj P
  let I : M ≃ₗ[U] N := by
    let : Algebra (tensorStage R A Γ i) U := (vtxHom R A Γ U eU i).toRingHom.toAlgebra
    exact Iraw
  obtain ⟨J, hJ⟩ := exists_scalarExtension_realization v M I rbc r hrbc
  exact ⟨i, P, hK, hfp, I, J, hJ⟩

theorem

exists_fg_gluingData

L2055 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.exists_fg_gluingData

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section AbstractSurj
open _root_.TwoChartCech
open _root_.AlgebraicGeometry.FGSubalgebra
section NativeChartGeneratorNotation
open scoped ChangeOfRings
open _root_.ModuleCat

Lean source · full declaration

theorem exists_fg_gluingData
    (R : Type u) [CommRing R] (A : Type u) [CommRing A] [Algebra R A]
    (𝒰 : Cover.{u, u} R) (UA : Cover.{u, u} A) (T : Sections.{u, u, w} UA)
    [Module.Invertible UA.A0 T.M0] [Module.Invertible UA.A1 T.M1]
    (eU0 : A ⊗[R] 𝒰.A0 ≃ₐ[A] UA.A0) (eU1 : A ⊗[R] 𝒰.A1 ≃ₐ[A] UA.A1) (eU01 : A ⊗[R] 𝒰.A01 ≃ₐ[A]
        UA.A01)
    (hρU0 : ∀ (a : A) (γ : 𝒰.A0), UA.ρ0 (eU0 (a ⊗ₜ[R] γ)) = eU01 (a ⊗ₜ[R] 𝒰.ρ0 γ))
    (hρU1 : ∀ (a : A) (γ : 𝒰.A1), UA.ρ1 (eU1 (a ⊗ₜ[R] γ)) = eU01 (a ⊗ₜ[R] 𝒰.ρ1 γ))
    (rbc0 :
      let _ : Algebra UA.A0 UA.A01 := UA.ρ0.toRingHom.toAlgebra
      UA.A01 ⊗[UA.A0] T.M0 ≃ₗ[UA.A01] T.M01)
    (hrbc0 :
      let _ : Algebra UA.A0 UA.A01 := UA.ρ0.toRingHom.toAlgebra
      ∀ m, rbc0 ((1 : UA.A01) ⊗ₜ[UA.A0] m) = T.r0 m)
    (rbc1 :
      let _ : Algebra UA.A1 UA.A01 := UA.ρ1.toRingHom.toAlgebra
      UA.A01 ⊗[UA.A1] T.M1 ≃ₗ[UA.A01] T.M01)
    (hrbc1 :
      let _ : Algebra UA.A1 UA.A01 := UA.ρ1.toRingHom.toAlgebra
      ∀ m, rbc1 ((1 : UA.A01) ⊗ₜ[UA.A1] m) = T.r1 m)
    (V : ∀ D : FGSubalgebra R A, Cover.{u, u} ↥D.1)
    (eV0 : ∀ D : FGSubalgebra R A, ↥D.1 ⊗[R] 𝒰.A0 ≃ₐ[↥D.1] (V D).A0)
    (eV1 : ∀ D : FGSubalgebra R A, ↥D.1 ⊗[R] 𝒰.A1 ≃ₐ[↥D.1] (V D).A1)
    (eV01 : ∀ D : FGSubalgebra R A, ↥D.1 ⊗[R] 𝒰.A01 ≃ₐ[↥D.1] (V D).A01)
    (hρV0 : ∀ (D : FGSubalgebra R A) (d : ↥D.1) (γ : 𝒰.A0), (V D).ρ0 (eV0 D (d ⊗ₜ[R] γ)) = eV01 D
        (d ⊗ₜ[R] 𝒰.ρ0 γ))
    (hρV1 : ∀ (D : FGSubalgebra R A) (d : ↥D.1) (γ : 𝒰.A1), (V D).ρ1 (eV1 D (d ⊗ₜ[R] γ)) = eV01 D
        (d ⊗ₜ[R] 𝒰.ρ1 γ))
    (φ0 : ∀ D : FGSubalgebra R A, (V D).A0 →+* UA.A0) (φ1 : ∀ D : FGSubalgebra R A, (V D).A1 →+*
        UA.A1)
    (φ01 : ∀ D : FGSubalgebra R A, (V D).A01 →+* UA.A01)
    (eφ0 : ∀ D : FGSubalgebra R A, A ⊗[↥D.1] (V D).A0 ≃ₐ[A] UA.A0)
    (heφ0 : ∀ (D : FGSubalgebra R A) (s : (V D).A0), eφ0 D ((1 : A) ⊗ₜ[↥D.1] s) = φ0 D s)
    (eφ1 : ∀ D : FGSubalgebra R A, A ⊗[↥D.1] (V D).A1 ≃ₐ[A] UA.A1)
    (heφ1 : ∀ (D : FGSubalgebra R A) (s : (V D).A1), eφ1 D ((1 : A) ⊗ₜ[↥D.1] s) = φ1 D s)
    (eφ01 : ∀ D : FGSubalgebra R A, A ⊗[↥D.1] (V D).A01 ≃ₐ[A] UA.A01)
    (heφ01 : ∀ (D : FGSubalgebra R A) (s : (V D).A01), eφ01 D ((1 : A) ⊗ₜ[↥D.1] s) = φ01 D s)
    (hφρ0 : ∀ (D : FGSubalgebra R A) (s : (V D).A0), φ01 D ((V D).ρ0 s) = UA.ρ0 (φ0 D s))
    (hφρ1 : ∀ (D : FGSubalgebra R A) (s : (V D).A1), φ01 D ((V D).ρ1 s) = UA.ρ1 (φ1 D s))
    (hφe0 : ∀ (D : FGSubalgebra R A) (d : ↥D.1) (γ : 𝒰.A0), φ0 D (eV0 D (d ⊗ₜ[R] γ)) = eU0 ((d :
        A) ⊗ₜ[R] γ))
    (hφe1 : ∀ (D : FGSubalgebra R A) (d : ↥D.1) (γ : 𝒰.A1), φ1 D (eV1 D (d ⊗ₜ[R] γ)) = eU1 ((d :
        A) ⊗ₜ[R] γ))
    (hφe01 : ∀ (D : FGSubalgebra R A) (d : ↥D.1) (γ : 𝒰.A01), φ01 D (eV01 D (d ⊗ₜ[R] γ)) = eU01
        ((d : A) ⊗ₜ[R] γ)) :
    ∃ (D : FGSubalgebra R A)
      (P0 : Type u) (_ : AddCommGroup P0) (_ : Module (V D).A0 P0) (_ : Module.Finite (V D).A0 P0)
      (_ : Module.Projective (V D).A0 P0)
      (P1 : Type u) (_ : AddCommGroup P1) (_ : Module (V D).A1 P1) (_ : Module.Finite (V D).A1 P1)
      (_ : Module.Projective (V D).A1 P1)
      (_ : ∀ (K : Type u) [Field K] [Algebra (V D).A0 K], Module.finrank K (K ⊗[(V D).A0] P0) = 1)
