Finiteness aspects of Gorenstein homological dimensions

Samir Bouchiba¹

¹ Department of Mathematics University Moulay Ismail Meknes, Morocco

Colloquium Mathematicum, Tome 131 (2013) no. 2, pp. 171-193

Résumé

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 $\hbox {leftG-gldim}(R)= \max\{{\rm leftsilp}(R), {\rm leftspli}(R)\}$, 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 $\aleph _0$-Noetherian rings and group rings.

Détail

DOI : 10.4064/cm131-2-2

Keywords: present alternative measuring gorenstein projective resp injective dimension modules via type complete projective resp injective resolutions application easily recover known theorems auslander bridger formula approach allows relate gorenstein global dimension ring cohomological invariants silp spli introduced gedrich gruenberg proving hbox leftg gldim max leftsilp leftspli recovering recent theorem emmanouil algebra moreover formula permits recover main theorem bennis mahdou proc amer math soc furthermore prove setting right noetherian ring gorenstein global dimension left right symmetric generalizing theorem enochs jenda finally using recent work emmanouil talelli compute gorenstein global dimension various types rings commutative aleph noetherian rings group rings