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## Colloquium Mathematicum

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## When the flat and Gorenstein flat dimensions coincide?

### Tom 147 / 2017

Colloquium Mathematicum 147 (2017), 77-85 MSC: 13D02, 13D05, 13D07, 16E05, 16E10. DOI: 10.4064/cm6833-3-2016 Opublikowany online: 9 December 2016

#### Streszczenie

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$_R(N) \lt \infty$.