Module

nonempty_pullback_sectionTwist_iso_of_range_subset.lean

Nyll.JacobianVarieties.Picard.nonempty_pullback_sectionTwist_iso_of_range_subset

3 declarations indexed from sourceView source ↗

Module documentation

Pullback of a section twist

The ideal of a section commutes with base change. Its powers and inverse modules therefore give the comparison isomorphism for section twists along a base morphism.

Imports10

Imported by12

theorem

comap_pow

L64 ↗

JacobianVarieties.Picard.Proofs.SectionTwistBaseChange.SectTwistBC.comap_pow

Source context 10 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.SectionTwistBaseChange
attribute [-instance]
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
attribute [-instance]
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
attribute [-instance]
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
attribute [-simp] PresheafOfModules.freeεIso_hom_app PresheafOfModules.freeμIso_hom_app
universe u
open CategoryTheory CategoryTheory.Limits AlgebraicGeometry
open AlgebraicGeometry.RelPicard NeronModelInfra
namespace SectTwistBC

Lean source · statement

theorem comap_pow {X Y : Scheme.{u}} (I : Y.IdealSheafData) (f : X ⟶ Y) (n : ℕ) :
    (I ^ n).comap f = (I.comap f) ^ n

theorem

sectionIdeal_comap

L74 ↗

JacobianVarieties.Picard.Proofs.SectionTwistBaseChange.SectTwistBC.sectionIdeal_comap

Source context 11 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.SectionTwistBaseChange
attribute [-instance]
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
attribute [-instance]
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
attribute [-instance]
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
attribute [-simp] PresheafOfModules.freeεIso_hom_app PresheafOfModules.freeμIso_hom_app
universe u
open CategoryTheory CategoryTheory.Limits AlgebraicGeometry
open AlgebraicGeometry.RelPicard NeronModelInfra
namespace SectTwistBC
variable {R : Type u} [CommRing R] {C : Scheme.{u}}

Lean source · statement

theorem sectionIdeal_comap (c : C ⟶ Spec (CommRingCat.of R)) [IsSeparated c]
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c) {T T' : Scheme.{u}}
    {t : T ⟶ Spec (CommRingCat.of R)} {t' : T' ⟶ Spec (CommRingCat.of R)}
    (ψ : SchemeHomOver t' t) :
    (RelPicard.sectionIdeal c ε t).comap (RelPicard.baseChangeSnd c ψ) =
      RelPicard.sectionIdeal c ε t'

theorem