NYLL
Module

Scheme_Modules_isLocallyFreeOfRank_pushforward_of_forall_exists_isPullback.lean

Nyll.AlgebraicCurves.Geometry.Sheaves.Scheme_Modules_isLocallyFreeOfRank_pushforward_of_forall_exists_isPullback
7 declarations indexed from sourceView source ↗

Module documentation

Local freeness of pushforward through pullback charts

Pullback charts of the base provide local trivializations that establish the rank of a pushed-forward module sheaf.

Imports8
Imported by4

Declarations

def

pushforwardIsoEquivalence

L56 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_Modules_isLocallyFreeOfRank_pushforward_of_forall_exists_isPullback.AlgebraicGeometry.Scheme.Modules.pushforwardIsoEquivalence
Documentation

The equivalence of module categories induced by pushing forward along the scheme isomorphism.

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_Modules_isLocallyFreeOfRank_pushforward_of_forall_exists_isPullback
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
open _root_.AlgebraicGeometry.Scheme.Modules
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
section iso
variable {Y Y' : Scheme.{u}} (e : Y ≅ Y')
Lean source · statement
def pushforwardIsoEquivalence : Y.Modules ≌ Y'.Modules
def

pullbackIsoPushforwardInv

L63 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_Modules_isLocallyFreeOfRank_pushforward_of_forall_exists_isPullback.AlgebraicGeometry.Scheme.Modules.pullbackIsoPushforwardInv
Documentation

The pullback along the isomorphism identified with pushforward along its inverse.

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_Modules_isLocallyFreeOfRank_pushforward_of_forall_exists_isPullback
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
open _root_.AlgebraicGeometry.Scheme.Modules
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
section iso
variable {Y Y' : Scheme.{u}} (e : Y ≅ Y')
Lean source · statement
def pullbackIsoPushforwardInv : Scheme.Modules.pullback e.hom ≅ pushforward e.inv
theorem

isLocallyFreeOfRank_pushforward_of_forall_exists_isPullback_aux

L69 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_Modules_isLocallyFreeOfRank_pushforward_of_forall_exists_isPullback.AlgebraicGeometry.Scheme.Modules.isLocallyFreeOfRank_pushforward_of_forall_exists_isPullback_aux
Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_Modules_isLocallyFreeOfRank_pushforward_of_forall_exists_isPullback
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
open _root_.AlgebraicGeometry.Scheme.Modules
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
Lean source · statement
theorem isLocallyFreeOfRank_pushforward_of_forall_exists_isPullback_aux
    {X T : Scheme.{u}} (π : X ⟶ T) (F : X.Modules) (n : ℕ)
    (h : ∀ y : T, ∃ (W : T.Opens), y ∈ W ∧ ∃ (W' X' : Scheme.{u}) (e : W' ≅ W.toScheme) (π' : X' ⟶
        W')
      (g' : X' ⟶ X), IsPullback g' π' π (e.hom ≫ W.ι) ∧
        IsLocallyFreeOfRank n ((pushforward π').obj ((Scheme.Modules.pullback g').obj F))) :
    IsLocallyFreeOfRank n ((pushforward π).obj F)
theorem

exists_iso_pullback_of_isPullback_aux

L112 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_Modules_isLocallyFreeOfRank_pushforward_of_forall_exists_isPullback.AlgebraicGeometry.Scheme.Modules.exists_iso_pullback_of_isPullback_aux
Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_Modules_isLocallyFreeOfRank_pushforward_of_forall_exists_isPullback
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
open _root_.AlgebraicGeometry.Scheme.Modules
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
Lean source · statement
theorem exists_iso_pullback_of_isPullback_aux
    {X T X' W' Z : Scheme.{u}} (π : X ⟶ T) (π' : X' ⟶ W') (g' : X' ⟶ X) (j : W' ⟶ T)
    (hcart : IsPullback g' π' π j) (s : Z ⟶ W') :
    ∃ φ : Limits.pullback π (s ≫ j) ≅ Limits.pullback π' s,
      φ.hom ≫ Limits.pullback.snd π' s = Limits.pullback.snd π (s ≫ j) ∧
      φ.hom ≫ Limits.pullback.fst π' s ≫ g' = Limits.pullback.fst π (s ≫ j) ∧
      ∀ G : X.Modules, Nonempty ((Scheme.Modules.pullback (Limits.pullback.fst π (s ≫ j))).obj G ≅
        (Scheme.Modules.pullback φ.hom).obj ((Scheme.Modules.pullback (Limits.pullback.fst π'
            s)).obj ((Scheme.Modules.pullback g').obj G)))
theorem

