The Stanley’s Conjecture on Cohen-Macaulay multigraded modules is studied especially in dimension 2. In codimension 2 similar results were obtained by Herzog, Soleyman-Jahan and Yassemi. As a consequence of our results Stanley’s Conjecture holds in 5 variables.

We give a criterion for the section ring of an ample line bundle to be Koszul in terms of multigraded regularity. We discuss applications to adjoint bundles on toric varieties as well as to polytopal semigroup rings.

We demonstrate a method to parallelize the computation of a Gröbner basis for a homogenous ideal in a multigraded polynomial ring. Our method uses anti-chains in the lattice N to separate mutually independent S-polynomials for reduction.

The goal of this paper is to provide homological descriptions of monomial ideals. The key concepts in these descriptions are the minimal free resolution, the Koszul homology and the multigraded Betti numbers. These three objects are strongly related, being Tor modules a simple way to describe this relation. We introduce a new tool, Mayer-Vietoris trees which provides a good way to compute the h...

In this paper we study the equations of elimination ideal associated with n+1 generic multihomogeneous polynomials defined over a product projective spaces dimension n. We first prove duality property and then make explicit by introducing multigraded Sylvester forms. These results provide partial generalization similar properties that are known in setting homogeneous polynomial systems single s...