Homological behavior of Auslander's $k$-Gorenstein rings

Abstract

In this paper we mainly study the homological properties of dual modules over $k$-Gorenstein rings. For a right quasi $k$-Gorenstein ring $\Lambda$, we show that the right self-injective dimension of $\Lambda$ is at most $k$ if and only if each $M\in$mod $\Lambda$ satisfying the condition that Ext${}_{\Lambda }^i(M,\Lambda )=0$ for any $1\le i\le k$ is reflexive. For an $\infty$-Gorenstein ring, we show that the big and small finitistic dimensions and the self-injective dimension of $\Lambda$ are identical. In addition, we show that if $\Lambda$ is a left quasi $\infty$-Gorenstein ring and $M\in$mod $\Lambda$ with grade$M$ finite, then Ext${}_{\Lambda }^i($Ext${}_{{\Lambda }^{op}}^i($Ext${}_{\Lambda }^{\mathrm{grade}M}(M,\Lambda ),\Lambda ),\Lambda )=0$ if and only if $i\ne$grade$M$. For a 2-Gorenstein ring $\Lambda$, we show that a non-zero proper left ideal $I$ of $\Lambda$ is reflexive if and only if $\Lambda /I$ has no non-zero pseudo-null submodule.

arXiv comment22 pages. The original title is "Duality in Auslander's $k$-Gorenstein rings". It has been accepted for publication in Algebras and Representation Theory