A variant theory for the Gorenstein flat dimension

Samir Bouchiba¹

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

Colloquium Mathematicum, Tome 140 (2015) no. 2, pp. 183-204

Résumé

This paper discusses a variant theory for the Gorenstein flat dimension. Actually, since it is not yet known whether the category $\mathcal {GF}(R)$ of Gorenstein flat modules over a ring $R$ is projectively resolving or not, it appears legitimate to seek alternate ways of measuring the Gorenstein flat dimension of modules which coincide with the usual one in the case where $\mathcal {GF}(R)$ is projectively resolving, on the one hand, and present nice behavior for an arbitrary ring $R$, on the other. In this paper, we introduce and study one of these candidates called the generalized Gorenstein flat dimension of a module $M$ and denoted by $\hbox{GGfd}_R(M)$ via considering exact sequences of modules of finite flat dimension. The new entity stems naturally from the very definition of Gorenstein flat modules. It turns out that the generalized Gorenstein flat dimension enjoys nice behavior in the general setting. First, for each $R$-module $M$, we prove that $\hbox{GGfd} _R(M)=\hbox{Gid} _R(\mathop{\rm Hom} _{\mathbb{Z}} (M,\mathbb{Q}/\mathbb{Z} ) )$ whenever GGf$_R(M)$ is finite. Also, we show that $\mathcal {GF}(R)$ is projectively resolving if and only if the Gorenstein flat dimension and the generalized Gorenstein flat dimension coincide. In particular, if $R$ is a right coherent ring, then $\hbox{GGfd} _R(M)=\hbox{Gfd}_R(M)$ for any $R$-module $M$. Moreover, the global dimension associated to the generalized Gorenstein flat dimension, called the generalized Gorenstein weak global dimension and denoted by $\hbox{GG-wgldim}(R)$, turns out to be the best counterpart of the classical weak global dimension in Gorenstein homological algebra. In fact, it is left-right symmetric and it is related to the cohomological invariants $\hbox{r-sfli}(R)$ and $\hbox{l-sfli}(R)$ by the formula $$ \hbox{GG-wgldim}(R)=\max\{\hbox{r-sfli}(R),\hbox{l-sfli}(R)\}. $$

Détail

DOI : 10.4064/cm140-2-3

Keywords: paper discusses variant theory gorenstein flat dimension actually since yet known whether category mathcal gorenstein flat modules ring projectively resolving appears legitimate seek alternate ways measuring gorenstein flat dimension modules which coincide usual where mathcal projectively resolving present nice behavior arbitrary ring other paper introduce study these candidates called generalized gorenstein flat dimension module denoted hbox ggfd via considering exact sequences modules finite flat dimension entity stems naturally definition gorenstein flat modules turns out generalized gorenstein flat dimension enjoys nice behavior general setting first each r module prove hbox ggfd hbox gid mathop hom mathbb mathbb mathbb whenever ggf finite mathcal projectively resolving only gorenstein flat dimension generalized gorenstein flat dimension coincide particular right coherent ring hbox ggfd hbox gfd r module moreover global dimension associated generalized gorenstein flat dimension called generalized gorenstein weak global dimension denoted hbox gg wgldim turns out best counterpart classical weak global dimension gorenstein homological algebra left right symmetric related cohomological invariants hbox r sfli hbox l sfli formula hbox gg wgldim max hbox r sfli hbox l sfli