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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm131-2-2,
author = {Samir Bouchiba},
title = {Finiteness aspects of Gorenstein homological dimensions},
journal = {Colloquium Mathematicae},
volume = {131},
year = {2013},
pages = {171-193},
zbl = {06195625},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm131-2-2}
}
Samir Bouchiba. Finiteness aspects of Gorenstein homological dimensions. Colloquium Mathematicae, Tome 131 (2013) pp. 171-193. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm131-2-2/