Gorenstein analogue of Auslander’s theorem on the global dimension. (English) Zbl 1317.13030
The authors prove that the Gorenstein analogue of the well-known theorem of Auslander on global dimension holds true. More precisely, the main result of this paper indicates that, for a commutative ring \(R\) and a positive integer \(n\), \(\text{Ggldim}(R)\leq n\) if and only if \(\text{Gpd}_RM\leq n\) for every finitely generated \(R\)-module \(M\), if and only if \(\text{Gpd}_R(R/I)\leq n\) for every ideal \(I\) of \(R\). Here \(\text{Gpd}_RM\) is denoted the Gorenstein projective dimension of \(M\) over \(R\), and \(\text{Ggldim}(R)\) is the denoted the Gorenstein global dimension of \(R\).
Reviewer: Parviz Sahandi (Tabriz)
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.) |
| 03E75 | Applications of set theory |
Keywords:
Gorenstein global dimension of rings; Gorenstein projective and injective dimensions of modules; \(n\)-Gorenstein ringsReferences:
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