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ISSN 1088-6850(e) ISSN 0002-9947(p)

Gorenstein projective dimension for complexes

Author: Oana Veliche
Journal: Trans. Amer. Math. Soc. 358 (2006), 1257-1283
MSC (2000): Primary 16E10, 18G25, 13D05; Secondary 13D25, 16E30, 16E45
Posted: May 26, 2005
MathSciNet review: 2187653
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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

1. Auslander, M., Bridger, M., Stable module theory, Mem. Amer. Math. Soc. 94 (1969). MR 0269685 (42:4580)
2. Avramov, L.L., Homological dimensions and related invariants of modules over local rings, Representations of Algebras, ICRA IX (Beijing, 2000), vol. I, Beijing Normal Univ. Press 2002, 1-39. MR 2067368
3. Avramov, L.L., Foxby, H.-B., Homological dimension of unbounded complexes, J. Pure Appl. Alg. 71 (1991), 129-155. MR 1117631 (93g:18017)
4. Avramov, L.L., Foxby, H.-B., Halperin, S., Differential graded homological algebra, Preprint, version of 09/11/2004.
5. Avramov, L.L., Gasharov, V.N., Peeva, I.V., Complete intersection dimension, Publ. Math. I.H.E.S., 86 (1997), 67-114. MR 1608565 (99c:13033)
6. Avramov, L.L., Martsinkovsky, A., Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension, Proc. London Math. Soc. (3) 85 (2002), 393-440.MR 1912056 (2003g:16009)
7. Bourbaki, N., Algèbre Chaptre 10, Algèbre Homologique, Masson, Paris, New York, 1980.MR 0610795 (82j:18022)
8. Bruns, W., Herzog, J., Cohen Macaylay ring, (revised edition), Advances in Mathematics, Vol. 39, Cambridge Univ. Press, Cambridge, UK, 1996.MR 1251956 (95h:13020)
9. Buchweitz, R.-O., Maximal Cohen-Macaulay modules and Tate cohomology over Gorenstein rings, Preprint, Univ. Hannover, 1986.
10. Cartan, H., Eilenberg, S., Homological Algebra, Princeton University Press, Princeton, NJ, reprint 1999. MR 1731415 (2000h:18022)
11. Cornick, J., Kropholler, P.H., On complete resolutions , Topology Appl. 78 (1997), 235-250. MR 1454602 (98k:20087)
12. Christensen, L.W., Gorenstein dimensions, Lecture Notes in Math. 1747, Springer, Berlin, 2000. MR 1799866 (2002e:13032)
13. Enochs, E.E., Jenda, O.M.G., Gorenstein injective and projective modules, Math. Z. 220, (1995), 611-633. MR 1363858 (97c:16011)
14. Enochs, E.E., Jenda, O.M.G., Relative Homological Algebra, De Gruyter Exp. Math. 30, De Gruyter, Berlin, 2000. MR 1753146 (2001h:16013)
15. Farrell, F.T., An extension of Tate cohomology to a class of infinite groups, J. Pure Appl. Alg. 10 (1977), 153-161. MR 0470103 (57:9870)
16. Gedrich, T.V., Gruenberg, K.W., Complete cohomological functors on groups, Topology Appl. 25 (1987), 203-223. MR 0884544 (89h:20073)
17. Gerko, A.A., On homological dimensions, Sb. Math. 192 (2001), 1165-1179. MR 1862245 (2002h:13024)
18. Goto, S., Vanishing of $\operatorname{Ext}_{A}^{i}(M,A)$ , J. Math. Kyoto Univ. 22 (1982), 481-484. MR 0674605 (84c:13019)
19. Holm, H., Gorenstein derived functors, Proc. Amer. Math. Soc., 132 (2004), 1913-1923. MR 2053961
20. Holm, H., Gorenstein homological dimensions, J. Pure Appl. Alg. 189 (2004), 167-193. MR 2038564 (2004k:16013)
21. Iversen, B., Cohomology of Sheaves, Springer-Verlag, Berlin-Heidelberg, 1986. MR 0842190 (87m:14013)
22. Sather-Wagstaff, S., Complete intersection dimension for complexes, J. Pure Appl. Alg. 190 (2004), 267-290. MR 2043332
23. Yassemi, S., Gorenstein dimension, Math. Scand. 77 (1995), 161-174. MR 1379262 (97d:13017)
24. Veliche, O., Construction of modules of finite Gorenstein dimension, J. Algebra 250 (2002), 427-449. MR 1899298 (2003e:13023)
25. Weibel, C.,A., An introduction to homological algebra, Cambridge Studies in advanced mathematics 38, Cambridge University Press, Cambridge, 1994. MR 1269324 (95f:18001)

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