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The essential ideal is a Cohen-Macaulay module

Author: David J. Green
Journal: Proc. Amer. Math. Soc. 133 (2005), 3191-3197
MSC (2000): Primary 20J06; Secondary 13C14
Published electronically: May 9, 2005
MathSciNet review: 2160180
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Abstract: Let $G$ be a finite $p$-group which does not contain a rank two elementary abelian $p$-group as a direct factor. Then the ideal of essential classes in the mod-$p$ cohomology ring of $G$is a Cohen-Macaulay module whose Krull dimension is the $p$-rank of the centre of $G$. This basically answers in the affirmative a question posed by J. F. Carlson (Question 5.4 in Problems in the calculation of group cohomology, 1999).

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  • 1. Alejandro Adem and Dikran Karagueuzian, Essential cohomology of finite groups, Comment. Math. Helv. 72 (1997), no. 1, 101-109. MR 1456319 (98f:20038)
  • 2. Alejandro Adem and R. James Milgram, The mod $2$ cohomology rings of rank $3$ simple groups are Cohen-Macaulay, Prospects in topology (Princeton, NJ, 1994) (Frank Quinn, ed.), Ann. of Math. Stud., vol. 138, Princeton Univ. Press, Princeton, NJ, 1995, pp. 3-12. MR 1368650 (96m:20084)
  • 3. D. J. Benson, Polynomial invariants of finite groups, London Math. Soc. Lecture Note Series, vol. 190, Cambridge University Press, Cambridge, 1993. MR 1249931 (94j:13003)
  • 4. -, Representations and cohomology. II, second ed., Cambridge Studies in Advanced Math., vol. 31, Cambridge University Press, Cambridge, 1998.MR 1634407 (99f:20001b)
  • 5. -, Dickson invariants, regularity and computation in group cohomology, Illinois J. Math. 48 (2004), no. 1, 171-197. MR 2048221
  • 6. Carlos Broto and Hans-Werner Henn, Some remarks on central elementary abelian $p$-subgroups and cohomology of classifying spaces, Quart. J. Math. Oxford Ser. (2) 44 (1993), no. 174, 155-163. MR 1222371 (94c:57060)
  • 7. Jon F. Carlson, Depth and transfer maps in the cohomology of groups, Math. Z. 218 (1995), no. 3, 461-468. MR 1324540 (95m:20058)
  • 8. -, Problems in the calculation of group cohomology, Computational methods for representations of groups and algebras (Essen, 1997) (P. Dräxler, G. O. Michler, and C. M. Ringel, eds.), Birkhäuser, Basel, 1999, pp. 107-120.MR 1714605 (2001i:20111)
  • 9. -, Calculating group cohomology: Tests for completion, J. Symbolic Comput. 31 (2001), no. 1-2, 229-242. MR 1806218 (2002c:20083)
  • 10. Leonard Evens, The cohomology of groups, Oxford Univ. Press, Oxford, 1991.MR 1144017 (93i:20059)
  • 11. David J. Green, On Carlson's depth conjecture in group cohomology, Math. Z. 244 (2003), no. 4, 711-723. MR 2000456 (2004h:20072)
  • 12. Huynh Mui, The mod $p$ cohomology algebra of the extra-special group $E(p^3)$, Unpublished essay, 1982.
  • 13. Moss E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969. MR 0252485 (40:5705)

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Additional Information

David J. Green
Affiliation: Department of Mathematics, University of Wuppertal, D-42097 Wuppertal, Germany

Received by editor(s): February 27, 2004
Received by editor(s) in revised form: June 24, 2004
Published electronically: May 9, 2005
Communicated by: Paul Goerss
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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