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ae00908 | 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.Encoding.PresentationReindex
import OAI.Algebra.Finitistic.Selection.SelectionAlgebra
namespace OAI
namespace LittleFinitistic.SelectionEncoding
open QuadraticPresentation QuadraticEncoding
/-- Concrete finite coefficient data, with an exact presentation equivalence. -/
structure Encoding (k R : Type*) [Field k] [Ring R] [Algebra k R] where
generators : ℕ
equations : ℕ
relations : Relations k generators equations
equiv : Original k (Fin generators) (QuotientAction.polynomial relations) ≃ₐ[k] R
universe u v w
variable {k : Type u} {R : Type v} [Field k] [Ring R] [Algebra k R] (E : Encoding k R)
noncomputable def generator (h : Fin E.generators) : R :=
E.equiv (originalMap k (Fin E.generators) (QuotientAction.polynomial E.relations)
(FreeAlgebra.ι k h))
/-- These concrete elements of R satisfy every coefficient equation. -/
theorem generator_relations (j : Fin E.equations) :
(∑ h, ∑ l, E.relations.quadratic j h l • (generator E h * generator E l)) +
(∑ h, E.relations.linear j h • generator E h) + E.relations.constant j • (1 : R) = 0 := by
have h := congrArg E.equiv (original_relation k (Fin E.generators)
(QuotientAction.polynomial E.relations) j)
simpa only [QuotientAction.polynomial,map_add,map_sum,map_smul,map_mul,map_one,map_zero,generator]
using h
abbrev directedAlgebra := B E.relations
noncomputable instance : Ring (directedAlgebra E) :=
ThreeVertexAlgebra.instRing (composition E.relations)
theorem directedAlgebra_finiteDimensional : FiniteDimensional k (directedAlgebra E) :=
QuadraticEncoding.finiteDimensional E.relations
open CategoryTheory in
theorem directedAlgebra_globalDimension_le_two :
(⨆ N : ModuleCat.{max u w} (directedAlgebra E), projectiveDimension N) ≤ (2 : ℕ) ∧
(⨆ N : ModuleCat.{max u w} (directedAlgebra E)ᵐᵒᵖ, projectiveDimension N) ≤ (2 : ℕ) :=
QuadraticEncoding.globalDimension_le_two E.relations
/-- The actual Section 2 selecting algebra has one fixed concrete finite encoding. -/
noncomputable def auxiliaryEncoding : Encoding ℂ SelectionRing :=
Classical.choice (by
obtain ⟨d,t,P,⟨E⟩⟩ := PresentationReindex.auxiliaryGroupAlgebra_has_encoding
exact ⟨⟨d,t,P,E⟩⟩)
end LittleFinitistic.SelectionEncoding
end OAI
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