      (_ : ∀ (K : Type u) [Field K] [Algebra (V D).A1 K], Module.finrank K (K ⊗[(V D).A1] P1) = 1)
      (σ :
        let _ : Algebra (V D).A0 (V D).A01 := (V D).ρ0.toRingHom.toAlgebra
        let _ : Algebra (V D).A1 (V D).A01 := (V D).ρ1.toRingHom.toAlgebra
        (V D).A01 ⊗[(V D).A0] P0 ≃ₗ[(V D).A01] (V D).A01 ⊗[(V D).A1] P1),
      ∀ (S : Sections.{u, u, w} (V D)) (e0 : S.M0 ≃ₗ[(V D).A0] P0)
        (e1 : S.M1 ≃ₗ[(V D).A1] P1)
        (e01 :
          let _ : Algebra (V D).A0 (V D).A01 := (V D).ρ0.toRingHom.toAlgebra
          S.M01 ≃ₗ[(V D).A01] (V D).A01 ⊗[(V D).A0] P0),
        (let _ : Algebra (V D).A0 (V D).A01 := (V D).ρ0.toRingHom.toAlgebra
          ∀ m, e01 (S.r0 m) = (1 : (V D).A01) ⊗ₜ[(V D).A0] e0 m) →
        (let _ : Algebra (V D).A0 (V D).A01 := (V D).ρ0.toRingHom.toAlgebra
         let _ : Algebra (V D).A1 (V D).A01 := (V D).ρ1.toRingHom.toAlgebra
          ∀ m, σ (e01 (S.r1 m)) = (1 : (V D).A01) ⊗ₜ[(V D).A1] e1 m) →
        Nonempty (ChartBC (k := ↥D.1) (A' := A) (φ0 D) (φ1 D) (φ01 D) S T) := by
  classical
  have : IsDirectedOrder (FGSubalgebra R A) := ⟨IsDirected.directed⟩
  let B0 : FGSubalgebra R A → Type u := tensorStage R A 𝒰.A0
  let B1 : FGSubalgebra R A → Type u := tensorStage R A 𝒰.A1
  let C : FGSubalgebra R A → Type u := tensorStage R A 𝒰.A01
  let tau0 : ∀ i j : FGSubalgebra R A, i ≤ j → B0 i →+* B0 j :=
    fun i j h => (tensorStageMap R A 𝒰.A0 i j h).toRingHom
  let tau1 : ∀ i j : FGSubalgebra R A, i ≤ j → B1 i →+* B1 j :=
    fun i j h => (tensorStageMap R A 𝒰.A1 i j h).toRingHom
  let tauC : ∀ i j : FGSubalgebra R A, i ≤ j → C i →+* C j :=
    fun i j h => (tensorStageMap R A 𝒰.A01 i j h).toRingHom
  let g0 : ∀ D : FGSubalgebra R A, B0 D →+* UA.A0 := by
    let : Algebra 𝒰.A0 UA.A0 := algebraOfPresent R A 𝒰.A0 UA.A0 eU0
    exact fun D => (vtxHom R A 𝒰.A0 UA.A0 eU0 D).toRingHom
  let g1 : ∀ D : FGSubalgebra R A, B1 D →+* UA.A1 := by
    let : Algebra 𝒰.A1 UA.A1 := algebraOfPresent R A 𝒰.A1 UA.A1 eU1
    exact fun D => (vtxHom R A 𝒰.A1 UA.A1 eU1 D).toRingHom
  let gC : ∀ D : FGSubalgebra R A, C D →+* UA.A01 := by
    let : Algebra 𝒰.A01 UA.A01 := algebraOfPresent R A 𝒰.A01 UA.A01 eU01
    exact fun D => (vtxHom R A 𝒰.A01 UA.A01 eU01 D).toRingHom
  let res0 : ∀ D : FGSubalgebra R A, B0 D →+* C D := fun D =>
    (Algebra.TensorProduct.map 𝒰.ρ0 (AlgHom.id R ↥D.1)).toRingHom
  let res1 : ∀ D : FGSubalgebra R A, B1 D →+* C D := fun D =>
    (Algebra.TensorProduct.map 𝒰.ρ1 (AlgHom.id R ↥D.1)).toRingHom
  have hg0 : ∀ i j (h : i ≤ j), (g0 j).comp (tau0 i j h) = g0 i := by
    intro i j h
    let : Algebra 𝒰.A0 UA.A0 := algebraOfPresent R A 𝒰.A0 UA.A0 eU0
    exact congrArg (fun f : tensorStage R A 𝒰.A0 i →ₐ[𝒰.A0] UA.A0 => f.toRingHom)
      (vtxHom_comp R A 𝒰.A0 UA.A0 eU0 i j h)
  have hg1 : ∀ i j (h : i ≤ j), (g1 j).comp (tau1 i j h) = g1 i := by
    intro i j h
    let : Algebra 𝒰.A1 UA.A1 := algebraOfPresent R A 𝒰.A1 UA.A1 eU1
    exact congrArg (fun f : tensorStage R A 𝒰.A1 i →ₐ[𝒰.A1] UA.A1 => f.toRingHom)
      (vtxHom_comp R A 𝒰.A1 UA.A1 eU1 i j h)
  have hgC : ∀ i j (h : i ≤ j), (gC j).comp (tauC i j h) = gC i := by
    intro i j h
    let : Algebra 𝒰.A01 UA.A01 := algebraOfPresent R A 𝒰.A01 UA.A01 eU01
    exact congrArg (fun f : tensorStage R A 𝒰.A01 i →ₐ[𝒰.A01] UA.A01 => f.toRingHom)
      (vtxHom_comp R A 𝒰.A01 UA.A01 eU01 i j h)
  have hres0 : ∀ i j (h : i ≤ j),
      (tauC i j h).comp (res0 i) = (res0 j).comp (tau0 i j h) := by
    exact fun i j h => tensorStage_restriction_natural R A 𝒰.A0 𝒰.A01 𝒰.ρ0 i j h
  have hlim0 : ∀ D, (gC D).comp (res0 D) = UA.ρ0.toRingHom.comp (g0 D) := by
    exact fun D => tensorStage_presentation_restriction R A 𝒰.A0 𝒰.A01
      UA.A0 UA.A01 𝒰.ρ0 eU0 eU01 UA.ρ0 hρU0 D
