Generalized Gorenstein Categories

TL;DR

This paper introduces a generalized framework for Gorenstein categories within abelian categories, providing new characterizations and conditions, and applies these to relevant categories, including insights into the Wakamatsu tilting conjecture.

Contribution

It generalizes Gorenstein categories by defining one-sided $n$-$( ext{C}, ext{D})$-Gorenstein categories and offers new equivalent characterizations based on projective and injective dimensions.

Findings

New characterization of Gorenstein categories.

Necessary condition for Wakamatsu tilting conjecture.

Application to categories of interest.

Abstract

Let $\mathscr{A}$ be an abelian category and let $\mathscr{C}$ and $\mathscr{D}$ be additive subcategories of $\mathscr{A}$. As a generalization of Gorenstein categories, we introduce one-sided $n$-$(\C,\D)$-Gorenstein categories with $n\geq 0$. Under certain conditions, we give some equivalent characterizations of one-sided $n$-$(\C,\D)$-Gorenstein categories in term of the finiteness of projective and injective dimensions relative to one-sided Gorenstein subcategories, which induce some new equivalent characterizations of Gorenstein categories. Then we apply these results to categories of interest. In particular, a necessary condition is obtained for the validity of the Wakamatsu tilting conjecture.