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.