Q. J. Math. (Oxford) 61 (2010), no. 1, 1--28.
Offprints (electronic) available on request.
Preprint version available at arXiv 0709.3325 (last updated Sept. 2008. Warning: not the final version)
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Building on the recent determination of the simplicial cohomology groups of the convolution algebra l1(Z+k) [F. Gourdeau, Z. A. Lykova and M. C. White, A Künneth formula in topological homology and its applications to the simplicial cohomology of l1(Z+k), Studia Math. 166 (2005), 29 -- 54], we investigate what can be said for cohomology of this algebra with more general symmetric coefficients. Our approach leads us to a discussion of Harrison homology and cohomology in the context of Banach algebras, and a development of some of its basic features. As an application of our techniques we reprove some known results on second-degree cohomology.
(incomplete draft comments to be updated)
The main results here are taken from my PhD thesis, but with some adjustments to the presentation and a closing section giving an illustrative application to a paticular case.
The underlying motivation for this paper comes not just from the paper of Gourdeau, Lykova & White, but from a spectral sequence in commutative algebra. Given a commutative algebra R and a symmetric R-bimodule M, this spectral sequence (which follows, as so many do, from that of Grothendieck*) converges to Hochschild cohomology H*(R,M) and has E2-terms given by
E2p,q = Hp( Hq(R,R), M)
(The alert reader will spot that we are omitting certain hypotheses on the algebra R and its ground field, but this is not overly restrictive.)
This unfortunately is ill-defined if we try to transport it over to the setting of continuous Hochschild cohomology groups of Banach modules. However, one can use it as a motivating heuristic, and this is indeed the guide behind the more nuts-and-bolts approach taken in the paper.
(*) That is, the Grothendieck spectral sequence for computing the derived functors of a composition of two functors between suitable categories.