Homological dimension in semigroup algebras

AbstractIf S is a semigroup with identity and k is a commutative ring with identity, let kS be the semigroup algebra. For M a left kS-module, let dkS(M) be the projective dimension of M. Then D(kS) = l.gl.dim(kS) = sup{dkS(M): M is a left kS-module}. For S a finite regular semigroup, we obtain an upper bound for D(kS) in terms of properties of the composition factors of S and of {D(kG): G ⊂S a maximal subgroup}. Using this we find that if k is a field of characteristic relatively prime to the orders of the maximal subgroups of S, then D(kS) is finite, and that if S is completely 0-simple, this bound is attained.We can also use this to show that if S is a finite, regular, combinatorial semigroup, then the cohomology groups Hn(S,−) vanish for sufficiently high dimension