Gorenstein cohomology in abelian categories

We investigate relative cohomology functors on subcategories of abelian categories via Auslander-Buchweitz approximations and the resulting strict resolutions. We verify that certain comparison maps between these functors are isomorphisms and introduce a notion of perfection for this context. Our main theorem is a balance result for relative cohomology that simultaneously recovers theorems of Holm and the current authors as special cases.

💡 Research Summary

The paper develops a unified framework for relative cohomology in an arbitrary abelian category A by exploiting Auslander‑Buchweitz approximations and the notion of “strict” resolutions. The authors fix four full additive subcategories X, Y, W, V with the following structural hypotheses: X and Y are closed under extensions; W is an injective cogenerator for X; V is a projective generator for Y; and orthogonality conditions W ⊥ Y and X ⊥ V hold (i.e., Ext¹_A(W,Y)=0 and Ext¹_A(X,V)=0).

Strict WX‑resolutions and properness.
For any object M∈A, a bounded strict W X‑resolution is a bounded X‑resolution whose terms in positive degrees belong to W. The authors show (Lemma 3.2) that when W is an injective cogenerator for X, every bounded strict W X‑resolution is automatically X‑proper, and the associated WX‑approximation (0→K→X₀→M→0 with K∈res W, X₀∈X) is Hom_A(X,–)‑exact. Dually, strict Y V‑coresolutions are Y‑proper when V is a projective generator for Y. These results guarantee the existence of proper resolutions for any object of finite X‑projective dimension (resp. finite Y‑injective dimension).

Comparison maps.
Section 4 studies the natural comparison morphisms
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