| import OAI.Algebra.Finitistic.Localization.PresentedActionCompatibility |
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|
| namespace OAI |
| noncomputable section |
| open CategoryTheory |
| namespace LittleFinitistic.QuadraticEncoding |
| universe u v w |
| variable {k : Type u} [Field k] {d t : ℕ} {P : Relations k d t} |
| variable {Y : Type v} [AddCommGroup Y] [Module k Y] (X : Representation P Y) |
|
|
| theorem quotientAction_square (r : PresentedAlgebra P) : |
| (boundedQuotientEvaluation X).map (quotientAction P r) ≫ (quotientVertexIso X 0).hom = |
| (quotientVertexIso X 0).hom ≫ (vectorStalk (k := k)).map |
| (liftedEndHom (givenAction X r)) := by |
| have h := congrArg (fun f => (quotientVertexIso X 0).hom ≫ f) |
| (AlgHom.congr_fun (quotientAction_evaluates X) r) |
| change (quotientVertexIso X 0).hom ≫ ((quotientVertexIso X 0).inv ≫ |
| (boundedQuotientEvaluation X).map (quotientAction P r) ≫ (quotientVertexIso X 0).hom) = |
| (quotientVertexIso X 0).hom ≫ (𝟙 _ ≫ (vectorStalk (k := k)).map |
| (liftedEndHom (givenAction X r)) ≫ 𝟙 _) at h |
| simpa only [← Category.assoc, Iso.hom_inv_id, Category.id_comp, Category.comp_id] using h |
|
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| variable {R : Type w} [Ring R] [Algebra k R] |
| variable (e : PresentedAlgebra P ≃ₐ[k] R) |
|
|
| def presentationRepresentation [Module R Y] [IsScalarTower k R Y] : Representation P Y := |
| Representation.ofAction (presentedGenerator P) (presentedGenerator_satisfies P) |
| ((Algebra.lsmul k k Y).comp e.toAlgHom) |
|
|
| lemma presentationRepresentation_givenAction [Module R Y] [IsScalarTower k R Y] : |
| givenAction (presentationRepresentation (Y := Y) e) = |
| (Algebra.lsmul k k Y).comp e.toAlgHom := by |
| apply RingQuot.ringQuot_ext' |
| apply FreeAlgebra.hom_ext |
| funext h |
| exact presentationLift_generator P _ _ h |
|
|
| /-- For a genuinely presented algebra R, every element of theta evaluates to its R-module action. -/ |
| theorem module_quotientAction_square [Module R Y] [IsScalarTower k R Y] (r : R) : |
| (boundedQuotientEvaluation (presentationRepresentation (Y := Y) e)).map |
| (quotientActionOfPresentation P e r) ≫ |
| (quotientVertexIso (presentationRepresentation (Y := Y) e) 0).hom = |
| (quotientVertexIso (presentationRepresentation (Y := Y) e) 0).hom ≫ |
| (vectorStalk (k := k)).map (liftedEndHom ((Algebra.lsmul k k Y) r)) := by |
| have h := quotientAction_square (presentationRepresentation (Y := Y) e) (e.symm r) |
| rw [presentationRepresentation_givenAction] at h |
| simpa only [AlgHom.comp_apply, AlgEquiv.coe_toAlgHom, AlgEquiv.apply_symm_apply, |
| quotientActionOfPresentation] using h |
| end LittleFinitistic.QuadraticEncoding |
|
|
| end |
|
|
| end OAI |
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|
