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216fb31 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 | import OAI.Algebra.Finitistic.Localization.PresentedActionCompatibility
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
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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