File size: 3,028 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 | import OAI.Algebra.Finitistic.Localization.FormalSummandAction
import OAI.Algebra.Finitistic.Localization.DiagramLifting
namespace OAI
namespace LittleFinitistic.FixedDiagram
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Idempotents
open FormalSummandAction
variable {k R : Type*} [CommRing k] [Ring R] [Algebra k R]
variable {C Q : Type*} [Category C] [Category Q] [Preadditive C] [Preadditive Q]
[Linear k C] [Linear k Q]
/-- Algebra relations are the homogeneous quadratic diagram relations under any action. -/
theorem action_relations {X : Q} (ρ : R →ₐ[k] End X) (na nb nr : ℕ)
(x : Fin na → R) (y : Fin nb → R) (c : Fin nr → Fin na → Fin nb → k)
(hc : ∀ t, ∑ a, ∑ b, c t a b • (y b * x a) = 0) :
∀ t, ∑ a, ∑ b, c t a b • ((ρ (x a) : X ⟶ X) ≫ ρ (y b)) = 0 := by
intro t
have h := congrArg ρ (hc t)
simpa only [map_sum,map_smul,map_mul,map_zero,End.mul_def] using h
variable (q : C ⥤ Q) (W : MorphismProperty C) [q.IsLocalization W]
[W.HasRightCalculusOfFractions] [q.Additive] [q.Linear k]
variable [HasShift Q ℤ] [HasZeroObject Q]
[∀ n : ℤ, (shiftFunctor Q n).Additive] [Pretriangulated Q]
[∀ n : ℤ, (shiftFunctor Q n).Linear k]
local instance : HasFiniteBiproducts (Karoubi Q) := Karoubi.karoubi_hasFiniteBiproducts
local instance : HasBinaryBiproducts (Karoubi Q) :=
hasBinaryBiproducts_of_finite_biproducts (Karoubi Q)
include W in
/-- Fixed source-category objects and arrows, with every quadratic relation exact,
represent the odd double of the selecting action in the localization. -/
theorem fixed_diagram {L : Q} (θ : R →ₐ[k] End L)
(e : R) (he : e*e=e) (hec : ∀ r : R, e*r=r*e) (α : R →ₐ[k] R)
(na nb nr : ℕ) (x : Fin na → R) (y : Fin nb → R)
(c : Fin nr → Fin na → Fin nb → k)
(hc : ∀ t, ∑ a, ∑ b, c t a b • (y b * x a) = 0) :
∃ (V₀ : Q) (ρ₀ : R →ₐ[k] End V₀)
(i : (toKaroubi Q).obj V₀ ≅
summand θ e he ⊞ OddDouble.formalShift (summand θ e he) 3)
(D₀ D₁ D₂ : C) (φ₀ : q.obj D₀ ≅ V₀) (φ₁ : q.obj D₁ ≅ V₀)
(φ₂ : q.obj D₂ ≅ V₀) (f : Fin na → (D₀ ⟶ D₁)) (g : Fin nb → (D₁ ⟶ D₂)),
(∀ r : R, (toKaroubi Q).map (ρ₀ r) ≫ i.hom =
i.hom ≫ oddDoubleAction θ e he hec α r) ∧
(∀ a, q.map (f a) ≫ φ₁.hom = φ₀.hom ≫ ρ₀ (x a)) ∧
(∀ b, q.map (g b) ≫ φ₂.hom = φ₁.hom ≫ ρ₀ (y b)) ∧
(∀ t, ∑ a, ∑ b, c t a b • (f a ≫ g b) = 0) := by
obtain ⟨V₀,ρ₀,i,hi⟩ := fixed_old_object_with_action θ e he hec α
obtain ⟨D₀,D₁,D₂,φ₀,φ₁,φ₂,f,g,hf,hg,hrel⟩ :=
DiagramLifting.finite_quadratic_diagram_lifting q W na nb nr V₀ V₀ V₀
(fun a => ρ₀ (x a)) (fun b => ρ₀ (y b)) c (action_relations ρ₀ na nb nr x y c hc)
exact ⟨V₀,ρ₀,i,D₀,D₁,D₂,φ₀,φ₁,φ₂,f,g,hi,hf,hg,hrel⟩
end LittleFinitistic.FixedDiagram
end OAI
|