Finiteness of Gorenstein injective dimension of modules

Abstract

The Chouinard formula for the injective dimension of a module over a noetherian ring is extended to Gorenstein injective dimension. Specifically, if $M$ is a module of finite positive Gorenstein injective dimension over a commutative noetherian ring $R$, then its Gorenstein injective dimension is the supremum of ${depth} R_{\mathfrak {p}}- {width} _{R_\mathfrak {p}}M_{\mathfrak {p}}$, where $\mathfrak {p}$ runs through all prime ideals of $R$. It is also proved that if $M$ is finitely generated and non-zero, then its Gorenstein injective dimension is equal to the depth of the base ring. This generalizes the classical Bass formula for injective dimension.