The essential ideal is a Cohen–Macaulay module

Green, David

doi:10.1090/S0002-9939-05-07887-1

The essential ideal is a Cohen–Macaulay module
HTML articles powered by AMS MathViewer

by David J. Green

Proc. Amer. Math. Soc. 133 (2005), 3191-3197

DOI: https://doi.org/10.1090/S0002-9939-05-07887-1

Published electronically: May 9, 2005

PDF | Request permission

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).

References

Alejandro Adem and Dikran Karagueuzian, Essential cohomology of finite groups, Comment. Math. Helv. 72 (1997), no. 1, 101–109. MR 1456319, DOI 10.1007/PL00000361
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) Ann. of Math. Stud., vol. 138, Princeton Univ. Press, Princeton, NJ, 1995, pp. 3–12. MR 1368650
D. J. Benson, Polynomial invariants of finite groups, London Mathematical Society Lecture Note Series, vol. 190, Cambridge University Press, Cambridge, 1993. MR 1249931, DOI 10.1017/CBO9780511565809
D. J. Benson, Representations and cohomology. I, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 30, Cambridge University Press, Cambridge, 1998. Basic representation theory of finite groups and associative algebras. MR 1644252
Dave Benson, Dickson invariants, regularity and computation in group cohomology, Illinois J. Math. 48 (2004), no. 1, 171–197. MR 2048221
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, DOI 10.1093/qmath/44.2.155
Jon F. Carlson, Depth and transfer maps in the cohomology of groups, Math. Z. 218 (1995), no. 3, 461–468. MR 1324540, DOI 10.1007/BF02571916
Jon F. Carlson, Problems in the calculation of group cohomology, Computational methods for representations of groups and algebras (Essen, 1997) Progr. Math., vol. 173, Birkhäuser, Basel, 1999, pp. 107–120. MR 1714605
Jon F. Carlson, Calculating group cohomology: tests for completion, J. Symbolic Comput. 31 (2001), no. 1-2, 229–242. Computational algebra and number theory (Milwaukee, WI, 1996). MR 1806218, DOI 10.1006/jsco.1999.1003
Leonard Evens, The cohomology of groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1991. Oxford Science Publications. MR 1144017
David J. Green, On Carlson’s depth conjecture in group cohomology, Math. Z. 244 (2003), no. 4, 711–723. MR 2000456, DOI 10.1007/s00209-003-0517-z
Huỳnh Mui, The mod $p$ cohomology algebra of the extra-special group $E(p^3)$, Unpublished essay, 1982.
Moss E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969. MR 0252485