Multigraded regularity: Coarsenings and resolutions

Abstract

Let S = k[x 1 , . . . , x n ] be a Z r -graded ring with deg(x i ) = a i ∈ 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.

Regularity and Resolutions for Multigraded Modules

“…The results of section 4 depend on the paper [19], whose main results depend on Maclagan and Smith's paper. We will argue that the main results, especially [13,Theorem 4.7,Theorem 5.4] are true as stated in this more general setting.…”

“…In effect this means that one is coarsening the multigrading to a simple, but generally nonstandard grading. This has been considered in a recent paper by Sidman-Van Tuyl-Wang [19] who extended some of the techniques used in [12]. They introduced a notion of positive coarsening vector which coarsens the multidegrees on M to a single nonstandard Z grading, and obtain finite bounds on the multidegrees of a minimal free resolution of M .…”

“…This case is studied in detailed in [19]. Often, the (B, C)-regular region can be estimated in terms of (B 0 , C 0 )-regularity regions, defined by various φ's, and the advantage of considering the φ's is that one can often give easily described constraints on the regions J p .…”

“…In Section 5, we will discuss this issue, and also we illustrate its use for multiprojective spaces P m1 × · · · × P mt . The example of multiprojective spaces shows the necessity of considering gradings φ which assign 0 to some of the variables, which was not treated in [19]. We can include such gradings by writing S = R[x j ] where R = K[x i ] where x i are the variables with φ(deg(x i )) = 0, and x j are the variables that are φ-positive, i.e., admissible for the theory of regularity.…”

Multigraded Hilbert functions and toric complete intersection codes

“…the Hilbert function stabilizes in that direction (see [Sidman and Van Tuyl (2006), Proposition 1.9]).…”