exists_isPullback_snd_specMap_algebraOfHom_aux

L145 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_Modules_isLocallyFreeOfRank_pushforward_of_forall_exists_isPullback.AlgebraicGeometry.Scheme.TwoAffineOpenCover.exists_isPullback_snd_specMap_algebraOfHom_aux
Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_Modules_isLocallyFreeOfRank_pushforward_of_forall_exists_isPullback
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace TwoAffineOpenCover
open _root_.AlgebraicGeometry.Scheme.TwoAffineOpenCover
open _root_.AlgebraicGeometry.Scheme.TwoAffineOpenCover
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
Lean source · full declaration
theorem exists_isPullback_snd_specMap_algebraOfHom_aux
    {R : Type u} [CommRing R] {C T : Scheme.{u}} (c : C ⟶ Spec (.of R)) (t : T ⟶ Spec (.of R))
    (W : T.Opens) (hW : IsAffineOpen W) :
    letI := algebraOfHom t W
    ∃ g' : Limits.pullback c (specMap R Γ(T, W)) ⟶ Limits.pullback c t,
      IsPullback g' (Limits.pullback.snd c (specMap R Γ(T, W))) (Limits.pullback.snd c t)
          hW.fromSpec ∧
        g' ≫ Limits.pullback.fst c t = Limits.pullback.fst c (specMap R Γ(T, W)) := by
  let := algebraOfHom t W
  have key : hW.fromSpec ≫ t = specMap R Γ(T, W) := by
    have h := IsAffineOpen.SpecMap_appLE_fromSpec t (isAffineOpen_top _) hW (le_top (a := W))
    rw [IsAffineOpen.fromSpec_top, Scheme.isoSpec_Spec_inv, ← Spec.map_comp] at h
    change _ = Spec.map (CommRingCat.ofHom ((Scheme.ΓSpecIso (.of R)).inv ≫ t.appLE ⊤ W le_top).hom)
    rw [CommRingCat.ofHom_hom]
    exact h.symm
  let e₁ : Limits.pullback (Limits.pullback.snd c t) hW.fromSpec ≅ Limits.pullback c (specMap R Γ(T,
      W)) :=
    pullbackLeftPullbackSndIso c t hW.fromSpec ≪≫ Limits.pullback.congrHom rfl key
  have h1 : IsPullback (Limits.pullback.fst (Limits.pullback.snd c t) hW.fromSpec)
      (Limits.pullback.snd (Limits.pullback.snd c t) hW.fromSpec) (Limits.pullback.snd c t)
          hW.fromSpec :=
    IsPullback.of_hasPullback _ _
  refine ⟨e₁.inv ≫ Limits.pullback.fst (Limits.pullback.snd c t) hW.fromSpec, ?_, ?_⟩
  swap
  · rw [Category.assoc, Iso.inv_comp_eq]
    simp only [e₁, Iso.trans_hom, Category.assoc, Limits.pullback.congrHom_hom]
    delta CategoryTheory.Limits.pullback.map
    rw [Limits.pullback.lift_fst, Category.comp_id, pullbackLeftPullbackSndIso_hom_fst]
  refine h1.of_iso e₁ (Iso.refl _) (Iso.refl _) (Iso.refl _) (by simp) ?_ (by simp) (by simp)
  simp only [e₁, Iso.trans_hom, Iso.refl_hom, Category.comp_id, Category.assoc,
      Limits.pullback.congrHom_hom]
  delta CategoryTheory.Limits.pullback.map
  rw [Limits.pullback.lift_snd, Category.comp_id, pullbackLeftPullbackSndIso_hom_snd]
theorem

finiteType_algebraOfHom_aux

L179 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_Modules_isLocallyFreeOfRank_pushforward_of_forall_exists_isPullback.AlgebraicGeometry.Scheme.TwoAffineOpenCover.finiteType_algebraOfHom_aux
Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_Modules_isLocallyFreeOfRank_pushforward_of_forall_exists_isPullback
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace TwoAffineOpenCover
open _root_.AlgebraicGeometry.Scheme.TwoAffineOpenCover
open _root_.AlgebraicGeometry.Scheme.TwoAffineOpenCover
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
Lean source · full declaration
theorem finiteType_algebraOfHom_aux {R : Type u} [CommRing R] {T : Scheme.{u}} (t : T ⟶ Spec (.of
  R))
    [LocallyOfFiniteType t] (W : T.Opens) (hW : IsAffineOpen W) :
    letI := algebraOfHom t W
    Algebra.FiniteType R Γ(T, W) := by
  let := algebraOfHom t W
  have h : (t.appLE ⊤ W le_top).hom.FiniteType :=
    t.finiteType_appLE (isAffineOpen_top _) hW le_top
  have h' : (Scheme.ΓSpecIso (.of R)).inv.hom.FiniteType :=
    RingHom.FiniteType.of_surjective _ (Scheme.ΓSpecIso (.of
        R)).symm.commRingCatIsoToRingEquiv.surjective
  rw [← RingHom.finiteType_algebraMap]
  exact h.comp h'
theorem

isLocallyFreeOfRank_pushforward_of_forall_exists_isPullback

L197 ↗
AlgebraicGeometry.Scheme.Modules.isLocallyFreeOfRank_pushforward_of_forall_exists_isPullback
Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_Modules_isLocallyFreeOfRank_pushforward_of_forall_exists_isPullback
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
open _root_.AlgebraicGeometry.Scheme.Modules AlgebraicGeometry.Scheme.Modules in
with_weak_namespace _root_.AlgebraicGeometry.Scheme.Modules
Lean source · statement
theorem isLocallyFreeOfRank_pushforward_of_forall_exists_isPullback
    {X T : Scheme.{u}} (π : X ⟶ T) (F : X.Modules) (n : ℕ)
    (h : ∀ y : T, ∃ (W : T.Opens), y ∈ W ∧ ∃ (W' X' : Scheme.{u}) (e : W' ≅ W.toScheme) (π' : X' ⟶
        W')
      (g' : X' ⟶ X), IsPullback g' π' π (e.hom ≫ W.ι) ∧
        Scheme.Modules.IsLocallyFreeOfRank n
          ((Scheme.Modules.pushforward π').obj ((Scheme.Modules.pullback g').obj F))) :
    Scheme.Modules.IsLocallyFreeOfRank n ((Scheme.Modules.pushforward π).obj F)

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