## Finiteness aspects of Gorenstein homological dimensions

### Volume 131 / 2013

#### Abstract

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.