The equality of Hochschild cohomology group and module cohomology group for semigroup algebras

Let $S$ be a commutative inverse semigroup with idempotent set $E$. In this paper, we show that for every $n\in \mathbb{N}_0$, $n$-th Hochschild cohomology group of semigroup algebra $\ell^1(S)$ with coefficients in $\ell^\infty(S)$ and its $n$-th $\ell^1(E)$-module cohomology group, are equal. Indeed, we prove that \[ \HH^{n}(\ell^1(S),\ell^\infty(S))=\HH^{n}_{\ell^1(E)}(\ell^1(S),\ell^\infty(S)),\] for all $n\geq 0$.