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 ${\mathcal{H}}_{\mathfrak{A}}^k(\mathcal{A},X^{\ast })$ and $ $\mathcal{H}$^k (\frac{$\mathcal{A}$}{J}, X^*) $areisomorphic,where$ 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 ${\mathrm{\ell }}^1(E)$ acts on ${\mathrm{\ell }}^1(S)$ by multiplication from right and trivially from left, the first module cohomology $$\mathcal{H}$^1_{\ell^1(E)} (\ell^1(S), \ell^1(G_S)^{(2n+1)})$istrivialforeach$ n \in $\mathbb{N}$ $. As a consequence we conclude that the second module cohomology ${\mathcal{H}}_{{\mathrm{\ell }}^1(E)}^2({\mathrm{\ell }}^1(S),{\mathrm{\ell }}^1(G_S{)}^{(2n+1)})$ is a Banach space, where $G_S$ is the maximal group homomorphic image of $S$.