When the flat and Gorenstein flat dimensions coincide?
Volume 147 / 2017
Abstract
It is well known that, given a ring $R$, if $M$ is an $R$-module such that pd$_R(M) \lt \infty $, then Gid$_R(M)= \mathrm {id}_R(M)$ (Holm, 2004). This shows in particular that if $R$ is a Noetherian ring such that Gid$(R) \lt \infty $, then $R$ is Gorenstein. Dually, if $M$ is an $R$-module such that $\mathrm {id}_R(M) \lt \infty $, then Gpd$_R(M)=$ pd$_R(M)$ (Holm, 2004). Regarding the Gorenstein flat dimension, there have been no appropriate analogs of these two theorems. The unique result, in this vein, states, under the strong hypothesis of $R$ being a left and right coherent ring with finite right finitistic projective dimension, that Gfd$_R(M)= \mathrm {fd}_R(M)$ for any $R$-module $M$ such that $\mathrm {id}_R(M) \lt \infty $ (Holm, 2004).
We give the appropriate analogs of the above two formulas for the Gorenstein flat dimension. Actually, in the general setting, we prove that if $M$ is an $R$-module admitting a short flat resolution $0\rightarrow K\rightarrow F_{n-1}\rightarrow F_{n-2}\rightarrow \cdots \rightarrow F_0\rightarrow M\rightarrow 0$ such that $K$ is Gorenstein flat and fd$_R(M^+) \lt \infty $, then $K$ is flat and Gfd$_R(M)=$ fd$_R(M)$, where $A^+$ stands for the Pontryagin dual Hom$_{\mathbb {Z}}(A,{\mathbb {Q}/\mathbb {Z}})$ of a module $A$. This implies, in particular, that if $R$ is a left GF-closed ring, then Gfd$_R(M)=$ fd$_R(M)$ for any $R$-module $M$ such that fd$_R(M^+) \lt \infty $. Dually, we prove that if $R$ is left GF-closed, then Gfd$_R(N^+)=$ fd$_R(N^+)$ for any $R$-module $N$ such that fd$_R(N) \lt \infty $.