  obtain ⟨i0, oldP0, hInv0, hFP0, oldI0Native, smallOverlap0, hSmallOverlap0⟩ :=
    exists_invertible_chart_and_overlap R A 𝒰.A0 UA.A0 UA.A01 eU0
      UA.ρ0.toRingHom rbc0 T.r0 hrbc0
  have : Module.Invertible (tensorStage R A 𝒰.A0 i0) oldP0 := hInv0
  let smallM0 : ModuleCat.{u} UA.A0 := (extendScalars (g0 i0)).obj oldP0
  have : Module.FinitePresentation (B0 i0) oldP0 := hFP0
  have hres1 : ∀ i j (h : i ≤ j),
      (tauC i j h).comp (res1 i) = (res1 j).comp (tau1 i j h) := by
    exact fun i j h => tensorStage_restriction_natural R A 𝒰.A1 𝒰.A01 𝒰.ρ1 i j h
  have hlim1 : ∀ D, (gC D).comp (res1 D) = UA.ρ1.toRingHom.comp (g1 D) := by
    exact fun D => tensorStage_presentation_restriction R A 𝒰.A1 𝒰.A01
      UA.A1 UA.A01 𝒰.ρ1 eU1 eU01 UA.ρ1 hρU1 D
  obtain ⟨i1, oldP1, hInv1, hFP1, oldI1Native, smallOverlap1, hSmallOverlap1⟩ :=
    exists_invertible_chart_and_overlap R A 𝒰.A1 UA.A1 UA.A01 eU1
      UA.ρ1.toRingHom rbc1 T.r1 hrbc1
  have : Module.Invertible (tensorStage R A 𝒰.A1 i1) oldP1 := hInv1
  let smallM1 : ModuleCat.{u} UA.A1 := (extendScalars (g1 i1)).obj oldP1
  have : Module.FinitePresentation (B1 i1) oldP1 := hFP1
  let sigmaLimit : (extendScalars UA.ρ0.toRingHom).obj smallM0 ≅
      (extendScalars UA.ρ1.toRingHom).obj smallM1 :=
    (smallOverlap0 ≪≫ₗ smallOverlap1.symm).toModuleIso
  have hSigmaLimit : ∀ z : (extendScalars UA.ρ0.toRingHom).obj smallM0,
      smallOverlap1 (sigmaLimit.hom z) = smallOverlap0 z :=
    fun z => smallOverlap1.apply_symm_apply (smallOverlap0 z)
  obtain ⟨D, h0, h1, E0, E1, F0, F1, stageIso, hF0unit, hF1unit, hreal⟩ :=
      exists_common_stage_clutching
    B0 B1 C tau0 tau1 tauC UA.A0 UA.A1 UA.A01 g0 g1 gC hg0 hg1 hgC
    (vtxHom_surj R A 𝒰.A01 UA.A01 eU01)
    (vtxHom_zero R A 𝒰.A01 UA.A01 eU01)
    res0 res1 hres0 hres1 UA.ρ0.toRingHom UA.ρ1.toRingHom hlim0 hlim1
    i0 i1 oldP0 oldP1 sigmaLimit
  let t0 : B0 D →+* (V D).A0 := stgHom R A 𝒰.A0 D (V D).A0 (eV0 D)
  let t1 : B1 D →+* (V D).A1 := stgHom R A 𝒰.A1 D (V D).A1 (eV1 D)
  let tC : C D →+* (V D).A01 := stgHom R A 𝒰.A01 D (V D).A01 (eV01 D)
  have hCompat0 : g0 D = (φ0 D).comp t0 :=
    vtxHom_eq_comp_stgHom R A 𝒰.A0 UA.A0 eU0 D (V D).A0
      (eV0 D) (φ0 D) (hφe0 D)
  have hCompat1 : g1 D = (φ1 D).comp t1 :=
    vtxHom_eq_comp_stgHom R A 𝒰.A1 UA.A1 eU1 D (V D).A1
      (eV1 D) (φ1 D) (hφe1 D)
  have hCompatC : gC D = (φ01 D).comp tC :=
    vtxHom_eq_comp_stgHom R A 𝒰.A01 UA.A01 eU01 D (V D).A01
      (eV01 D) (φ01 D) (hφe01 D)
  let L0 : ModuleCat.{u} (B0 D) := (extendScalars (tau0 i0 D h0)).obj oldP0
  let Q0 : ModuleCat.{u} (C D) := (extendScalars (res0 D)).obj L0
  have : Module.Invertible (B0 D) L0 :=
    invertible_extendScalars_of_invertible (tau0 i0 D h0) oldP0
  obtain ⟨newI0, Xi0, Xi0Native, hXi0real, hXi0Native, hXi0gen⟩ :=
    exists_algebraTensor_realization_of_maps (g0 D) (res0 D) (gC D) UA.ρ0.toRingHom
      L0 smallM0 E0 F0 oldI0Native smallOverlap0 T.r0 hF0unit hSmallOverlap0
  let L1 : ModuleCat.{u} (B1 D) := (extendScalars (tau1 i1 D h1)).obj oldP1
  let Q1 : ModuleCat.{u} (C D) := (extendScalars (res1 D)).obj L1
  have : Module.Invertible (B1 D) L1 :=
    invertible_extendScalars_of_invertible (tau1 i1 D h1) oldP1
  obtain ⟨newI1, Xi1, Xi1Native, hXi1real, hXi1Native, hXi1gen⟩ :=
    exists_algebraTensor_realization_of_maps (g1 D) (res1 D) (gC D) UA.ρ1.toRingHom
      L1 smallM1 E1 F1 oldI1Native smallOverlap1 T.r1 hF1unit hSmallOverlap1
  let P0 : ModuleCat.{u} (V D).A0 := algebraTensorObj t0 L0
  have hSquare0 : (V D).ρ0.toRingHom.comp t0 = tC.comp (res0 D) := by
    exact stgHom_restriction R A 𝒰.A0 𝒰.A01 𝒰.ρ0 D (V D).A0 (V D).A01
      (eV0 D) (eV01 D) (V D).ρ0 (hρV0 D)
  let P1 : ModuleCat.{u} (V D).A1 := algebraTensorObj t1 L1
