Gorenstein injective dimension, Bass formula and Gorenstein rings

TL;DR

This paper characterizes Gorenstein rings via the finiteness of Gorenstein injective dimension of the residue field and extends the Bass formula to modules with finite Gorenstein injective dimension.

Contribution

It establishes a new characterization of Gorenstein rings using Gorenstein injective dimension and generalizes the Bass formula for finitely generated modules.

Findings

R is Gorenstein iff the Gorenstein injective dimension of k is finite.

A generalized Bass formula is proved for modules with finite Gorenstein injective dimension.

Improves Christensen's generalized Bass formula.

Abstract

Let $(R,\frak m, k)$ be a noetherian local ring. It is well-known that $R$ is regular if and only if the injective dimension of $k$ is finite. In this paper it is shown that $R$ is Gorenstein if and only if the Gorenstein injective dimension of $k$ is finite. On the other hand a generalized version of the so-called Bass formula is proved for finitely generated modules of finite Gorenstein injective dimension. It also improves Christensen's generalized Bass formula (cf. "Gorenstein dimensions", volume 1747 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000).