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Finiteness aspects of Gorenstein homological dimensions

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Bouchiba S.

Colloquium Mathematicum

2013

tom 131

nr 2

171-193

We present an alternative way of measuring the Gorenstein projective (resp., injective) dimension of modules via a new type of complete projective (resp., injective) resolutions. As an application, we easily recover well known theorems such as the Auslander-Bridger formula. Our approach allows us to relate the Gorenstein global dimension of a ring R to the cohomological invariants silp(R) and spli(R) introduced by Gedrich and Gruenberg by proving that leftG-gldim(R) = maxleftsilp(R), leftspli(R), recovering a recent theorem of [I. Emmanouil, J. Algebra 372 (2012), 376-396]. Moreover, this formula permits to recover the main theorem of [D. Bennis and N. Mahdou, Proc. Amer. Math. Soc. 138 (2010), 461-465]. Furthermore, we prove that, in the setting of a left and right Noetherian ring, the Gorenstein global dimension is left-right symmetric, generalizing a theorem of Enochs and Jenda. Finally, using recent work of I. Emmanouil and O. Talelli, we compute the Gorenstein global dimension for various types of rings such as commutative ℵ₀-Noetherian rings and group rings.

A variant theory for the Gorenstein flat dimension

100%

Bouchiba S.

Colloquium Mathematicum

2015

tom 140

nr 2

183-204

This paper discusses a variant theory for the Gorenstein flat dimension. Actually, since it is not yet known whether the category 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 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 $GGfd_{R}(M)$ 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 $GGfd_{R}(M) = Gid_{R}(Hom_{ℤ} (M,ℚ /ℤ))$ whenever $GGf_{R}(M)$ is finite. Also, we show that 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 $GGfd_{R}(M) = Gfd_{R}(M)$ 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 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 r-sfli(R) and l-sfli(R) by the formula GG-wgldim(R) = max{r-sfli(R),l-sfli(R)}.

Ograniczanie wyników

2 Colloquium Mathematicum

2 Bouchiba S.

1 2015

1 2013

Wyniki wyszukiwania

Finiteness aspects of Gorenstein homological dimensions

A variant theory for the Gorenstein flat dimension