  have hSquare1 : (V D).ρ1.toRingHom.comp t1 = tC.comp (res1 D) := by
    exact stgHom_restriction R A 𝒰.A1 𝒰.A01 𝒰.ρ1 D (V D).A1 (V D).A01
      (eV1 D) (eV01 D) (V D).ρ1 (hρV1 D)
  let algB0V0 : Algebra (B0 D) (V D).A0 := t0.toAlgebra
  let algB1V1 : Algebra (B1 D) (V D).A1 := t1.toAlgebra
  let algB0U0 : Algebra (B0 D) UA.A0 := (g0 D).toAlgebra
  let algB1U1 : Algebra (B1 D) UA.A1 := (g1 D).toAlgebra
  let algV0V01 : Algebra (V D).A0 (V D).A01 := (V D).ρ0.toRingHom.toAlgebra
  let algV1V01 : Algebra (V D).A1 (V D).A01 := (V D).ρ1.toRingHom.toAlgebra
  let algCV01 : Algebra (C D) (V D).A01 := tC.toAlgebra
  let algCU01 : Algebra (C D) UA.A01 := (gC D).toAlgebra
  have : Module.Invertible (V D).A0 P0 :=
    invertible_algebraTensorObj_of_invertible t0 L0
  have : Module.Invertible (V D).A1 P1 :=
    invertible_algebraTensorObj_of_invertible t1 L1
  obtain ⟨kappa0, kappa1, sigmaV, hKappa0, hKappa1, hKappaSigma⟩ :=
    exists_algebraTensor_clutching_of_commuting_squares t0 t1 (res0 D) (res1 D)
      (V D).ρ0.toRingHom (V D).ρ1.toRingHom tC hSquare0 hSquare1 L0 L1 stageIso
  let sigmaV' : (V D).A01 ⊗[C D] Q0 ≃ₗ[(V D).A01] (V D).A01 ⊗[C D] Q1 :=
    LinearEquiv.baseChange (C D) (V D).A01 Q0 Q1 stageIso.toLinearEquiv
  change ∀ z : (V D).A01 ⊗[(V D).A0] P0,
    kappa1 (sigmaV z) = sigmaV' (kappa0 z) at hKappaSigma
  let sigmaU : UA.A01 ⊗[C D] Q0 ≃ₗ[UA.A01] UA.A01 ⊗[C D] Q1 :=
    LinearEquiv.baseChange (C D) UA.A01 Q0 Q1 stageIso.toLinearEquiv
  have hXiSigmaNative := realization_extendScalars_compat (gC D) Q0 Q1
    ((extendScalars UA.ρ0.toRingHom).obj smallM0) ((extendScalars UA.ρ1.toRingHom).obj smallM1)
    stageIso F0 F1 sigmaLimit Xi0Native Xi1Native smallOverlap0 smallOverlap1
    hXi0real hXi1real hSigmaLimit hreal
  have hXiSigma : ∀ z : UA.A01 ⊗[C D] Q0, Xi1 (sigmaU z) = Xi0 z := by
    exact algebraTensor_realization_baseChange_compat (gC D) Q0 Q1 stageIso
      Xi0Native Xi1Native Xi0 Xi1 hXi0Native hXi1Native hXiSigmaNative
  have : SMulCommClass (B0 D) ↥D.1 (V D).A0 := smulCommClass_of_algebras _ _ _
  have : SMulCommClass (B1 D) ↥D.1 (V D).A1 := smulCommClass_of_algebras _ _ _
  have : SMulCommClass (V D).A0 ↥D.1 (V D).A01 := smulCommClass_of_algebras _ _ _
  have : SMulCommClass (V D).A1 ↥D.1 (V D).A01 :=
    smulCommClass_of_algebras (V D).A1 ↥D.1 (V D).A01
  have : SMulCommClass (B0 D) A UA.A0 := smulCommClass_of_algebras _ _ _
  have : SMulCommClass (B1 D) A UA.A1 := smulCommClass_of_algebras _ _ _
  have : SMulCommClass (C D) ↥D.1 (V D).A01 := smulCommClass_of_algebras _ _ _
  have : SMulCommClass (C D) A UA.A01 := smulCommClass_of_algebras _ _ _
  refine ⟨D, (P0 : Type u), inferInstance, inferInstance, inferInstance, inferInstance,
    (P1 : Type u), inferInstance, inferInstance, inferInstance, inferInstance,
    finrank_baseChange_eq_one_of_invertible (A := (V D).A0) (P0 : Type u),
    finrank_baseChange_eq_one_of_invertible (A := (V D).A1) (P1 : Type u),
    sigmaV, fun S e0 e1 e01 hC0 hC1 => ?_⟩
  obtain ⟨Z0', hZ0', hZ0's⟩ := exists_presentEquiv ↥D.1 A (V D).A0 UA.A0 (B0 D) L0
    (φ0 D) (eφ0 D) (heφ0 D) (fun b => RingHom.congr_fun hCompat0 b)
  obtain ⟨Z1', hZ1', hZ1's⟩ := exists_presentEquiv ↥D.1 A (V D).A1 UA.A1 (B1 D) L1
    (φ1 D) (eφ1 D) (heφ1 D) (fun b => RingHom.congr_fun hCompat1 b)
  obtain ⟨present0, present1, hPresent0, hPresent0s, hPresent1, hPresentSigma⟩ :=
    exists_presentEquivs_natural ↥D.1 A (V D).A01 UA.A01 (C D) Q0 Q1
      (φ01 D) (eφ01 D) (heφ01 D) (fun b => RingHom.congr_fun hCompatC b) stageIso.toLinearEquiv
  obtain ⟨H0, hH0, hH0s⟩ := TensorProduct.exists_realized_tensorPresentation_with_scalar
    ↥D.1 A UA.A0 (S := (V D).A0) (P := (V D).A0 ⊗[B0 D] L0)
    (Q := UA.A0 ⊗[B0 D] L0) (N := T.M0) (φ0 D) Z0'
