File size: 2,847 Bytes
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 55 56 57 58 59 60 61 62 63 64 | import OAI.Algebra.Finitistic.Localization.NormalizedQuotientAction
import Mathlib.Algebra.FreeAlgebra
import Mathlib.Algebra.RingQuot
namespace OAI
noncomputable section
open CategoryTheory
namespace LittleFinitistic.QuadraticEncoding
universe u v
variable {k : Type u} [Field k] {d t : ℕ} (P : Relations k d t)
/-- The literal inhomogeneous relation polynomial. -/
def relationPolynomial (j : Fin t) : FreeAlgebra k (Fin d) :=
(∑ h, ∑ l, P.quadratic j h l • (FreeAlgebra.ι k h * FreeAlgebra.ι k l)) +
(∑ h, P.linear j h • FreeAlgebra.ι k h) + P.constant j • 1
def presentationRelation (a b : FreeAlgebra k (Fin d)) : Prop :=
∃ j, a = relationPolynomial P j ∧ b = 0
/-- The algebra with exactly the displayed finite quadratic presentation. -/
abbrev PresentedAlgebra := RingQuot (presentationRelation P)
def presentedGenerator (h : Fin d) : PresentedAlgebra P :=
RingQuot.mkAlgHom k (presentationRelation P) (FreeAlgebra.ι k h)
theorem presentedGenerator_satisfies : P.Satisfies (presentedGenerator P) := by
intro j
have h := RingQuot.mkAlgHom_rel k (s := presentationRelation P) ⟨j,rfl,rfl⟩
simpa only [relationPolynomial,map_add,map_sum,map_smul,map_mul,map_one,map_zero,
presentedGenerator] using h
variable {A : Type v} [Ring A] [Algebra k A]
/-- Descent through the actual presentation, with every relation checked. -/
def presentationLift (x : Fin d → A) (hx : P.Satisfies x) : PresentedAlgebra P →ₐ[k] A :=
RingQuot.liftAlgHom k ⟨FreeAlgebra.lift k x, by
rintro a b ⟨j,rfl,rfl⟩
simpa only [relationPolynomial,map_add,map_sum,map_smul,map_mul,map_one,map_zero,
FreeAlgebra.lift_ι_apply] using hx j⟩
@[simp] theorem presentationLift_generator (x : Fin d → A) (hx : P.Satisfies x) (h : Fin d) :
presentationLift P x hx (presentedGenerator P h) = x h := by
simp [presentationLift,presentedGenerator,RingQuot.liftAlgHom_mkAlgHom_apply]
/-- The manuscript's quotient action for its displayed presentation. -/
def quotientAction : PresentedAlgebra P →ₐ[k] End (vertexQ P 0) :=
presentationLift P (normalizedGenerator P) (normalizedGenerator_satisfies P)
@[simp] theorem quotientAction_generator (h : Fin d) :
quotientAction P (presentedGenerator P h) = normalizedGenerator P h :=
presentationLift_generator _ _ _ _
/-- Transport the action along a genuine presentation isomorphism for R. -/
def quotientActionOfPresentation (e : PresentedAlgebra P ≃ₐ[k] A) :
A →ₐ[k] End (vertexQ P 0) := (quotientAction P).comp e.symm.toAlgHom
@[simp] theorem quotientActionOfPresentation_generator (e : PresentedAlgebra P ≃ₐ[k] A)
(h : Fin d) : quotientActionOfPresentation P e (e (presentedGenerator P h)) =
normalizedGenerator P h := by simp [quotientActionOfPresentation]
end LittleFinitistic.QuadraticEncoding
end
end OAI
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