Bockstein homomorphisms for Hochschild cohomology of group algebras and of block algebras of finite groups
-
Constantin-Cosmin Todea
Abstract
We give an explicit approach for Bockstein homomorphisms of the Hochschild cohomology of a group algebra and of a block algebra of a finite group and we show some properties. To give explicit definitions for these maps we use an additive decomposition and a product formula for the Hochschild cohomology of group algebras given by Siegel and Witherspoon in 1999. For an algebraically closed field k of characteristic p and a finite group G we prove an additive decomposition and a product formula for the cohomology algebra of a defect group of a block ideal of kG with coefficients in the source algebra of this block, and we define similar Bockstein homomorphisms.
References
[1] Benson D. J., Representations and Cohomology II: Cohomology of Groups and Modules, Cambridge University Press, Cambridge, 1991. 10.1017/CBO9780511623622Search in Google Scholar
[2] Chouinard L., Projectivity and relative projectivity over group rings, J. Pure Appl. Algebra 7 (1976), 278–302. 10.1016/0022-4049(76)90055-4Search in Google Scholar
[3] Eilenberg S. and Mac Lane S., Cohomology theory in abstract groups. I, Ann. of Math. (2) 48 (1947), 51–78. 10.2307/1969215Search in Google Scholar
[4] Evens L., The Cohomology of Groups, Clarendon Press, Oxford, 1991. 10.1093/oso/9780198535805.001.0001Search in Google Scholar
[5] Linckelmann M., Transfer in Hochschild cohomology of blocks of finite groups, Algebr. Represent. Theory 2 (1999), 107–135. 10.1023/A:1009979222100Search in Google Scholar
[6] Linckelmann M., Varieties in block theory, J. Algebra 215 (1999), 460–480. 10.1006/jabr.1998.7724Search in Google Scholar
[7] Linckelmann M., Quillen stratification for block varieties, J. Pure Appl. Algebra 172 (2002), 257–270. 10.1016/S0022-4049(01)00143-8Search in Google Scholar
[8] Linckelmann M., The Hochschild and block cohomology varieties are isomorphic, J. Lond. Math. Soc. (2) 81 (2010), 381–411. 10.1112/jlms/jdp078Search in Google Scholar
[9] Pakianathan J. and Witherspoon S. J., Hochschild cohomolgy and Linckelmann cohomology for blocks of finite groups, J. Pure Appl. Algebra 178 (2003), 87–100. 10.1016/S0022-4049(02)00189-5Search in Google Scholar
[10] Siegel S. and Witherspoon S. J., The Hochschild cohomology ring of a group algebra, Proc. Lond. Math. Soc. (3) 79 (1999), no. 1, 131–157. 10.1112/S0024611599011958Search in Google Scholar
[11] Thévenaz J., G-Algebras and Modular Representation Theory, Clarendon Press, Oxford, 1995. Search in Google Scholar
© 2017 by De Gruyter
Articles in the same Issue
- Frontmatter
- Probabilistic trace and Poisson summation formulae on locally compact abelian groups
- The Selberg trace formula as a Dirichlet series
- On the cohomology and their torsion of real toric objects
- Outer automorphism groups of simple Lie algebras and symmetries of painted diagrams
- Non-abelian tensor and exterior products of multiplicative Lie rings
- On the infimum of the spectrum of a relativistic Schrödinger operator
- Character correspondences for symmetric groups and wreath products
- Univalence in locally cartesian closed ∞-categories
- On subordinate random walks
- Tame combings and easy groups
- On Belk’s classifying space for Thompson’s group F
- Subspace confinement for switched linear systems
- Exceptional bundles of homological dimension ${k}$
- Representation zeta functions of some nilpotent groups associated to prehomogeneous vector spaces
- Bockstein homomorphisms for Hochschild cohomology of group algebras and of block algebras of finite groups
Articles in the same Issue
- Frontmatter
- Probabilistic trace and Poisson summation formulae on locally compact abelian groups
- The Selberg trace formula as a Dirichlet series
- On the cohomology and their torsion of real toric objects
- Outer automorphism groups of simple Lie algebras and symmetries of painted diagrams
- Non-abelian tensor and exterior products of multiplicative Lie rings
- On the infimum of the spectrum of a relativistic Schrödinger operator
- Character correspondences for symmetric groups and wreath products
- Univalence in locally cartesian closed ∞-categories
- On subordinate random walks
- Tame combings and easy groups
- On Belk’s classifying space for Thompson’s group F
- Subspace confinement for switched linear systems
- Exceptional bundles of homological dimension ${k}$
- Representation zeta functions of some nilpotent groups associated to prehomogeneous vector spaces
- Bockstein homomorphisms for Hochschild cohomology of group algebras and of block algebras of finite groups