    (show UA.A0 ⊗[B0 D] L0 ≃ₗ[UA.A0] T.M0 from newI0) (hZ0's (1 : A))
  obtain ⟨Z0, hZ0, hZ0s⟩ := TensorProduct.exists_coordinate_realization_with_scalar
    ↥D.1 A (V D).A0 (M := S.M0) (P := (V D).A0 ⊗[B0 D] L0)
    (N := T.M0) (U := UA.A0) (X := UA.A0 ⊗[B0 D] L0)
    (φ0 D) (show S.M0 ≃ₗ[(V D).A0] (V D).A0 ⊗[B0 D] L0 from e0) H0 Z0' newI0 hH0 hH0s
  obtain ⟨H1, hH1, hH1s⟩ := TensorProduct.exists_realized_tensorPresentation_with_scalar
    ↥D.1 A UA.A1 (S := (V D).A1) (P := (V D).A1 ⊗[B1 D] L1)
    (Q := UA.A1 ⊗[B1 D] L1) (N := T.M1) (φ1 D) Z1'
    (show UA.A1 ⊗[B1 D] L1 ≃ₗ[UA.A1] T.M1 from newI1) (hZ1's (1 : A))
  obtain ⟨Z1, hZ1, hZ1s⟩ := TensorProduct.exists_coordinate_realization_with_scalar
    ↥D.1 A (V D).A1 (M := S.M1) (P := (V D).A1 ⊗[B1 D] L1)
    (N := T.M1) (U := UA.A1) (X := UA.A1 ⊗[B1 D] L1)
    (φ1 D) (show S.M1 ≃ₗ[(V D).A1] (V D).A1 ⊗[B1 D] L1 from e1) H1 Z1' newI1 hH1 hH1s
  obtain ⟨H01, hH01, hH01s⟩ := TensorProduct.exists_realized_tensorPresentation_with_scalar
    ↥D.1 A UA.A01 (S := (V D).A01) (P := (V D).A01 ⊗[C D] Q0)
    (Q := UA.A01 ⊗[C D] Q0) (N := T.M01) (φ01 D) present0
    (show UA.A01 ⊗[C D] Q0 ≃ₗ[UA.A01] T.M01 from Xi0) (hPresent0s (1 : A))
  obtain ⟨Z01, hZ01, hZ01s⟩ := TensorProduct.exists_coordinate_realization_with_scalar
    ↥D.1 A (V D).A01 (M := S.M01) (P := (V D).A01 ⊗[C D] Q0)
    (N := T.M01) (U := UA.A01) (X := UA.A01 ⊗[C D] Q0)
    (φ01 D) (e01 ≪≫ₗ kappa0) H01 present0 Xi0 hH01 hH01s
  simp only [LinearEquiv.trans_apply] at hZ01
  have htransition := TensorProduct.realized_presentation_coordinate_transition
    ↥D.1 A (V D).A01 (C D) Q0 Q1 stageIso.toLinearEquiv
    kappa0 kappa1 sigmaV present0 present1 sigmaU Xi0 Xi1
    hKappaSigma hPresentSigma hXiSigma
  have hZ01_1 : ∀ (a : A) (m : S.M01),
      Z01 (a ⊗ₜ[↥D.1] m) = Xi1 (present1 (a ⊗ₜ[↥D.1] kappa1 (sigmaV (e01 m)))) := by
    intro a m
    exact (hZ01 a m).trans (htransition a (e01 m)).symm
  refine ⟨{ β0 := Z0, β1 := Z1, β01 := Z01, hr0 := ?_, hr1 := ?_, hs0 := hZ0s, hs1 := hZ1s, hs01
      := hZ01s }⟩
  · exact TensorProduct.baseChange_square_of_native_chart_coordinates
      (k := ↥D.1) (A := A) (B := B0 D) (J := (V D).A0)
      (C := C D) (D := (V D).A01) (U := UA.A0) (V := UA.A01)
      (algBJ := algB0V0) (algBU := algB0U0)
      (algJD := algV0V01) (algCD := algCV01) (algCV := algCU01)
      (X := L0) (Q := Q0) (M := S.M0) (N := S.M01)
      (T0 := T.M0) (T1 := T.M01)
      e0.toAddEquiv S.r0 T.r0 Z0 Z01 e01 kappa0 newI0 Xi0
      (fun p : (V D).A0 ⊗[B0 D] L0 => Z0' ((1 : A) ⊗ₜ[↥D.1] p))
      (fun q : (V D).A01 ⊗[C D] Q0 => present0 ((1 : A) ⊗ₜ[↥D.1] q))
      (fun x : L0 =>
        (show Q0 from (extendRestrictScalarsAdj (res0 D)).unit.app L0 x))
      (V D).ρ0.toRingHom UA.ρ0.toRingHom (φ0 D) (φ01 D)
      hKappa0 (hPresent0 (1 : A)) hXi0gen (hZ0' (1 : A))
      (hZ01 (1 : A)) (hZ0 (1 : A)) hC0 T.r0_smul (hφρ0 D)
  · exact TensorProduct.baseChange_square_of_native_chart_coordinates
      (k := ↥D.1) (A := A) (B := B1 D) (J := (V D).A1)
      (C := C D) (D := (V D).A01) (U := UA.A1) (V := UA.A01)
      (algBJ := algB1V1) (algBU := algB1U1)
      (algJD := algV1V01) (algCD := algCV01) (algCV := algCU01)
      (X := L1) (Q := Q1) (M := S.M1) (N := S.M01)
      (T0 := T.M1) (T1 := T.M01)
      e1.toAddEquiv S.r1 T.r1 Z1 Z01 (fun n : S.M01 => sigmaV (e01 n)) kappa1 newI1 Xi1
      (fun p : (V D).A1 ⊗[B1 D] L1 => Z1' ((1 : A) ⊗ₜ[↥D.1] p))
      (fun q : (V D).A01 ⊗[C D] Q1 => present1 ((1 : A) ⊗ₜ[↥D.1] q))
      (fun x : L1 =>
        (show Q1 from (extendRestrictScalarsAdj (res1 D)).unit.app L1 x))
      (V D).ρ1.toRingHom UA.ρ1.toRingHom (φ1 D) (φ01 D)
      hKappa1 (hPresent1 (1 : A)) hXi1gen (hZ1' (1 : A))
      (hZ01_1 (1 : A)) (hZ1 (1 : A)) hC1 T.r1_smul (hφρ1 D)

