When the flat and Gorenstein flat dimensions coincide?

Samir Bouchiba¹

¹ Department of Mathematics Faculty of Sciences University Moulay Ismail 50000 Meknes, Morocco

Colloquium Mathematicum, Tome 147 (2017) no. 1, pp. 77-85.

Résumé

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 $.

Détail

DOI : 10.4064/cm6833-3-2016

Keywords: known given ring r module infty gid mathrm holm shows particular noetherian ring gid infty gorenstein dually r module mathrm infty gpd holm regarding gorenstein flat dimension there have appropriate analogs these theorems unique result vein states under strong hypothesis being right coherent ring finite right finitistic projective dimension gfd mathrm r module mathrm infty holm appropriate analogs above formulas gorenstein flat dimension actually general setting prove r module admitting short flat resolution rightarrow rightarrow n rightarrow n rightarrow cdots rightarrow rightarrow rightarrow gorenstein flat infty flat gfd where stands pontryagin dual hom mathbb mathbb mathbb module nbsp implies particular gf closed ring gfd r module infty dually prove gf closed gfd r module infty