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Global Gorenstein dimensions of polynomial rings and of direct products of rings. (English) Zbl 1186.13007

Over any ring \(R\), E. E. Enochs and O. M. G. Jenda [Math. Z. 220, No. 4, 611–633 (1995; Zbl 0845.16005)] introduced the class of Gorenstein projective (left) \(R\)-modules. The Gorenstein projective dimension \(\text{Gpd}_RM\) of a (left) \(R\)-module \(M\) is the length of the shortest resolution of \(M\) consisting of Gorenstein projective (left) \(R\)-modules. The (left) Gorenstein global dimension of \(R\), denoted by \(\text{G-gldim}(R)\), is the supremum of \(\text{Gpd}_RM\), where \(M\) ranges over all (left) \(R\)-modules.
In this paper it is proved that \(\text{G-gldim}(R[X]) = \text{G-gldim}(R)+1\) if \(R\) is commutative. It is also proved that \(\text{G-gldim}(R\times S) = \max\{\text{G-gldim}(R),\text{G-gldim}(S)\}\) if \(R\) and \(S\) are commutative.
Similar results are obtained for the weak Gorenstein global dimension.

MSC:

13D02 Syzygies, resolutions, complexes and commutative rings
13D05 Homological dimension and commutative rings
13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.)

Citations:

Zbl 0845.16005