Gorenstein homological dimension and Ext-depth of modules

Abstract

Let $(R,{\frak{m}},k)$ be a commutative Noetherian local ring. It is well-known that $R$ is regular if and only if the flat dimension of $k$ is finite. In this paper, we show that $R$ is Gorenstein if and only if the Gorenstein flat dimension of $k$ is finite. Also, we will show that if $R$ is a Cohen-Macaulay ring and $M$ is a Tor-finite $R$-module of finite Gorenstein flat dimension, then the depth of the ring is equal to the sum of the Gorenstein flat dimension and Ext-depth of $M$. As a consequence, we get that this formula holds for every syzygy of a finitely generated $R$-module over a Gorenstein local ring.