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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 41 42 43 44 45 | import OAI.Algebra.AuslanderReiten.HomComplex.HomCochainExt
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Abelian
open scoped ModuleCat.Algebra
namespace ArExplicit.ExternalTensor
universe u
variable (k R : Type u) [Field k] [Ring R] [Algebra k R]
attribute [local instance 2000] moduleCatBaseModule moduleCatBaseTower
variable {X Y : ModuleCat.{u} R} (P : ProjectiveResolution X)
def homCochainQuotientEquiv (n : ℕ) :
(homCochain k R P.complex Y).homology n ≃ₗ[k]
(LinearMap.ker (homShort k R (Y := Y) P n).g.hom ⧸
LinearMap.range (homShort k R (Y := Y) P n).moduleCatToCycles) :=
((homCochain k R P.complex Y).homologyIsoSc' (n-1) n (n+1)
(by cases n <;> simp) (by simp)).toLinearEquiv.trans
(homShort k R (Y := Y) P n).moduleCatHomologyIso.toLinearEquiv
def homCochainCycleClass (n : ℕ)
(f : LinearMap.ker (homShort k R (Y := Y) P n).g.hom) :
(homCochain k R P.complex Y).homology n :=
(homCochainQuotientEquiv k R P n).symm (Submodule.Quotient.mk f)
theorem homCochainExtEquiv_cycleClass (n : ℕ)
(f : LinearMap.ker (homShort k R (Y := Y) P n).g.hom) :
homCochainExtEquiv k R P n (homCochainCycleClass k R P n f) =
P.extMk (ModuleCat.ofHom (X := P.complex.X n) (Y := Y) f.val)
(n+1) rfl (homCycle_closed k R P n f) := by
change ((Submodule.quotEquivOfEq _ _ (homCyclesToExt_kernel k R P n).symm).trans
((homCyclesToExt k R P n).quotKerEquivOfSurjective
(homCyclesToExt_surjective k R P n)))
((homCochainQuotientEquiv k R P n)
((homCochainQuotientEquiv k R P n).symm (Submodule.Quotient.mk f))) = _
rw [LinearEquiv.apply_symm_apply]
rfl
end ArExplicit.ExternalTensor
end
end OAI
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