The Equality of Hochschild Cohomology Group and Module Cohomology Group for Semigroup Algebras

Let S be a (not necessarily unital) commutative inverse semigroup with idempotent set E. In this paper, we show that for every n is an element of N-0, n-th Hochschild cohomology group of semigroup algebra l(1)(S) with coefficients in l(infinity)(S) and its n-th l(1) (E)-module cohomology group, are equal. Indeed, we prove that H-n(l(1)(S), l(infinity) (S)) = H-l1(E)(n) (l(1)(S), l(infinity)(S)), for all n >= 0.