COLLOQUIUM MATHEMATICUM VOL. 131 2013 NO. 2 FINITENESS ASPECTS OF GORENSTEIN HOMOLOGICAL DIMENSIONS BY SAMIR BOUCHIBA (Meknes) Abstract. We present an alternative way of measuring the Gorenstein projective (resp., injective) dimension of modules via a new type of complete projective (resp., in- jective) 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) = max{leftsilp(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. 1. Introduction. Throughout this paper, R denotes an associative ring with identity element. All modules, if not otherwise specified, are assumed to be left R-modules. Also, for any R-module A, Z (A) denotes the set of all zerodivisors of A. Recall that Gorenstein projective (resp., injective) modules originate from the classical notion of projective (resp., injective) modules, being im- ages and kernels of the differentials of complete projective (resp., injective) resolutions. Specifically, a module M is said to be Gorenstein projective if there exists an exact sequence of projective modules, called a complete projective resolution, P := ···→ P 2 → P 1 → P 0 → P −1 → P −2 →··· , such that P remains exact after applying the functor Hom R (−,P ) for each projective module P and M := Im(P 0 → P −1 ). Gorenstein injective mod- ules are defined dually. These new concepts allow Enochs and Jenda [17] to 2010 Mathematics Subject Classification : Primary 13D02, 13D05; Secondary 16E05, 16E10. Key words and phrases : Gorenstein projective dimension, Gorenstein injective dimension, Gorenstein global dimension, Gorenstein weak global dimension. DOI: 10.4064/cm131-2-2 [171] c  Instytut Matematyczny PAN, 2013