theorem

cover_ρ0_present

L2360 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.cover_ρ0_present

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section GeoDict
open _root_.AlgebraicGeometry.Scheme.TwoAffineOpenCover
open _root_.TwoChartCech
variable {R : Type u} [CommRing R] {C : Scheme.{u}} (𝒱 : C.TwoAffineOpenCover) (c : C ⟶ Spec (.of
    R))

Lean source · statement

theorem cover_ρ0_present (B : Type u) [CommRing B] [Algebra R B]
    (e0 : B ⊗[R] (𝒱.cover c).A0 ≃ₐ[B] ((𝒱.pullback c B).cover (Limits.pullback.snd c (specMap R
        B))).A0)
    (e01 : B ⊗[R] (𝒱.cover c).A01 ≃ₐ[B] ((𝒱.pullback c B).cover (Limits.pullback.snd c (specMap R
        B))).A01)
    (he0 : ∀ s, e0 ((1 : B) ⊗ₜ[R] s) = ((Limits.pullback.fst c (specMap R B)).app 𝒱.U0).hom s)
    (he01 : ∀ s, e01 ((1 : B) ⊗ₜ[R] s) = ((Limits.pullback.fst c (specMap R B)).app (𝒱.U0 ⊓
        𝒱.U1)).hom s)
    (b : B) (γ : (𝒱.cover c).A0) :
    ((𝒱.pullback c B).cover (Limits.pullback.snd c (specMap R B))).ρ0 (e0 (b ⊗ₜ[R] γ)) = e01 (b
        ⊗ₜ[R] (𝒱.cover c).ρ0 γ)

theorem

cover_ρ1_present

L2382 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.cover_ρ1_present

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section GeoDict
open _root_.AlgebraicGeometry.Scheme.TwoAffineOpenCover
open _root_.TwoChartCech
variable {R : Type u} [CommRing R] {C : Scheme.{u}} (𝒱 : C.TwoAffineOpenCover) (c : C ⟶ Spec (.of
    R))

Lean source · statement

theorem cover_ρ1_present (B : Type u) [CommRing B] [Algebra R B]
    (e1 : B ⊗[R] (𝒱.cover c).A1 ≃ₐ[B] ((𝒱.pullback c B).cover (Limits.pullback.snd c (specMap R
        B))).A1)
    (e01 : B ⊗[R] (𝒱.cover c).A01 ≃ₐ[B] ((𝒱.pullback c B).cover (Limits.pullback.snd c (specMap R
        B))).A01)
    (he1 : ∀ s, e1 ((1 : B) ⊗ₜ[R] s) = ((Limits.pullback.fst c (specMap R B)).app 𝒱.U1).hom s)
    (he01 : ∀ s, e01 ((1 : B) ⊗ₜ[R] s) = ((Limits.pullback.fst c (specMap R B)).app (𝒱.U0 ⊓
        𝒱.U1)).hom s)
    (b : B) (γ : (𝒱.cover c).A1) :
    ((𝒱.pullback c B).cover (Limits.pullback.snd c (specMap R B))).ρ1 (e1 (b ⊗ₜ[R] γ)) = e01 (b
        ⊗ₜ[R] (𝒱.cover c).ρ1 γ)

abbrev

φU0

L2407 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.φU0

Documentation

The ring map from first chart sections at a finite stage to the limit chart sections.

