Semigroup rings and simplicial complexes Original Research Article

We study the minimal free resolution image of a ring T = S/I where S is a positive affine semi-group ring over a field K, and I is an ideal in S generated by monomials. We will essentially use the fact that the multigraded Betti numbers of T can be computed from the relative homology of simplicial complexes that we shall call squarefree divisor complexes. In a sense, these simplicial complexes represent the divisibility relations in S if one neglects the multiplicities with which the irreducible 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, every simplicial complex can be ‘realized’ in a normal semigroup ring and also in a one-dimensional semigroup ring. Furthermore, we describe all the graphs among the squarefree divisor complexes. In Section 3 we deduce assertions about certain simplicial complexes of chessboard type from information about free resolutions of well-understood semigroup rings.