| import OAI.Algebra.Finitistic.Bar.Bar | |
| import Mathlib.LinearAlgebra.FiniteDimensional.Defs | |
| import Mathlib.RingTheory.Finiteness.Basic | |
| import Mathlib.RingTheory.TensorProduct.Finite | |
| namespace OAI | |
| noncomputable section | |
| open scoped TensorProduct | |
| open CategoryTheory | |
| namespace DerivedBar | |
| universe u | |
| variable (k A N : Type u) [Field k] [Ring A] [Algebra k A] | |
| [AddCommGroup N] [Module k N] [Module A N] [IsScalarTower k A N] | |
| [FiniteDimensional k A] [FiniteDimensional k N] | |
| /-- Every absolute bar word remains finite dimensional over the base field. -/ | |
| instance term_finiteDimensional : (n : ℕ) → FiniteDimensional k (term k A N n) | |
| | 0 => inferInstanceAs (FiniteDimensional k N) | |
| | n+1 => by | |
| let := term_finiteDimensional n | |
| exact inferInstanceAs (FiniteDimensional k (A ⊗[k] term k A N n)) | |
| instance term_finite (n : ℕ) : Module.Finite A (term k A N n) := | |
| Module.Finite.of_restrictScalars_finite k A (term k A N n) | |
| /-- The actual bar projective resolution has finitely generated free terms. -/ | |
| theorem projectiveResolution_finite (n : ℕ) : | |
| Module.Finite A ((projectiveResolution k A N).complex.X n) := | |
| term_finite k A N (n+1) | |
| omit [FiniteDimensional k A] [FiniteDimensional k N] in | |
| theorem projectiveResolution_free (n : ℕ) : | |
| Module.Free A ((projectiveResolution k A N).complex.X n) := | |
| term_free k A N n | |
| end DerivedBar | |
| end | |
| end OAI | |