Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 
 

 

Finiteness criteria in Gorenstein homological algebra


Authors: Ioannis Emmanouil and Olympia Talelli
Journal: Trans. Amer. Math. Soc. 366 (2014), 6329-6351
MSC (2010): Primary 16E10, 18G20; Secondary 20J05
DOI: https://doi.org/10.1090/S0002-9947-2014-06007-8
Published electronically: July 2, 2014
MathSciNet review: 3267012
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] Anneaux de Gorenstein, et torsion en algèbre commutative, Séminaire d'Algèbre Commutative dirigé par Pierre Samuel, 1966/67. Texte rédigé, d'après des exposés de Maurice Auslander, Marquerite Mangeney, Christian Peskine et Lucien Szpiro. École Normale Supérieure de Jeunes Filles, Secrétariat mathématique, Paris, 1967 (French). MR 0225844 (37 #1435)
  • [2] 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 (42 #4580)
  • [3] 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 (91h:13010)
  • [4] 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 (2003g:16009), https://doi.org/10.1112/S0024611502013527
  • [5] Luchezar L. Avramov and Oana Veliche, Stable cohomology over local rings, Adv. Math. 213 (2007), no. 1, 93-139. MR 2331239 (2008f:13020), https://doi.org/10.1016/j.aim.2006.11.012
  • [6] Abdolnaser Bahlekeh, Fotini Dembegioti, and Olympia Talelli, Gorenstein dimension and proper actions, Bull. Lond. Math. Soc. 41 (2009), no. 5, 859-871. MR 2557467 (2010k:20084), https://doi.org/10.1112/blms/bdp063
  • [7] D. J. Benson and Jon F. Carlson, Products in negative cohomology, J. Pure Appl. Algebra 82 (1992), no. 2, 107-129. MR 1182934 (93i:20058), https://doi.org/10.1016/0022-4049(92)90116-W
  • [8] Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York, 1982. MR 672956 (83k:20002)
  • [9] Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N.J., 1956. MR 0077480 (17,1040e)
  • [10] Lars Winther Christensen, Gorenstein dimensions, Lecture Notes in Mathematics, vol. 1747, Springer-Verlag, Berlin, 2000. MR 1799866 (2002e:13032)
  • [11] 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 (2007h:13022), https://doi.org/10.1016/j.jalgebra.2005.12.007
  • [12] 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 (99k:20105), https://doi.org/10.1112/S0024610798005729
  • [13] Ioannis Emmanouil, On certain cohomological invariants of groups, Adv. Math. 225 (2010), no. 6, 3446-3462. MR 2729012 (2011m:20122), https://doi.org/10.1016/j.aim.2010.06.007
  • [14] Edgar E. Enochs and Overtoun M. G. Jenda, Gorenstein injective and projective modules, Math. Z. 220 (1995), no. 4, 611-633. MR 1363858 (97c:16011), https://doi.org/10.1007/BF02572634
  • [15] 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 0470103 (57 #9870)
  • [16] 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 (89h:20073), https://doi.org/10.1016/0166-8641(87)90015-0
  • [17] François Goichot, Homologie de Tate-Vogel équivariante, J. Pure Appl. Algebra 82 (1992), no. 1, 39-64 (French, with English summary). MR 1181092 (94d:55014), https://doi.org/10.1016/0022-4049(92)90009-5
  • [18] Henrik Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra 189 (2004), no. 1-3, 167-193. MR 2038564 (2004k:16013), https://doi.org/10.1016/j.jpaa.2003.11.007
  • [19] Henrik Holm, Rings with finite Gorenstein injective dimension, Proc. Amer. Math. Soc. 132 (2004), no. 5, 1279-1283 (electronic). MR 2053331 (2005a:13031), https://doi.org/10.1090/S0002-9939-03-07466-5
  • [20] Peter H. Kropholler, On groups of type $ ({\rm FP})_\infty $, J. Pure Appl. Algebra 90 (1993), no. 1, 55-67. MR 1246274 (94j:20051b), https://doi.org/10.1016/0022-4049(93)90136-H
  • [21] Guido Mislin, Tate cohomology for arbitrary groups via satellites, Topology Appl. 56 (1994), no. 3, 293-300. MR 1269317 (95c:20072), https://doi.org/10.1016/0166-8641(94)90081-7
  • [22] Brita E. A. Nucinkis, Complete cohomology for arbitrary rings using injectives, J. Pure Appl. Algebra 131 (1998), no. 3, 297-318. MR 1637023 (99h:16012), https://doi.org/10.1016/S0022-4049(97)00082-0
  • [23] Olympia Talelli, Periodicity in group cohomology and complete resolutions, Bull. London Math. Soc. 37 (2005), no. 4, 547-554. MR 2143734 (2006d:20095), https://doi.org/10.1112/S0024609305004273
  • [24] C. T. C. Wall, Finiteness conditions for $ {\rm CW}$-complexes, Ann. of Math. (2) 81 (1965), 56-69. MR 0171284 (30 #1515)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 16E10, 18G20, 20J05

Retrieve articles in all journals with MSC (2010): 16E10, 18G20, 20J05


Additional Information

Ioannis Emmanouil
Affiliation: Department of Mathematics, University of Athens, Athens 15784, Greece
Email: emmanoui@math.uoa.gr

Olympia Talelli
Affiliation: Department of Mathematics, University of Athens, Athens 15784, Greece
Email: otalelli@math.uoa.gr

DOI: https://doi.org/10.1090/S0002-9947-2014-06007-8
Received by editor(s): March 6, 2012
Received by editor(s) in revised form: October 14, 2012
Published electronically: July 2, 2014
Additional Notes: This research was supported by a GSRT/Greece excellence grant, cofunded by the ESF/EU and National Resources
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society