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2c7d975 | 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 | import OAI.Algebra.AuslanderReiten.Complete.CompleteTailMap
import OAI.Algebra.AuslanderReiten.Stable.StableExtNaturality
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Abelian
open scoped ModuleCat.Algebra
namespace ArExplicit
open CompleteComparison
universe u
variable {k R : Type u} [Field k] [Ring R] [Algebra k R]
variable {P Q : ChainComplex (ModuleCat.{u} R) ℤ}
variable (hP : TotallyAcyclicComplex P) (hQ : TotallyAcyclicComplex Q)
variable (f : P ⟶ Q) (N : ModuleCat.{u} R)
theorem positiveExtStableEquiv_precomp (n : ℕ)
(x : Ext (ModuleCat.of R (Cokernel Q 0)) N (n+1)) :
positiveExtStableEquiv (k := k) hP N n
((Ext.mk₀ (ModuleCat.ofHom (cokernelMap f 0))).comp x (zero_add _)) =
StableHom.precomp (cokernelMap f ((n+1 : ℕ) : ℤ))
(positiveExtStableEquiv (k := k) hQ N n x) := by
obtain ⟨g, hg, rfl⟩ := (completeExtResolution hQ).extMk_surjective x (n+2) rfl
rw [ProjectiveResolution.mk₀_comp_extMk g (n+2) rfl hg (completeTailHom f hP hQ),
positiveExtStableEquiv_extMk, positiveExtStableEquiv_extMk]
change StableHom.classOf R _ _ _ = StableHom.precomp _ (StableHom.classOf R _ _ _)
rw [StableHom.precomp_classOf]
apply congrArg (StableHom.classOf R _ _)
apply LinearMap.ext
intro z
obtain ⟨z, rfl⟩ := Submodule.mkQ_surjective
(LinearMap.range (P.d (((n+1 : ℕ) : ℤ)+1) ((n+1 : ℕ) : ℤ)).hom) z
rfl
end ArExplicit
end
end OAI
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