Quillen stratification for Hochschild cohomology of blocks

Pakianathan, Jonathan; Witherspoon, Sarah

doi:10.1090/S0002-9947-05-04012-2

Quillen stratification for Hochschild cohomology of blocks
HTML articles powered by AMS MathViewer

by Jonathan Pakianathan and Sarah Witherspoon; and with an appendix by Stephen F. Siegel PDF

Trans. Amer. Math. Soc. 358 (2006), 2897-2916 Request permission

Abstract:

We decompose the maximal ideal spectrum of the Hochschild cohomology ring of a block of a finite group into a disjoint union of subvarieties corresponding to elementary abelian $p$-subgroups of a defect group. These subvarieties are described in terms of group cohomological varieties and the Alperin-Broué correspondence on blocks. Our description leads in particular to a homeomorphism between the Hochschild variety of the principal block and the group cohomological variety. The proofs require a result of Stephen F. Siegel, given in the Appendix, which states that nilpotency in Hochschild cohomology is detected on elementary abelian $p$-subgroups.

References

J. Alperin and Michel Broué, Local methods in block theory, Ann. of Math. (2) 110 (1979), no. 1, 143–157. MR 541333, DOI 10.2307/1971248
J. L. Alperin and L. Evens, Representations, resolutions and Quillen’s dimension theorem, J. Pure Appl. Algebra 22 (1981), no. 1, 1–9. MR 621284, DOI 10.1016/0022-4049(81)90079-7
J. L. Alperin and L. Evens, Varieties and elementary abelian groups, J. Pure Appl. Algebra 26 (1982), no. 3, 221–227. MR 678520, DOI 10.1016/0022-4049(82)90106-2
M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
George S. Avrunin and Leonard L. Scott, Quillen stratification for modules, Invent. Math. 66 (1982), no. 2, 277–286. MR 656624, DOI 10.1007/BF01389395
D. J. Benson, Representations and cohomology. I, Cambridge Studies in Advanced Mathematics, vol. 30, Cambridge University Press, Cambridge, 1991. Basic representation theory of finite groups and associative algebras. MR 1110581
D. J. Benson, Representations and cohomology. II, Cambridge Studies in Advanced Mathematics, vol. 31, Cambridge University Press, Cambridge, 1991. Cohomology of groups and modules. MR 1156302
Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. II, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1987. With applications to finite groups and orders; A Wiley-Interscience Publication. MR 892316
Karin Erdmann and Nicole Snashall, On Hochschild cohomology of preprojective algebras. I, II, J. Algebra 205 (1998), no. 2, 391–412, 413–434. MR 1632808, DOI 10.1006/jabr.1998.7547
Leonard Evens, The cohomology of groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1991. Oxford Science Publications. MR 1144017
Murray Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. (2) 78 (1963), 267–288. MR 161898, DOI 10.2307/1970343
Edward L. Green, Nicole Snashall, and Øyvind Solberg, The Hochschild cohomology ring of a selfinjective algebra of finite representation type, Proc. Amer. Math. Soc. 131 (2003), no. 11, 3387–3393. MR 1990627, DOI 10.1090/S0002-9939-03-06912-0
Markus Linckelmann, Transfer in Hochschild cohomology of blocks of finite groups, Algebr. Represent. Theory 2 (1999), no. 2, 107–135. MR 1702272, DOI 10.1023/A:1009979222100
Markus Linckelmann, Varieties in block theory, J. Algebra 215 (1999), no. 2, 460–480. MR 1686201, DOI 10.1006/jabr.1998.7724
Markus Linckelmann, Quillen stratification for block varieties, J. Pure Appl. Algebra 172 (2002), no. 2-3, 257–270. MR 1906878, DOI 10.1016/S0022-4049(01)00143-8
Jonathan Pakianathan and Sarah Witherspoon, Hochschild cohomology and Linckelmann cohomology for blocks of finite groups, J. Pure Appl. Algebra 178 (2003), no. 1, 87–100. MR 1947968, DOI 10.1016/S0022-4049(02)00189-5
Daniel Quillen, The spectrum of an equivariant cohomology ring. I, II, Ann. of Math. (2) 94 (1971), 549–572; ibid. (2) 94 (1971), 573–602. MR 298694, DOI 10.2307/1970770
D. Quillen and B. B. Venkov, Cohomology of finite groups and elementary abelian subgroups, Topology 11 (1972), 317–318. MR 294506, DOI 10.1016/0040-9383(72)90017-1
Jean-Pierre Serre, Sur la dimension cohomologique des groupes profinis, Topology 3 (1965), 413–420 (French). MR 180619, DOI 10.1016/0040-9383(65)90006-6
S. F. Siegel, Varieties for Hochschild cohomology of group algebras and blocks, unpublished notes, 1997.
Stephen F. Siegel and Sarah J. Witherspoon, The Hochschild cohomology ring of a group algebra, Proc. London Math. Soc. (3) 79 (1999), no. 1, 131–157. MR 1687539, DOI 10.1112/S0024611599011958
Stephen F. Siegel and Sarah J. Witherspoon, The Hochschild cohomology ring of a cyclic block, Proc. Amer. Math. Soc. 128 (2000), no. 5, 1263–1268. MR 1691003, DOI 10.1090/S0002-9939-00-05466-6
Nicole Snashall and Øyvind Solberg, Support varieties and Hochschild cohomology rings, Proc. London Math. Soc. (3) 88 (2004), no. 3, 705–732. MR 2044054, DOI 10.1112/S002461150301459X
A. Suslin, The detection theorem for finite group schemes, to appear in J. Pure Appl. Algebra.
Jacques Thévenaz, Some remarks on $G$-functors and the Brauer morphism, J. Reine Angew. Math. 384 (1988), 24–56. MR 929977, DOI 10.1515/crll.1988.384.24