Homological properties of modules over semigroup algebras

Let S be a semigroup. In this paper we investigate the injectivity of ℓ1(S) as a Banach right module over ℓ1(S). For weakly cancellative S this is the same as studying the flatness of the predual left module c0(S). For such semigroups S, we also investigate the projectivity of c0(S). We prove that for many semigroups S for which the Banach algebra ℓ1(S) is non-amenable, the ℓ1(S)-module ℓ1(S) is not injective. The main result about the projectivity of c0(S) states that for a weakly cancellative inverse semigroup S, c0(S) is projective if and only if S is finite.