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section GeoDict
open _root_.AlgebraicGeometry.Scheme.TwoAffineOpenCover
open _root_.TwoChartCech
variable {R : Type u} [CommRing R] {C : Scheme.{u}} (𝒱 : C.TwoAffineOpenCover) (c : C ⟶ Spec (.of
    R))
variable (A : Type u) [CommRing A] [Algebra R A] (D : Subalgebra R A)

Lean source · statement

abbrev φU0 : ((𝒱.pullback c ↥D).cover (Limits.pullback.snd c (specMap R ↥D))).A0 →+*
    ((𝒱.pullback c A).cover (Limits.pullback.snd c (specMap R A))).A0

abbrev

φU1

L2414 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.φU1

Documentation

The ring map from second chart sections at a finite stage to the limit chart sections.

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section GeoDict
open _root_.AlgebraicGeometry.Scheme.TwoAffineOpenCover
open _root_.TwoChartCech
variable {R : Type u} [CommRing R] {C : Scheme.{u}} (𝒱 : C.TwoAffineOpenCover) (c : C ⟶ Spec (.of
    R))
variable (A : Type u) [CommRing A] [Algebra R A] (D : Subalgebra R A)

Lean source · statement

abbrev φU1 : ((𝒱.pullback c ↥D).cover (Limits.pullback.snd c (specMap R ↥D))).A1 →+*
    ((𝒱.pullback c A).cover (Limits.pullback.snd c (specMap R A))).A1

abbrev

φU01

L2421 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.φU01

Documentation

The ring map from overlap sections at a finite stage to the limit overlap sections.

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section GeoDict
open _root_.AlgebraicGeometry.Scheme.TwoAffineOpenCover
open _root_.TwoChartCech
variable {R : Type u} [CommRing R] {C : Scheme.{u}} (𝒱 : C.TwoAffineOpenCover) (c : C ⟶ Spec (.of
    R))
variable (A : Type u) [CommRing A] [Algebra R A] (D : Subalgebra R A)

Lean source · statement

abbrev φU01 : ((𝒱.pullback c ↥D).cover (Limits.pullback.snd c (specMap R ↥D))).A01 →+*
    ((𝒱.pullback c A).cover (Limits.pullback.snd c (specMap R A))).A01

theorem

φU01_ρ0

L2427 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.φU01_ρ0

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section GeoDict
open _root_.AlgebraicGeometry.Scheme.TwoAffineOpenCover
open _root_.TwoChartCech
variable {R : Type u} [CommRing R] {C : Scheme.{u}} (𝒱 : C.TwoAffineOpenCover) (c : C ⟶ Spec (.of
    R))
variable (A : Type u) [CommRing A] [Algebra R A] (D : Subalgebra R A)

Lean source · statement

theorem φU01_ρ0 (s : ((𝒱.pullback c ↥D).cover (Limits.pullback.snd c (specMap R ↥D))).A0) :
    φU01 𝒱 c A D (((𝒱.pullback c ↥D).cover (Limits.pullback.snd c (specMap R ↥D))).ρ0 s) =
      ((𝒱.pullback c A).cover (Limits.pullback.snd c (specMap R A))).ρ0 (φU0 𝒱 c A D s)

theorem

φU01_ρ1

L2433 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.φU01_ρ1

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section GeoDict
open _root_.AlgebraicGeometry.Scheme.TwoAffineOpenCover
open _root_.TwoChartCech
variable {R : Type u} [CommRing R] {C : Scheme.{u}} (𝒱 : C.TwoAffineOpenCover) (c : C ⟶ Spec (.of
    R))
variable (A : Type u) [CommRing A] [Algebra R A] (D : Subalgebra R A)

Lean source · statement

theorem φU01_ρ1 (s : ((𝒱.pullback c ↥D).cover (Limits.pullback.snd c (specMap R ↥D))).A1) :
    φU01 𝒱 c A D (((𝒱.pullback c ↥D).cover (Limits.pullback.snd c (specMap R ↥D))).ρ1 s) =
      ((𝒱.pullback c A).cover (Limits.pullback.snd c (specMap R A))).ρ1 (φU1 𝒱 c A D s)

theorem

φU0_present

L2439 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.φU0_present

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section GeoDict
open _root_.AlgebraicGeometry.Scheme.TwoAffineOpenCover
open _root_.TwoChartCech
variable {R : Type u} [CommRing R] {C : Scheme.{u}} (𝒱 : C.TwoAffineOpenCover) (c : C ⟶ Spec (.of
    R))
variable (A : Type u) [CommRing A] [Algebra R A] (D : Subalgebra R A)

Lean source · statement

theorem φU0_present
    (eD : ↥D ⊗[R] (𝒱.cover c).A0 ≃ₐ[↥D] ((𝒱.pullback c ↥D).cover (Limits.pullback.snd c (specMap R
        ↥D))).A0)
    (heD : ∀ s, eD ((1 : ↥D) ⊗ₜ[R] s) = ((Limits.pullback.fst c (specMap R ↥D)).app 𝒱.U0).hom s)
    (eA : A ⊗[R] (𝒱.cover c).A0 ≃ₐ[A] ((𝒱.pullback c A).cover (Limits.pullback.snd c (specMap R
        A))).A0)
    (heA : ∀ s, eA ((1 : A) ⊗ₜ[R] s) = ((Limits.pullback.fst c (specMap R A)).app 𝒱.U0).hom s)
    (eφ : A ⊗[↥D] ((𝒱.pullback c ↥D).cover (Limits.pullback.snd c (specMap R ↥D))).A0 ≃ₐ[A]
      ((𝒱.pullback c A).cover (Limits.pullback.snd c (specMap R A))).A0)
    (heφ : ∀ s, eφ ((1 : A) ⊗ₜ[↥D] s) = φU0 𝒱 c A D s)
    (d : ↥D) (γ : (𝒱.cover c).A0) :
    φU0 𝒱 c A D (eD (d ⊗ₜ[R] γ)) = eA ((d : A) ⊗ₜ[R] γ)

theorem

φU1_present

L2477 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.φU1_present

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section GeoDict
open _root_.AlgebraicGeometry.Scheme.TwoAffineOpenCover
open _root_.TwoChartCech
variable {R : Type u} [CommRing R] {C : Scheme.{u}} (𝒱 : C.TwoAffineOpenCover) (c : C ⟶ Spec (.of
    R))
variable (A : Type u) [CommRing A] [Algebra R A] (D : Subalgebra R A)

