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Hochschild homology and cohomology of \(\ell ^1 (\mathbb Z_+^k)\). (English) Zbl 1194.46110
This paper deals with the Hochschild homology and cohomology of \(\ell^1(\mathbb Z^k_+)\). Let us recall that the simplicial cohomology groups of the convolution algebra \(\ell^1(\mathbb Z^k_+)\) have been determined by F. Gourdeau, Z. A. Lykova and M. C. White [Stud. Math. 166, No. 1, 29–54 (2005; Zbl 1065.46030)]. It was proved that this cohomology vanishes in degrees \(2\) and above, which tells us the importance of the coefficients. In this work, the author investigates the computation for more general symmetric coefficients. This leads to a study of the Harrison homology and cohomology. As a byproduct, new results concerning second-degree cohomology are found.

MSC:
46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
46H05 General theory of topological algebras
43A20 \(L^1\)-algebras on groups, semigroups, etc.
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