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Gorenstein dimensions over Cohen-Macaulay rings. (English) Zbl 0834.13014

Bruns, Winfried (ed.) et al., Commutative algebra: Extended abstracts of an international conference, July 27 - August 1, 1994, Vechta, Germany. Cloppenburg: Runge. Vechtaer Universitätsschriften. 13, 59-63 (1994).
This is a résumé of part of the author’s paper “Gorenstein dimensions over Noetherian rings” (in preparation). For any ring \(R\) the projective dimension \(\text{pd}_RM\) has proved to be a very important invariant of \(R\)-modules \(M\); M. Auslander has for finite (that is, finitely generated) modules \(M\) over a commutative noetherian ring \(R\) introduced the Gorenstein dimension \(G\text{-dim}_RM\), which is a finer invariant than \(\text{pd}_RM\) in the sense, that there is always an inequality \(G\text{-dim}_R M \leq \text{pd}_RM\) and equality holds when \(\text{pd}_RM\) is finite. Furthermore, \(G\text{- dim}_RM\) has many of the nice properties of \(\text{pd}_RM\). In section 2 the Gorenstein class \({\mathcal G}_0 (R)\) of \(R\) is introduced. It contains also non-finite modules, and a finite module \(M\) belongs to it if and only if \(G\text{-dim}_R M\) is finite. The modules in \({\mathcal G}_0 (R)\) have properties similar to finite modules of finite \(G\)- dimension. – Section 3 begins with the definition of Gorenstein- projective modules. This notion was introduced by E. Enochs and O. Jenda, and it gives rise to the definition of the Gorenstein- projective dimension \(\text{Gpd}_R M\) of an \(R\)-module \(M\). This is an extension of Auslander’s \(G\)-dimension: if \(M\) is finite then \(\text{Gdp}_R M = G\text{-dim}_R M\). Enochs and Xu have proved that \(\text{Gpd}_R M\) is finite if and only if \(M\) belongs to \({\mathcal G}_0 (R)\), and this enables us to prove that Gorenstein-projective dimensions do not increase under localization. – Section 4 is devoted to the study of the Gorenstein-flat dimension \(\text{Gfd}_R M\) of \(R\)- modules \(M\). In section 5 the dual Gorenstein class \({\mathcal I}_0 (R)\) is introduced. The duality is given by the functors \(D \otimes_R - : {\mathcal G}_0 (R) \to {\mathcal I}_0 (R)\) and \(\operatorname{Hom}_R (D,-) : {\mathcal I}_0 (R) \to {\mathcal G}_0 (R)\) providing an equivalence of the categories \({\mathcal G}_0 (R)\) and \({\mathcal I}_0 (R)\). Restrictions of these functors provide an equivalence of the category of modules of finite flat dimension and that of modules of finite injective dimension.
For the entire collection see [Zbl 0799.00021].

MSC:

13D05 Homological dimension and commutative rings
13C14 Cohen-Macaulay modules
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