SEMIGROUP RINGS AND SIMPLICIAL COMPLEXES

We study the minimal free resolution F of a ring T = S/I where S is a positive affine semigroup ring over a field K, and I is an ideal in S generated by monomials. We will essentially use the fact that the mult igraded Betti numbers of T can be computed from the relative homology of simplicial complexes that we shall call squarefree divisor complexe s. In a sense, these simplicial complexes represent the divisibility r elations in S if one neglects the multiplicities with which the irredu cible elements appear in the representation of an element. In Section 1 we study the dependence of the free resolution on the characteristic of K. In Section 2 we show that, up to an equivalence in homotopy, ev ery simplicial complex can be 'realized' in a normal semigroup ring an d also in a one-dimensional semigroup ring. Furthermore, we describe a ll the graphs among the squarefree divisor complexes. In Section 3 we deduce assertions about certain simplicial complexes of chessboard typ e from information about free resolutions of well-understood semigroup rings. (C) 1997 Elsevier Science B.V.