Proper Resolutions and Gorenstein Categories

TL;DR

This paper develops a method to construct proper resolutions in abelian categories, proving the stability and closure properties of Gorenstein categories, and providing criteria for computing related dimensions.

Contribution

It introduces a new method for constructing proper resolutions in abelian categories and proves the stability and closure of Gorenstein categories, answering an open question.

Findings

Gorenstein category $ extbf{G}( extbf{C})$ is stable under certain conditions.

$ extbf{G}( extbf{C})$ is closed under direct summands.

Criteria for computing $ extbf{C}$-dimension and $ extbf{G}( extbf{C})$-dimension.

Abstract

Let $\mathscr{A}$ be an abelian category and $\mathscr{C}$ an additive full subcategory of $\mathscr{A}$. We provide a method to construct a proper $\mathscr{C}$-resolution (resp. coproper $\mathscr{C}$-coresolution) of one term in a short exact sequence in $\mathscr{A}$ from that of the other two terms. By using these constructions, we answer affirmatively an open question on the stability of the Gorenstein category $\mathcal{G}(\mathscr{C})$ posed by Sather-Wagstaff, Sharif and White; and also prove that $\mathcal{G}(\mathscr{C})$ is closed under direct summands. In addition, we obtain some criteria for computing the $\mathscr{C}$-dimension and the $\mathcal{G}(\mathscr{C)}$-dimension of an object in $\mathscr{A}$.