Gorenstein projective dimension for complexes
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Abstract:
We define and study a notion of Gorenstein projective dimension for complexes of left modules over associative rings. For complexes of finite Gorenstein projective dimension we define and study a Tate cohomology theory. Tate cohomology groups have a natural transformation to classical Ext groups. In the case of module arguments, we show that these maps fit into a long exact sequence, where every third term is a relative cohomology group defined for left modules of finite Gorenstein projective dimension.References
- 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
- Luchezar L. Avramov, Homological dimensions and related invariants of modules over local rings, Representations of algebra. Vol. I, II, Beijing Norm. Univ. Press, Beijing, 2002, pp. 1–39. MR 2067368
- Luchezar L. Avramov and Hans-Bjørn Foxby, Homological dimensions of unbounded complexes, J. Pure Appl. Algebra 71 (1991), no. 2-3, 129–155. MR 1117631, DOI 10.1016/0022-4049(91)90144-Q
- Avramov, L.L., Foxby, H.-B., Halperin, S., Differential graded homological algebra, Preprint, version of 09/11/2004.
- Luchezar L. Avramov, Vesselin N. Gasharov, and Irena V. Peeva, Complete intersection dimension, Inst. Hautes Études Sci. Publ. Math. 86 (1997), 67–114 (1998). MR 1608565, DOI 10.1007/BF02698901
- 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
- Nicolas Bourbaki, Éléments de mathématique, Masson, Paris, 1980 (French). Algèbre. Chapitre 10. Algèbre homologique. [Algebra. Chapter 10. Homological algebra]. MR 610795
- Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
- Buchweitz, R.-O., Maximal Cohen-Macaulay modules and Tate cohomology over Gorenstein rings, Preprint, Univ. Hannover, 1986.
- Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1999. With an appendix by David A. Buchsbaum; Reprint of the 1956 original. MR 1731415
- Jonathan Cornick and Peter H. Kropholler, On complete resolutions, Topology Appl. 78 (1997), no. 3, 235–250. MR 1454602, DOI 10.1016/S0166-8641(96)00126-5
- Lars Winther Christensen, Gorenstein dimensions, Lecture Notes in Mathematics, vol. 1747, Springer-Verlag, Berlin, 2000. MR 1799866, DOI 10.1007/BFb0103980
- 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
- Edgar E. Enochs and Overtoun M. G. Jenda, Relative homological algebra, De Gruyter Expositions in Mathematics, vol. 30, Walter de Gruyter & Co., Berlin, 2000. MR 1753146, DOI 10.1515/9783110803662
- 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
- A. A. Gerko, On homological dimensions, Mat. Sb. 192 (2001), no. 8, 79–94 (Russian, with Russian summary); English transl., Sb. Math. 192 (2001), no. 7-8, 1165–1179. MR 1862245, DOI 10.1070/SM2001v192n08ABEH000587
- Shiro Goto, Vanishing of $\textrm {Ext}^{i}_{A}(M,\,A)$, J. Math. Kyoto Univ. 22 (1982/83), no. 3, 481–484. MR 674605, DOI 10.1215/kjm/1250521731
- Henrik Holm, Gorenstein derived functors, Proc. Amer. Math. Soc. 132 (2004), no. 7, 1913–1923. MR 2053961, DOI 10.1090/S0002-9939-04-07317-4
- 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
- Birger Iversen, Cohomology of sheaves, Universitext, Springer-Verlag, Berlin, 1986. MR 842190, DOI 10.1007/978-3-642-82783-9
- Sean Sather-Wagstaff, Complete intersection dimensions for complexes, J. Pure Appl. Algebra 190 (2004), no. 1-3, 267–290. MR 2043332, DOI 10.1016/j.jpaa.2003.09.002
- Siamak Yassemi, G-dimension, Math. Scand. 77 (1995), no. 2, 161–174. MR 1379262, DOI 10.7146/math.scand.a-12557
- Oana Veliche, Construction of modules with finite homological dimensions, J. Algebra 250 (2002), no. 2, 427–449. MR 1899298, DOI 10.1006/jabr.2001.9100
- Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324, DOI 10.1017/CBO9781139644136
Additional Information
- Oana Veliche
- Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
- Email: oveliche@math.purdue.edu, oveliche@math.utah.edu
- Received by editor(s): October 8, 2003
- Received by editor(s) in revised form: May 8, 2004
- Published electronically: May 26, 2005
- © Copyright 2005 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 358 (2006), 1257-1283
- MSC (2000): Primary 16E10, 18G25, 13D05; Secondary 13D25, 16E30, 16E45
- DOI: https://doi.org/10.1090/S0002-9947-05-03771-2
- MathSciNet review: 2187653