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Strongly Gorenstein projective, injective, and flat modules. (English) Zbl 1118.13014
Let \(R\) be a commutative ring with identity. A complete projective resolution of the form P \(= \ldots ^{f \atop \rightarrow} P ^{f \atop \rightarrow} P ^{f \atop \rightarrow} P ^{f \atop \rightarrow} \ldots \) is said to be a strongly complete projective resolution and is denoted by (P\(, f).\) Let \(M\) be an \(R\)-module. If \(M \cong\ker f\) for some strongly complete projective resolution (P \(, f)\) , then \(M\) is said to be strongly Gorenstein projective (\(SG\)-projective). Strongly complete injective resolutions and strongly Gorenstein injective (\(SG\)-injective modules are defined in a dual way. Direct sums (resp. direct products) of strongly Gorenstein projectives (resp. strongly Gorenstein injectives) are shown to be strongly Gorenstein projective (resp. strongly Gorenstein injective). The class of all strongly Gorenstein projectives (resp. strongly Gorenstein injectives) is shown to be strictly between the classes of all projectives (resp. injectives) and the class of all Gorenstein projectives (resp. Gorenstein injectives).
In one of the principal theorems of the paper, the Gorenstein projectives (resp. Gorenstein injectives) are characterized as the direct summands of the strongly Gorenstein projectives (resp. strongly Gorenstein injectives). This result is reminiscent of the characterization of projectives as direct summands of free modules. Both the strongly Gorenstein projectives and the finitely generated strongly Gorenstein projectives are characterized in terms of the existence of certain short exact sequences and properties of Ext or Hom.
Strongly complete flat resolutions (F \(, f)\) are defined in a similar fashion as strongly complete projective resolutions and an \(R\)-module \(M\) is called strongly Gorenstein flat (\(SG\)-flat) if \(M \cong\ker f\) for some strongly complete resolution (F \(, f)\). Then every flat module is strongly Gorenstein flat, but the converse is shown to be false in general but true if \(R\) has finite weak global dimension. Direct sums of strongly Gorenstein flat modules are shown to be strongly Gorenstein flat. Several characterizations of strongly Gorenstein flat modules in terms of the existence of certain short exact sequences and properties of Tor and \(\otimes\) are given.
The concepts of strongly Gorenstein flatness and projectivity are shown to coincide for finitely generated \(R\)-modules if \(R\) is local or an integral domain. \(S\)-rings are characterized as those rings for which every finitely generated strongly Gorenstein flat \(R\)-module is strongly Gorenstein projective.

MSC:
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
13C10 Projective and free modules and ideals in commutative rings
13C11 Injective and flat modules and ideals in commutative rings
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