Multigraded regularity: Coarsenings and resolutions

Let S=k[x1,…,xn] be a Zr-graded ring with deg(xi)=ai∈Zr for each i and suppose that M is a finitely generated Zr-graded S-module. In this paper we describe how to find finite subsets of Zr containing the multidegrees of the minimal multigraded syzygies of M. To find such a set, we first coarsen the grading of M so that we can view M as a Z-graded S-module. We use a generalized notion of Castelnuovo–Mumford regularity, which was introduced by D. Maclagan and G. Smith, to associate to M a number which we call the regularity number of M. The minimal degrees of the multigraded minimal syzygies are bounded in terms of this invariant.