Generalized Gorenstein Categories

Abstract

Let $\mathcal{A}$ be an abelian category and let $\mathcal{C}$ and $\mathcal{D}$ be additive subcategories of $\mathcal{A}$. As a generalization of Gorenstein categories, we introduce one-sided $n$-$(\C,\D)$-Gorenstein categories with $n\ge 0$. 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.

31 pages; accepted for publication in Journal of Algebra; typos corrected when proofreading