Hochschild duality, localization, and smash products

Abstract

In this work we study the class of algebras satisfying a duality property with respect to Hochschild homology and cohomology, as in [Proc. Amer. Math. Soc. 126 (1998) 1345-1348]. More precisely, we consider the class of algebras A such that there exists an invertible bimodule U and an integer number d with the property H • (A, M) ∼ = H d−• (A, U ⊗ A M), for all A-bimodules M. We show that this class is closed under localization and under smash products with respect to Hopf algebras satisfying also the duality property.We also illustrate the subtlety on dualities with smash products developing in detail the example S(V ) # G, the crossed product of the symmetric algebra on a vector space and a finite group acting linearly on V .  2004 Elsevier Inc. All rights reserved.

Skew Derivations and Deformations of a Family of Group Crossed Products

“…The deformations in that case are nontrivial since their corresponding Hochschild two-cocycles are not coboundaries, a consequence of the computations in [4]. Similarly, we now show that the same is true in the more general setting of a finite group G, based on a computation of the Hochschild cohomology of S(V )# α G. The Hochschild cohomology was computed by Farinati, Ginzburg and Kaledin in the case α = 1 [9,13]. The addition of a nontrivial cocycle α poses no difficulties, however we need to use some of the details from an explicit algebraic computation.…”

Untitled

“…Hochschild cohomology of S(V )#G. Farinati [2005] and Ginzburg and Kaledin [2004] determined the graded vector space structure of HH q (S(V )#G) when G acts faithfully on V . The same techniques apply to nonfaithful actions.…”

“…For noncommutative algebras, Hochschild homology provides a generalization of the notion of "differential forms". It is interesting to note that for some types of algebras (in particular for S(V )#G), Hochschild homology and cohomology are dual (see [van den Bergh 1998] for the general theory and [Farinati 2005] for the case S(V )#G). In this section, we work over an arbitrary field initially, then over ‫ރ‬ in Theorem 6-4.…”

“…Now we restrict our choice of field again to ‫.ރ‬ Let M = S(V )#G, and note that Hochschild homology decomposes just as does Hochschild cohomology: [Farinati 2005;Ştefan 1995]). Thus one is interested in the components…”

Hochschild cohomology and graded Hecke algebras

“…Fix a finite group G and a representation V = C n of G. The Hochschild cohomology of S(V )#G is given in [12,14] when G acts faithfully on V . In this section, we reformulate this result to aid our explicit computations and our determination of graded Hecke algebras.…”