Let (R,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.
