[Return to main list of publications]## Hochschild homology and cohomology of l^{1}(**Z**_{+}^{k})

#### Authors

Yemon Choi

#### Keywords

Hochschild homology, Hochschild cohomology, Harrison cohomology, Banach algebras, Hodge decomposition, semigroup algebra
#### Status

Appeared as
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)

#### Reviews

[ Math Reviews | Zentralblatt ]

#### Abstract

Building on the recent determination of the simplicial cohomology groups of the convolution algebra
l^{1}(**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 l^{1}(**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.

#### Updates/comments

(*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 E_{2}-terms given by

E_{2}^{p,q} = H^{p}( H_{q}(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.*

Yemon Choi
Last modified: Sun Jan 9 19:29:07 CT 2011