zbMATH — the first resource for mathematics

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
Keywords:
strongly; Gorenstein; projective; injective; flat
Full Text:
References:
 [1] Anderson, F.W.; Fuller, K.R., () [2] M. Auslander, Anneaux de Gorenstein et torsion en algèbre commutative, Secrétariat mathématique, Paris, 1967, in: Séminaire d’algèbre commutative dirigé par Pierre Samuel, 1966/67. Texte rédigé, d’après des exposés de Maurice Auslander, par Marquerite Mangeney, Christian Peskine et Lucien Szpiro, Ecole Normale Superieure de Jeunes Filles [3] Auslander, M.; Bridger, M., Stable module theory, () · Zbl 0204.36402 [4] Avramov, L.L.; Martsinkovsky, A., Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension, Proc. London math. soc. (3), 85, 2, 393-440, (2002) · Zbl 1047.16002 [5] Bass, H., Finitistic dimension and a homological generalization of semi-primary rings, Trans. amer. math. soc., 95, 466-488, (1960) · Zbl 0094.02201 [6] Bourbaki, N., Algèbre homologique, (1980), Masson Paris, (Chapitre 10) · Zbl 0455.18010 [7] Christensen, L.W., () [8] Christensen, L.W.; Frankild, A.; Holm, H., On Gorenstein projective, injective and flat dimensions — a functorial description with applications, J. algebra, 302, 231-279, (2006) · Zbl 1104.13008 [9] Endo, Sh., On flat modules over commutative rings, J. math. soc. Japan, 14, 3, 284-291, (1962) · Zbl 0108.26301 [10] Enochs, E.; Jenda, O., On Gorenstein injective modules, Comm. algebra, 21, 10, 3489-3501, (1993) · Zbl 0783.13011 [11] Enochs, E.; Jenda, O., Gorenstein injective and projective modules, Math. Z., 220, 4, 611-633, (1995) · Zbl 0845.16005 [12] Enochs, E.; Jenda, O.; Torrecillas, B., Gorenstein flat modules, Nanjing daxue xuebao shuxue Bannian kan, 10, 1, 1-9, (1993) · Zbl 0794.16001 [13] Enochs, E.; Jenda, O.; Xu, J., Foxby duality and Gorenstein injective and projective modules, Trans. amer. math. soc., 348, 8, 3223-3234, (1996) · Zbl 0862.13004 [14] Facchini, A.; Herbera, D.; Sakhajev, I., Finitely generated modules and a characterization of semiperfect rings, Comm. algebra, 31, 4195-4215, (2003) · Zbl 1032.16003 [15] H.-B. Foxby, Gorenstein dimension over Cohen-Macaulay rings, in: W. Bruns (Ed.), Proceedings of International Conference on Commutative Algebra, Universität Onsabrück, 1994 · Zbl 0834.13014 [16] Glaz, S., () [17] Holm, H., Gorenstein homological dimensions, J. pure appl. algebra, 189, 167-193, (2004) · Zbl 1050.16003 [18] Puninski, G.; Rothmaler, P., When every finitely generated flat module is projective, J. algebra, 277, 542-558, (2004) · Zbl 1077.16003 [19] Rotman, J., () [20] Sakhajev, I., On a criterion of projectivity of finitely generated flat modules, Izv. vuzov, 10, 68-75, (1991) [21] Vasconcelos, W.V., On finitely generated flat modules, Trans. amer. math. soc., 138, 505-512, (1969) · Zbl 0175.03603 [22] Xu, J., ()
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.