| import OAI.Algebra.AuslanderReiten.Stable.RawSourceCoordinates |
| import OAI.Algebra.AuslanderReiten.Stable.StableIsomorphism |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace ArExplicit.CompleteComparison |
| open CategoryTheory |
| open CochainComplex CochainComplex.HomComplex |
| open scoped ModuleCat.Algebra |
| universe u |
| variable {k R : Type u} [Field k] [Ring R] [Algebra k R] |
| variable (K L : CochainComplex (ModuleCat.{u} R) ℤ) |
| variable (hK : TotallyAcyclicComplex (chainOfCochain K)) |
| variable (hL : ExactProjectiveComplex (chainOfCochain L)) |
|
|
| def rawSourceStableEquiv : CohomologyClass K L 1 ≃+ StableHom R (RawSyzygy K) (CochainCokernel L) := |
| (fullHomSourceStableLinearEquiv (k := k) K L 1 hK hL).toAddEquiv.trans |
| (((StableIsomorphism.ofLinearEquiv (rawSyzygyEquiv K).symm).sourceAddEquiv).trans |
| (StableIsomorphism.ofLinearEquiv (cochainCokernelLinearEquiv L)).targetAddEquiv) |
|
|
| @[simp] |
| theorem rawSourceStableEquiv_mk (x : Cocycle K L 1) : |
| rawSourceStableEquiv (k := k) K L hK hL (CohomologyClass.mk x) = |
| StableHom.classOf R _ _ (rawSourceCocycleMap x) := by |
| simp only [rawSourceStableEquiv, AddEquiv.trans_apply, |
| LinearEquiv.coe_toAddEquiv, |
| StableIsomorphism.sourceAddEquiv, StableIsomorphism.targetAddEquiv, |
| StableIsomorphism.ofLinearEquiv, AddEquiv.coe_mk, Equiv.coe_fn_mk] |
| erw [fullHomSourceStableLinearEquiv_mk hK hL x] |
| rw [StableHom.comp_classOf, StableHom.comp_classOf] |
| rfl |
|
|
| end ArExplicit.CompleteComparison |
|
|
| end |
|
|
| end OAI |
|
|
