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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 | import OAI.Algebra.Finitistic.Localization.FixedDiagram
import OAI.Algebra.Finitistic.Localization.SourceShiftLinear
import OAI.Algebra.Finitistic.Encoding.RelationBasisAction
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Limits CategoryTheory.Idempotents
namespace LittleFinitistic.ConcreteFixedDiagram
open SelectionEncoding RelationBasisCoefficients RelationBasisAction FormalSummandAction
universe u v
variable {k : Type u} [Field k] {R : Type v} [Ring R] [Algebra k R]
variable (E : Encoding k R)
local instance : HasFiniteBiproducts (Karoubi (SourceVerdier.Q E.relations)) :=
Karoubi.karoubi_hasFiniteBiproducts
local instance : HasBinaryBiproducts (Karoubi (SourceVerdier.Q E.relations)) :=
hasBinaryBiproducts_of_finite_biproducts _
/-- The fixed odd-double diagram in the actual Kᵇ(proj-B) and its actual Verdier
quotient. Every independent relation of the constructed path algebra is included. -/
theorem fixed_diagram {L : SourceVerdier.Q E.relations} (θ : R →ₐ[k] End L)
(e : R) (he : e*e=e) (hec : ∀ r : R, e*r=r*e) (α : R →ₐ[k] R) :
∃ (V₀ : SourceVerdier.Q E.relations) (ρ₀ : R →ₐ[k] End V₀)
(i : (toKaroubi (SourceVerdier.Q E.relations)).obj V₀ ≅
summand θ e he ⊞ OddDouble.formalShift (summand θ e he) 3)
(D₀ D₁ D₂ : SourceVerdier.K E.relations)
(φ₀ : (SourceVerdier.q E.relations).obj D₀ ≅ V₀)
(φ₁ : (SourceVerdier.q E.relations).obj D₁ ≅ V₀)
(φ₂ : (SourceVerdier.q E.relations).obj D₂ ≅ V₀)
(f : Fin (E.generators+1) → (D₀ ⟶ D₁))
(g : Fin (E.generators+1) → (D₁ ⟶ D₂)),
(∀ r : R, (toKaroubi (SourceVerdier.Q E.relations)).map (ρ₀ r) ≫ i.hom =
i.hom ≫ oddDoubleAction θ e he hec α r) ∧
(∀ a, (SourceVerdier.q E.relations).map (f a) ≫ φ₁.hom =
φ₀.hom ≫ ρ₀ (arrow (generator E) a)) ∧
(∀ b, (SourceVerdier.q E.relations).map (g b) ≫ φ₂.hom =
φ₁.hom ≫ ρ₀ (arrow (generator E) b)) ∧
(∀ r : RelationIndex E.relations,
∑ a, ∑ b, basisCoefficient E.relations r a b • (f a ≫ g b) = 0) :=
FixedDiagram.fixed_diagram (SourceVerdier.q E.relations) (SourceVerdier.Sigma E.relations)
θ e he hec α (E.generators+1) (E.generators+1)
(Module.finrank k (QuadraticEncoding.relationSpace E.relations))
(arrow (generator E)) (arrow (generator E)) (basisCoefficient E.relations)
(encoding_basis_action E)
end LittleFinitistic.ConcreteFixedDiagram
end
end OAI
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