Module and Hochschild cohomology of certain semigroup algebras

We study the relation between module and Hochschild cohomology groups of Banach algebras with a compatible module structure. More precisely, we show that for every commutative Banach $\mathcal{A}$-$\mathfrak{A}$-bimodule $X$ and every $k \in \mathbb{N}$, the seminormed spaces $\mathcal{H}^{k}_{\mathfrak{A}} (\mathcal{A},X^*)$ and $\mathcal{H}^k (\frac{\mathcal{A}}{J}, X^*)$ are isomorphic, where $J$ is the closed ideal of $\mathcal{A}$ generated by the elements of the form $a (\alpha \cdot b)-(a\cdot \alpha)b$ with $a,b \in \mathcal{A}$ and $\alpha \in \mathfrak{A}.$ As an example, we calculate the module cohomologies of inverse semigroup algebras with coefficients in some related function algebras. In particular, we show that for an inverse semigroup $S$ with the set of idempotents $E$, when $\ell^1(E)$ acts on $\ell^1(S)$ by multiplication from right and trivially from left, the first module cohomology $\mathcal{H}^1_{\ell^1(E)} (\ell^1(S), \ell^1(G_S)^{(2n+1)})$ is trivial for each $n \in \mathbb{N}$. As a consequence we conclude that the second module cohomology $\mathcal{H}^2_{\ell^1(E)} (\ell^1(S),\ell^1(G_S)^{(2n+1)})$ is a Banach space, where $G_S$ is the maximal group homomorphic image of $S$.