Module

Action.lean

Nyll.AlgebraicCurves.Geometry.AbelianSchemes.GroupLaw.Action
8 declarations indexed from sourceView source ↗

Action

Imports3
Imported by1

Declarations

GoodReductionJacobian.RelativeGroupLaw.actionSource

The structure morphism of the source of the restricted group action.

Source context 5 commands

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

noncomputable section
universe u
open CategoryTheory CategoryTheory.Limits AlgebraicGeometry NeronModelInfra
namespace GoodReductionJacobian.RelativeGroupLaw
variable {R : Type u} [CommRing R] {J : Scheme.{u}} {f : J ⟶ Spec (CommRingCat.of R)}
abbrev actionSource (f : J ⟶ Spec (CommRingCat.of R)) {E : Scheme.{u}} (ι : E ⟶ J) :
    pullback (ι ≫ f) f ⟶ Spec (CommRingCat.of R)
GoodReductionJacobian.RelativeGroupLaw.actionFstPoint

The point given by the first projection followed by the inclusion into the group scheme.

Source context 5 commands

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

noncomputable section
universe u
open CategoryTheory CategoryTheory.Limits AlgebraicGeometry NeronModelInfra
namespace GoodReductionJacobian.RelativeGroupLaw
variable {R : Type u} [CommRing R] {J : Scheme.{u}} {f : J ⟶ Spec (CommRingCat.of R)}
def actionFstPoint (_L : RelativeGroupLaw R f) {E : Scheme.{u}} (ι : E ⟶ J) :
    SchemeHomOver (actionSource f ι) f
GoodReductionJacobian.RelativeGroupLaw.actionSndPoint

The point given by the second projection of the action source.

Source context 5 commands

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

noncomputable section
universe u
open CategoryTheory CategoryTheory.Limits AlgebraicGeometry NeronModelInfra
namespace GoodReductionJacobian.RelativeGroupLaw
variable {R : Type u} [CommRing R] {J : Scheme.{u}} {f : J ⟶ Spec (CommRingCat.of R)}
def actionSndPoint (_L : RelativeGroupLaw R f) {E : Scheme.{u}} (ι : E ⟶ J) :
    SchemeHomOver (actionSource f ι) f
GoodReductionJacobian.RelativeGroupLaw.action

The action obtained by multiplying the included point by the second group coordinate.

Source context 5 commands

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

noncomputable section
universe u
open CategoryTheory CategoryTheory.Limits AlgebraicGeometry NeronModelInfra
namespace GoodReductionJacobian.RelativeGroupLaw
variable {R : Type u} [CommRing R] {J : Scheme.{u}} {f : J ⟶ Spec (CommRingCat.of R)}
def action (L : RelativeGroupLaw R f) {E : Scheme.{u}} (ι : E ⟶ J) : pullback (ι ≫ f) f ⟶ J
GoodReductionJacobian.RelativeGroupLaw.action_def
Source context 5 commands

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

noncomputable section
universe u
open CategoryTheory CategoryTheory.Limits AlgebraicGeometry NeronModelInfra
namespace GoodReductionJacobian.RelativeGroupLaw
variable {R : Type u} [CommRing R] {J : Scheme.{u}} {f : J ⟶ Spec (CommRingCat.of R)}
theorem action_def (L : RelativeGroupLaw R f) {E : Scheme.{u}} (ι : E ⟶ J) :
    L.action ι = (L.mul (actionSource f ι) (L.actionFstPoint ι) (L.actionSndPoint ι)).1
GoodReductionJacobian.RelativeGroupLaw.actionFstPoint_coe
Source context 5 commands

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

noncomputable section
universe u
open CategoryTheory CategoryTheory.Limits AlgebraicGeometry NeronModelInfra
namespace GoodReductionJacobian.RelativeGroupLaw
variable {R : Type u} [CommRing R] {J : Scheme.{u}} {f : J ⟶ Spec (CommRingCat.of R)}
theorem actionFstPoint_coe (L : RelativeGroupLaw R f) {E : Scheme.{u}} (ι : E ⟶ J) :
    (L.actionFstPoint ι).1 = pullback.fst (ι ≫ f) f ≫ ι
GoodReductionJacobian.RelativeGroupLaw.actionSndPoint_coe
Source context 5 commands

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

noncomputable section
universe u
open CategoryTheory CategoryTheory.Limits AlgebraicGeometry NeronModelInfra
namespace GoodReductionJacobian.RelativeGroupLaw
variable {R : Type u} [CommRing R] {J : Scheme.{u}} {f : J ⟶ Spec (CommRingCat.of R)}
theorem actionSndPoint_coe (L : RelativeGroupLaw R f) {E : Scheme.{u}} (ι : E ⟶ J) :
    (L.actionSndPoint ι).1 = pullback.snd (ι ≫ f) f
GoodReductionJacobian.RelativeGroupLaw.action_over
Source context 5 commands

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

noncomputable section
universe u
open CategoryTheory CategoryTheory.Limits AlgebraicGeometry NeronModelInfra
namespace GoodReductionJacobian.RelativeGroupLaw
variable {R : Type u} [CommRing R] {J : Scheme.{u}} {f : J ⟶ Spec (CommRingCat.of R)}
theorem action_over (L : RelativeGroupLaw R f) {E : Scheme.{u}} (ι : E ⟶ J) :
    L.action ι ≫ f = actionSource f ι