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Multigraded regularity: coarsenings and resolutions

Jessica Sidman·Adam Van Tuyl·Haohao Wang
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Abstract

Let S = k[x_1,...,x_n] be a Z^r-graded ring with deg (x_i) = a_i \in Z^r for each i and suppose that M is a finitely generated Z^r-graded S-module. In this paper we describe how to find finite subsets of Z^r 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.

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Jessica Sidman, Adam Van Tuyl, Haohao Wang. 2005-10-01. Multigraded regularity: coarsenings and resolutions. https://arxiv.org/abs/math/0505421

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