Lean source · statement

theorem φU1_present
    (eD : ↥D ⊗[R] (𝒱.cover c).A1 ≃ₐ[↥D] ((𝒱.pullback c ↥D).cover (Limits.pullback.snd c (specMap R
        ↥D))).A1)
    (heD : ∀ s, eD ((1 : ↥D) ⊗ₜ[R] s) = ((Limits.pullback.fst c (specMap R ↥D)).app 𝒱.U1).hom s)
    (eA : A ⊗[R] (𝒱.cover c).A1 ≃ₐ[A] ((𝒱.pullback c A).cover (Limits.pullback.snd c (specMap R
        A))).A1)
    (heA : ∀ s, eA ((1 : A) ⊗ₜ[R] s) = ((Limits.pullback.fst c (specMap R A)).app 𝒱.U1).hom s)
    (eφ : A ⊗[↥D] ((𝒱.pullback c ↥D).cover (Limits.pullback.snd c (specMap R ↥D))).A1 ≃ₐ[A]
      ((𝒱.pullback c A).cover (Limits.pullback.snd c (specMap R A))).A1)
    (heφ : ∀ s, eφ ((1 : A) ⊗ₜ[↥D] s) = φU1 𝒱 c A D s)
    (d : ↥D) (γ : (𝒱.cover c).A1) :
    φU1 𝒱 c A D (eD (d ⊗ₜ[R] γ)) = eA ((d : A) ⊗ₜ[R] γ)

theorem

φU01_present

L2515 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.φU01_present

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section GeoDict
open _root_.AlgebraicGeometry.Scheme.TwoAffineOpenCover
open _root_.TwoChartCech
variable {R : Type u} [CommRing R] {C : Scheme.{u}} (𝒱 : C.TwoAffineOpenCover) (c : C ⟶ Spec (.of
    R))
variable (A : Type u) [CommRing A] [Algebra R A] (D : Subalgebra R A)

Lean source · statement

theorem φU01_present
    (eD : ↥D ⊗[R] (𝒱.cover c).A01 ≃ₐ[↥D] ((𝒱.pullback c ↥D).cover (Limits.pullback.snd c (specMap
        R ↥D))).A01)
    (heD : ∀ s, eD ((1 : ↥D) ⊗ₜ[R] s) = ((Limits.pullback.fst c (specMap R ↥D)).app (𝒱.U0 ⊓
        𝒱.U1)).hom s)
    (eA : A ⊗[R] (𝒱.cover c).A01 ≃ₐ[A] ((𝒱.pullback c A).cover (Limits.pullback.snd c (specMap R
        A))).A01)
    (heA : ∀ s, eA ((1 : A) ⊗ₜ[R] s) = ((Limits.pullback.fst c (specMap R A)).app (𝒱.U0 ⊓
        𝒱.U1)).hom s)
    (eφ : A ⊗[↥D] ((𝒱.pullback c ↥D).cover (Limits.pullback.snd c (specMap R ↥D))).A01 ≃ₐ[A]
      ((𝒱.pullback c A).cover (Limits.pullback.snd c (specMap R A))).A01)
    (heφ : ∀ s, eφ ((1 : A) ⊗ₜ[↥D] s) = φU01 𝒱 c A D s)
    (d : ↥D) (γ : (𝒱.cover c).A01) :
    φU01 𝒱 c A D (eD (d ⊗ₜ[R] γ)) = eA ((d : A) ⊗ₜ[R] γ)

theorem

exists_fg_nonempty_iso_pullbackAlong

L2563 ↗

JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard.LFP.exists_fg_nonempty_iso_pullbackAlong

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Final
open _root_.AlgebraicGeometry.Scheme.TwoAffineOpenCover
open _root_.TwoChartCech
open _root_.AlgebraicGeometry.FGSubalgebra

Lean source · statement

theorem exists_fg_nonempty_iso_pullbackAlong
    (R : Type u) [CommRing R] {C : Scheme.{u}} (𝒱 : C.TwoAffineOpenCover) (c : C ⟶ Spec
        (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (A : Type u) [CommRing A] [Algebra R A] (M : RigidifiedLineBundle c ε (specMap R A)) :
    ∃ (A₀ : Subalgebra R A) (_ : A₀.FG) (M₀ : RigidifiedLineBundle c ε (specMap R A₀)),
      Nonempty ((M₀.pullbackAlong (stageHom R A₀.val)).L ≅ M.L)

theorem

exists_fg_nonempty_iso_pullbackAlong

L2642 ↗

AlgebraicGeometry.RelPicard.LFP.exists_fg_nonempty_iso_pullbackAlong

Source context 13 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.LFP_exists_fg_nonempty_iso_pullbackAlong
attribute [-instance]
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
open LFP_exists_fg_nonempty_iso_pullbackAlong.AlgebraicGeometry.RelPicard in

Lean source · statement

theorem _root_.AlgebraicGeometry.RelPicard.LFP.exists_fg_nonempty_iso_pullbackAlong
    (R : Type u) [CommRing R] {C : Scheme.{u}} (𝒱 : C.TwoAffineOpenCover) (c : C ⟶ Spec
        (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (A : Type u) [CommRing A] [Algebra R A] (M : RigidifiedLineBundle c ε
        (Scheme.TwoAffineOpenCover.specMap R A)) :
    ∃ (A₀ : Subalgebra R A) (_ : A₀.FG) (M₀ : RigidifiedLineBundle c ε
        (Scheme.TwoAffineOpenCover.specMap R A₀)),
      Nonempty ((M₀.pullbackAlong (RelPicard.LFP.stageHom R A₀.val)).L ≅ M.L)

Module documentation

Descent of a rigidified line bundle to a finitely generated stage

Declarations