File size: 3,776 Bytes
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 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 | import OAI.Algebra.Finitistic.Ordinary.OrdinaryCochain
import Mathlib.Algebra.Category.ModuleCat.ChangeOfRings
import Mathlib.Algebra.Homology.Additive
import Mathlib.RingTheory.TensorProduct.Maps
namespace OAI
noncomputable section
open CategoryTheory
open scoped TensorProduct
namespace LittleFinitistic.OrdinarySimulation
universe u
variable {k A : Type u} [Field k] [Ring A] [Algebra k A]
/-- Forgetting to k uses the actual restriction-of-scalars functor on complexes. -/
abbrev underlyingComplex (P : CochainComplex (ModuleCat.{u} A) ℤ) :
CochainComplex (ModuleCat.{u} k) ℤ :=
((ModuleCat.restrictScalars (algebraMap k A)).mapHomologicalComplex (.up ℤ)).obj P
/-- Ordinary finite compression of the actual dg A-module P on [a,a+l].
Take A=B tensor Bᵐᵒᵖ for a genuine dg B-bimodule. -/
def Compression (P : CochainComplex (ModuleCat.{u} A) ℤ) (a : ℤ) (l : ℕ) :=
∀ i : Fin (l+1), P.X (a + (i.val : ℤ))
namespace Compression
variable (P : CochainComplex (ModuleCat.{u} A) ℤ) (a : ℤ) (l : ℕ)
instance : AddCommGroup (Compression P a l) :=
inferInstanceAs (AddCommGroup (∀ i : Fin (l+1), P.X (a + (i.val : ℤ))))
instance : Module k (Compression P a l) :=
inferInstanceAs (Module k (OrdinaryTerms (intervalTerms (underlyingComplex (k := k) P) a l)))
instance : Module A (Compression P a l) :=
inferInstanceAs (Module A (∀ i : Fin (l+1), P.X (a + (i.val : ℤ))))
instance : Module (ChainAlgebra k l) (Compression P a l) :=
cochainModule (underlyingComplex (k := k) P) a l
instance : IsScalarTower k (ChainAlgebra k l) (Compression P a l) :=
OrdinaryTerms.scalarTower _ _ (intervalDifferential_square (underlyingComplex (k := k) P) a l)
instance scalarTowerA : IsScalarTower k A (Compression P a l) where
smul_assoc c r v := by
funext i
change (c • r) • v i = (algebraMap k A c) • r • v i
rw [Algebra.smul_def, mul_smul]
instance : SMulCommClass (ChainAlgebra k l) A (Compression P a l) where
smul_comm x r v := by
let (n : ℤ) : Module k (P.X n) := Module.compHom _ (algebraMap k A)
let (n : ℤ) : IsScalarTower k A (P.X n) := IsScalarTower.of_compHom k A (P.X n)
let (i : Fin (l+1)) : Module A (intervalTerms (underlyingComplex (k := k) P) a l i) := inferInstanceAs (Module A (P.X (a + (i.val : ℤ))))
let (i : Fin (l+1)) : SMulCommClass k A (intervalTerms (underlyingComplex (k := k) P) a l i) := inferInstanceAs (SMulCommClass k A (P.X (a + (i.val : ℤ))))
change OrdinaryTerms.action (intervalTerms (underlyingComplex (k := k) P) a l) (intervalDifferential (underlyingComplex (k := k) P) a l) x (r • v) = r • OrdinaryTerms.action (intervalTerms (underlyingComplex (k := k) P) a l) (intervalDifferential (underlyingComplex (k := k) P) a l) x v
apply OrdinaryTerms.action_commutes
intro i j r v
exact (P.d (a + (i.val : ℤ)) (a + (j.val : ℤ))).hom.map_smul r v
/-- The commuting actions give the actual tensor-algebra representation. -/
def tensorAction : (A ⊗[k] ChainAlgebra k l) →ₐ[k] Module.End k (Compression P a l) :=
Algebra.TensorProduct.lift (Algebra.lsmul k k (Compression P a l))
(Algebra.lsmul k k (Compression P a l)) (by
intro r x
ext v
change r • (x • v) = x • (r • v)
exact (smul_comm x r v).symm)
@[simp] theorem tensorAction_tmul (r : A) (x : ChainAlgebra k l) (v : Compression P a l) :
tensorAction (k := k) P a l (r ⊗ₜ[k] x) v = r • (x • v) := rfl
instance finiteDimensional [∀ n, FiniteDimensional k ((underlyingComplex (k := k) P).X n)] :
FiniteDimensional k (Compression P a l) :=
inferInstanceAs (FiniteDimensional k (∀ i : Fin (l+1), (underlyingComplex (k := k) P).X (a+(i.val:ℤ))))
end Compression
end LittleFinitistic.OrdinarySimulation
end
end OAI
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