Modules with injective cohomology, and local duality for a finite group

New York Journal of Mathematics

Volumn 7, Issue , 2001, Pages 201-215

  Benson, David ^a  

^a UNIVERSITY OF GEORGIA   (United States)

Author keywords

Cohen Macaulay ring; Depth; Grothendieck duality; Idempotent module; Injective module; Koszul complex; Local cohomology; Matlis duality; Molecular representation; Spectral sequence; Tate cohomology

Indexed keywords

EID: 3042601159     PISSN: 10769803     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (23)

1. A. Adem and R. J. Milgram, The mod 2 cohomology rings of rank 3 simple groups are Cohen-Macaulay, Prospects in Topology (Princeton, NJ, 1994), Ann. of Math. Studies, vol. 138, Princeton Univ. Press, 1995, pp. 3-12, MR 96m:20084, Zbl 0928.20042.
2. D. J. Benson and J. F. Carlson, Projective resolutions and Poincaré duality complexes, Trans. Amer. Math. Soc. 132 (1994), 447-488, MR 94f:20100, Zbl 0816.20044.
3. D. J. Benson, J. F. Carlson, and J. Rickard, Complexity and varieties for infinitely generated modules, II, Math. Proc. Camb. Phil. Soc. 120 (1996), 597-615, MR 97f:20008, Zbl 0888.20003.
4. D. J. Benson and G. Ph. Gnacadja, Phantom maps and purity in modular representation theory, I, Fundamenta Mathematicae 161 (1999), 37-91, MR 2000k:20013, Zbl 0944.20004.
5. D. J. Benson and H. Krause, Pure injectives and the spectrum of the cohomology ring of a finite group, to appear in Journal für die Reine und Angewandte Mathematik.
6. M. P. Brodmann and R. Y. Sharp, Local Cohomology, Cambridge Studies in Advanced Mathematics, vol. 60, Cambridge University Press, 1998, Zbl 0903.13006.
7. W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, 1993, MR 95h:13020, Zbl 0909.13005.
8. J. F. Carlson and W. W. Wheeler, Homomorphisms in higher complexity quotient categories, Group representations: cohomology, group actions and topology (Seattle, WA, 1996), Proc. Symp. Pure Math., vol. 63, American Math. Society, 1998, pp. 115-155, MR 99k:20019, Zbl 0899.20001.
9. J. Duflot, Depth and equivariant cohomology, Comment. Math. Helvetici 56 (1981), 627-637, MR 83g:57029, Zbl 0493.55003.
10. L. Evens, The cohomology ring of a finite group, Trans. Amer. Math. Soc. 101 (1961), 224-239, MR 25 #1191, Zbl 0104.25101.
11. J. P. C. Greenlees, Commutative algebra in group cohomology, J. Pure & Applied Algebra 98 (1995), 151-162, MR 96f:20084, Zbl 0826.55012.
12. J. P. C. Greenlees and G. Lyubeznik, Rings with a local cohomology theorem and applications to cohomology rings of groups, J. Pure Appl. Algebra 149 (2000), no. 3, 267-285, MR 2001f:13025, Zbl 0965.13012.
13. A. Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA2), North Holland, Amsterdam, 1962, MR 57 #16294, Zbl 0197.47202.
14. A. Grothendieck, Local Cohomology (notes by R. Hartshorne), Lecture Notes in Mathematics, vol. 41, Springer-Verlag, Berlin/New York, 1967, MR 37 #219, Zbl 0185.49202.
15. E. Matlis, Injective modules over Noetherian rings, Pacific Journal of Math. 8 (1958), 511-528, MR 20 #5800, Zbl 0084.26601.
16. H. Matsumura, Commutative Ring Theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, 1986, MR 90i:13001, Zbl 0603.13001.
17. M. Nagata, Local Rings, Wiley, New York, 1962, MR 27 #5790, Zbl 0123.03402.
18. D. G. Quillen, The mod 2 cohomology rings of extra-special 2-groups and the spinor groups, Math. Ann. 194 (1971), 197-212, MR 44 #7582, Zbl 0225.55015.
19. D. G. Quillen, The spectrum of an equivariant cohomology ring, I, Ann. of Math. 94 (1971), 549-572, MR 45 #7743, Zbl 0247.57013.
20. D. G. Quillen, The spectrum of an equivariant cohomology ring, II, Ann. of Math. 94 (1971), 573-602, MR 45 #7743, Zbl 0247.57013.
21. D. G. Quillen, On the cohomology and K-theory of the general linear groups over a finite field, Ann. of Math. 96 (1972), 552-586, MR 47 #3565, Zbl 0249.18022.
22. J. Rickard, Idempotent modules in the stable category, J. London Math. Soc. 56 (1997), 149-170, MR 98d:20058, Zbl 0910.20034.
23. B. B. Venkov, Cohomology algebras for some classifying spaces, Dokl. Akad. Nauk. SSSR 127 (1959), 943-944, MR 21 #7500, Zbl 0099.38802.