Naturality of the reverse coherent-defect Ext² calculation #
theorem
MagnitudeConjecture.RightModule.FiniteIndecomposableSkeleton.finiteCovariantDefectPresentationLinearEquiv_postcomp
{k A : Type u}
[Field k]
[IsAlgClosed k]
[Ring A]
[Algebra k A]
[FiniteDimensional k A]
[IsNoetherianRing Aᵐᵒᵖ]
(S : FiniteIndecomposableSkeleton k A)
[CategoryTheory.HasExt S.FiniteCovariantFunctor]
{K : CategoryTheory.ShortComplex FG}
(hK : K.ShortExact)
{X Y : S.IndecCategoryᵒᵖ}
(a : X ⟶ Y)
(q :
((S.finiteCovariantDefectLeftShortComplex K).X₁ ⟶ S.finiteCovariantRepresentableOnSkeleton.obj X) ⧸ ProjectivePresentationExt.presentationRange (S.finiteCovariantDefectLeftShortComplex K)
(S.finiteCovariantRepresentableOnSkeleton.obj X))
:
(S.finiteCovariantDefectPresentationLinearEquiv hK Y)
((ProjectivePresentationExt.postcompQuotient (S.finiteCovariantDefectLeftShortComplex K)
(S.finiteCovariantRepresentableOnSkeleton.map a))
q) = ((CategoryTheory.Abelian.Ext.mk₀ (S.finiteCovariantRepresentableOnSkeleton.map a)).postcompOfLinear k
(S.finiteCovariantDefect K) ⋯)
((S.finiteCovariantDefectPresentationLinearEquiv hK X) q)
Naturality of the reverse quotient model for Ext².