A numerical semigroup $S$ is an additive subsemigroup of the non-negative integers with finite complement, and squarefree divisor complex element $m \in S$ a simplicial $\Delta_m$ that arises in study multigraded Betti numbers. We compute complexes for certain classes semigroups, exhibit new family are occur as some element.

We provide a complete description of normal affine varieties with effective algebraic torus action in terms of what we call proper polyhedral divisors on semiprojective varieties. Our theory extends classical cone constructions of Dolgachev, Demazure and Pinkham to the multigraded case, and it comprises the theory of affine toric varieties.

Projective resolutions of modules over a ring R are constructed starting from appropriate projective resolutions over a ring Q mapping to R. It is shown that such resolutions may be chosen to be minimal in codimension ≤ 2, but not in codimension 3. This is used to obtain minimal resolutions for essentially all modules over local (or graded) rings R with codimension ≤ 2. Explicit resolutions are...