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Hochschild constructions for Green functors. (English) Zbl 1070.16010

Let \(G\) be a finite group and let \(R\) be a commutative ring. The Hochschild cohomology ring \(HH^*(RG,RG)\) has an additive decomposition \(HH^*(RG,RG)=\bigoplus_{x\in X}H^*(C_G(x),R)\) where \(X\) is a transversal for the conjugacy classes of \(G\) and where \(H^*(C_G(x),R)\) denotes the cohomology ring of the group \(C_G(x)\), for \(x\in X\). S. F. Siegel and S. J. Witherspoon [Proc. Lond. Math. Soc., III. Ser. 79, No. 1, 131-157 (1999; Zbl 1044.16005)] have computed how the product in \(HH^*(RG,RG)\) is reflected by this decomposition. In the paper under review, the author shows that their result is a special case of a more general construction for Green functors.

MSC:

16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
19A22 Frobenius induction, Burnside and representation rings
20J06 Cohomology of groups
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16S34 Group rings

Citations:

Zbl 1044.16005
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References:

[1] Bouc S., Lecture Notes in Mathematics 1671 (1997)
[2] DOI: 10.1090/S0002-9939-97-03727-1 · Zbl 0865.18005 · doi:10.1090/S0002-9939-97-03727-1
[3] DOI: 10.1007/PL00000389 · Zbl 0872.16004 · doi:10.1007/PL00000389
[4] DOI: 10.2307/1970343 · Zbl 0131.27302 · doi:10.2307/1970343
[5] DOI: 10.1112/S0024611599011958 · Zbl 1044.16005 · doi:10.1112/S0024611599011958
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