On Gorenstein projective, injective and flat dimensions – a functorial description with applications. (English) Zbl 1104.13008
An extension of the classical concept of homological dimension is constituted by Gorenstein homological dimension. The present paper contributes to the study of Gorenstein projective, injective and flat dimensions by presenting some functorial descriptions and by enlarging the class of rings known to admit good criteria for finiteness with respect to these dimensions. There is proved that this class includes the rings encountered in commutative algebraic geometry and, in the noncommutative realm, the \(k\)-algebras with a dualizing complex. On the other hand, the paper gives some applications of the Gorenstein dimension. The most important are the following:
Theorem I. If the ring \(R\) has a dualizing complex, then for an \(R \)-module \(M\) the next two conditions are equivalent:
(i) \(M\) has finite Gorenstein projective dimension.
(ii) \(M\) has finite Gorenstein flat dimension.
Theorem II. If the ring \(R\) has a dualizing complex and \(N\) is a nonzero finitely generated \(R\)-module of finite Gorenstein dimension, then Gid\(_{R}N=\)depth\(R.\)
Theorem III. If the ring \(R\) has a dualizing complex, then any direct product of Gorenstein flat \(R\)-modules is Gorenstein flat, as well as any direct sum of Gorenstein injective \(R\)-modules is Gorenstein injective.
Theorem I. If the ring \(R\) has a dualizing complex, then for an \(R \)-module \(M\) the next two conditions are equivalent:
(i) \(M\) has finite Gorenstein projective dimension.
(ii) \(M\) has finite Gorenstein flat dimension.
Theorem II. If the ring \(R\) has a dualizing complex and \(N\) is a nonzero finitely generated \(R\)-module of finite Gorenstein dimension, then Gid\(_{R}N=\)depth\(R.\)
Theorem III. If the ring \(R\) has a dualizing complex, then any direct product of Gorenstein flat \(R\)-modules is Gorenstein flat, as well as any direct sum of Gorenstein injective \(R\)-modules is Gorenstein injective.
Reviewer: Marius Tarnauceanu (Iaşi)
MSC:
| 13D05 | Homological dimension and commutative rings |
| 16E10 | Homological dimension in associative algebras |
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