Finiteness criteria in Gorenstein homological algebra

Emmanouil, Ioannis; Talelli, Olympia

doi:10.1090/S0002-9947-2014-06007-8

Finiteness criteria in Gorenstein homological algebra
HTML articles powered by AMS MathViewer

by Ioannis Emmanouil and Olympia Talelli PDF

Trans. Amer. Math. Soc. 366 (2014), 6329-6351 Request permission

Abstract:

In this paper, we examine the class of modules of finite Gorenstein projective dimension and study approximations of modules in that class by modules which are either Gorenstein projective or else have finite projective dimension. We also examine the relevance of complete cohomology in the study of modules of finite Gorenstein projective dimension and obtain, over group rings, such a finiteness criterion that involves only complete cohomology.

References

Anneaux de Gorenstein, et torsion en algèbre commutative, École Normale Supérieure de Jeunes Filles, Secrétariat mathématique, Paris, 1967 (French). Séminaire d’Algèbre Commutative dirigé par Pierre Samuel, 1966/67; Texte rédigé, d’après des exposés de Maurice Auslander, Marguerite Mangeney, Christian Peskine et Lucien Szpiro. MR 0225844
Maurice Auslander and Mark Bridger, Stable module theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, R.I., 1969. MR 0269685
Maurice Auslander and Ragnar-Olaf Buchweitz, The homological theory of maximal Cohen-Macaulay approximations, Mém. Soc. Math. France (N.S.) 38 (1989), 5–37 (English, with French summary). Colloque en l’honneur de Pierre Samuel (Orsay, 1987). MR 1044344
Luchezar L. Avramov and Alex Martsinkovsky, Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension, Proc. London Math. Soc. (3) 85 (2002), no. 2, 393–440. MR 1912056, DOI 10.1112/S0024611502013527
Luchezar L. Avramov and Oana Veliche, Stable cohomology over local rings, Adv. Math. 213 (2007), no. 1, 93–139. MR 2331239, DOI 10.1016/j.aim.2006.11.012
Abdolnaser Bahlekeh, Fotini Dembegioti, and Olympia Talelli, Gorenstein dimension and proper actions, Bull. Lond. Math. Soc. 41 (2009), no. 5, 859–871. MR 2557467, DOI 10.1112/blms/bdp063
D. J. Benson and Jon F. Carlson, Products in negative cohomology, J. Pure Appl. Algebra 82 (1992), no. 2, 107–129. MR 1182934, DOI 10.1016/0022-4049(92)90116-W
Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956, DOI 10.1007/978-1-4684-9327-6
Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
Lars Winther Christensen, Gorenstein dimensions, Lecture Notes in Mathematics, vol. 1747, Springer-Verlag, Berlin, 2000. MR 1799866, DOI 10.1007/BFb0103980
Lars Winther Christensen, Anders Frankild, and Henrik Holm, On Gorenstein projective, injective and flat dimensions—a functorial description with applications, J. Algebra 302 (2006), no. 1, 231–279. MR 2236602, DOI 10.1016/j.jalgebra.2005.12.007
Jonathan Cornick and Peter H. Kropholler, Homological finiteness conditions for modules over group algebras, J. London Math. Soc. (2) 58 (1998), no. 1, 49–62. MR 1666074, DOI 10.1112/S0024610798005729
Ioannis Emmanouil, On certain cohomological invariants of groups, Adv. Math. 225 (2010), no. 6, 3446–3462. MR 2729012, DOI 10.1016/j.aim.2010.06.007
Edgar E. Enochs and Overtoun M. G. Jenda, Gorenstein injective and projective modules, Math. Z. 220 (1995), no. 4, 611–633. MR 1363858, DOI 10.1007/BF02572634
F. Thomas Farrell, An extension of Tate cohomology to a class of infinite groups, J. Pure Appl. Algebra 10 (1977/78), no. 2, 153–161. MR 470103, DOI 10.1016/0022-4049(77)90018-4
T. V. Gedrich and K. W. Gruenberg, Complete cohomological functors on groups, Topology Appl. 25 (1987), no. 2, 203–223. Singapore topology conference (Singapore, 1985). MR 884544, DOI 10.1016/0166-8641(87)90015-0
François Goichot, Homologie de Tate-Vogel équivariante, J. Pure Appl. Algebra 82 (1992), no. 1, 39–64 (French, with English summary). MR 1181092, DOI 10.1016/0022-4049(92)90009-5
Henrik Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra 189 (2004), no. 1-3, 167–193. MR 2038564, DOI 10.1016/j.jpaa.2003.11.007
Henrik Holm, Rings with finite Gorenstein injective dimension, Proc. Amer. Math. Soc. 132 (2004), no. 5, 1279–1283. MR 2053331, DOI 10.1090/S0002-9939-03-07466-5
Peter H. Kropholler, On groups of type $(\textrm {FP})_\infty$, J. Pure Appl. Algebra 90 (1993), no. 1, 55–67. MR 1246274, DOI 10.1016/0022-4049(93)90136-H
Guido Mislin, Tate cohomology for arbitrary groups via satellites, Topology Appl. 56 (1994), no. 3, 293–300. MR 1269317, DOI 10.1016/0166-8641(94)90081-7
Brita E. A. Nucinkis, Complete cohomology for arbitrary rings using injectives, J. Pure Appl. Algebra 131 (1998), no. 3, 297–318. MR 1637023, DOI 10.1016/S0022-4049(97)00082-0
Olympia Talelli, Periodicity in group cohomology and complete resolutions, Bull. London Math. Soc. 37 (2005), no. 4, 547–554. MR 2143734, DOI 10.1112/S0024609305004273
C. T. C. Wall, Finiteness conditions for $\textrm {CW}$-complexes, Ann. of Math. (2) 81 (1965), 56–69. MR 171284, DOI 10.2307/1970382