| import OAI.Algebra.Finitistic.Selected.SelectedWindow |
| import OAI.Algebra.Finitistic.Ordinary.OrdinaryBlockBimodule |
| import OAI.Algebra.Finitistic.Ordinary.OrdinaryRightProjective |
| import Mathlib.Algebra.TrivSqZeroExt.Basic |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits |
| namespace LittleFinitistic.SelectedOrdinaryAlgebra |
| open SelectedFixedDiagram QuadraticEncoding |
| open OrdinarySimulation |
|
|
| /-- A fixed support interval containing degree zero, chosen before any test module. -/ |
| def a : ℤ := min (SelectedWindow.lo-2) 0 |
| def b : ℤ := max SelectedWindow.hi 0 |
| def length : ℕ := (b-a).toNat |
|
|
| theorem a_le_zero : a ≤ 0 := min_le_right _ _ |
| theorem b_nonneg : 0 ≤ b := le_max_right _ _ |
| theorem endpoints : a + (length : ℤ) = b := by |
| have ha := a_le_zero |
| have hb := b_nonneg |
| dsimp [length] |
| omega |
|
|
| theorem support (n : ℤ) (hn : n < a ∨ b < n) : |
| IsZero (SelectedRectification.bimodule.X n) := by |
| apply SelectedWindow.bounded n |
| dsimp [a,b] at hn |
| omega |
| |
| instance term_finite (n : ℤ) : |
| FiniteDimensional ℂ ((underlyingComplex (k := ℂ) SelectedRectification.bimodule).X n) := |
| SelectedWindow.verified.finite n |
| instance term_projective (n : ℤ) : |
| Module.Projective (B P)ᵐᵒᵖ ((underlyingRightComplex SelectedRectification.bimodule).X n) := by |
| let : Projective ((underlyingRightComplex SelectedRectification.bimodule).X n) := |
| SelectedWindow.verified.projective n |
| exact ModuleCat.projective_of_module_projective _ |
|
|
| /-- The actual finite-dimensional ordinary algebra and off-diagonal bimodule. -/ |
| abbrev D := AlgebraD ℂ length (B P) |
| abbrev O := BimoduleO SelectedRectification.bimodule a length |
| abbrev X := BimoduleX SelectedRectification.bimodule a length |
| instance base_finite : FiniteDimensional ℂ (B P) := QuadraticEncoding.finiteDimensional P |
| instance D_finite : FiniteDimensional ℂ D := inferInstance |
| instance O_finite : FiniteDimensional ℂ O := inferInstance |
| instance X_finite : FiniteDimensional ℂ X := ordinary_finite _ _ _ |
| instance O_right_projective : Module.Projective (B P)ᵐᵒᵖ O := |
| ordinaryO_right_projective _ _ _ |
|
|
| /-- The manuscript's one fixed square-zero algebra. No projective-dimension |
| conclusion is asserted here. -/ |
| abbrev A := TrivSqZeroExt D X |
| abbrev augmentation : A →ₐ[ℂ] D := TrivSqZeroExt.fstHom ℂ D X |
| |
| instance A_finite : FiniteDimensional ℂ A := |
| inferInstanceAs (FiniteDimensional ℂ (D × X)) |
| end LittleFinitistic.SelectedOrdinaryAlgebra |
|
|
| end |
|
|
| end OAI |
|
|
