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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 | 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
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