A variant theory for the Gorenstein flat dimension
Tom 140 / 2015
Streszczenie
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 $$ \hbox{GG-wgldim}(R)=\max\{\hbox{r-sfli}(R),\hbox{l-sfli}(R)\}. $$
