The $\ell^{1}$-convolution algebra of a semilattice is known to have trivial cohomology in degrees 1, 2 and 3 whenever the coefficient bimodule is symmetric. We extend this result to all cohomology groups of degree $\geq 1$ with symmetric coefficients. Our techniques prove a stronger splitting result, namely that the splitting can be made natural with respect to the underlying semilattice.

It is well known that c0(Z) is amenable and so its global dimension is zero. In this paper we will investigate the cyclic and Hochschild cohomology of Banach algebra c0 (Z, ω-1) and its unitisation with coefficients in its dual space, where ω is a weight on Z which satisfies inf {ω(i)} = 0.Moreover we show that the weak homological bi-dimension of c0 (Z, ω-1) is infinity.

Given two topological algebra sheaves, we seek that their tensor product be an (algebra) sheaf of the same type. We further study the latter sheaf in connection with the set of morphisms which are defined on it. As an application, we finally consider fundamental notions and results related to algebras of holomorphic functions in the framework of topological algebra sheaves.