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Macaulay Constants and Vanishing of Cohomology

Uwe Nagel

Abstract

Dubé introduced cone decompositions and their Macaulay constants and used them to obtain an upper bound on the degrees of the generators in a Gröbner basis of an ideal. Liang extended the theory to submodules of a free module. In this paper, Macaulay constants of any finitely generated graded module $M$ over a polynomial ring are introduced by adapting the concept of a cone decomposition to $M$. It is shown that these constants provide upper bounds for the degrees in which the local cohomology modules of $M$ are not zero. The results include an upper bound on the Castelnuovo-Mumford regularity of $M$ and a generalization of Gotzmann's Regularity Theorem from ideals to modules. As an application, an upper bound on the Castelnuovo-Mumford regularity of any coherent sheaf on projective space is established. The mentioned bounds are sharp even for cyclic modules. Furthermore, Macaulay constants are utilized to provide a characterization of Hilbert polynomials of finitely generated graded modules.

Macaulay Constants and Vanishing of Cohomology

Abstract

Dubé introduced cone decompositions and their Macaulay constants and used them to obtain an upper bound on the degrees of the generators in a Gröbner basis of an ideal. Liang extended the theory to submodules of a free module. In this paper, Macaulay constants of any finitely generated graded module over a polynomial ring are introduced by adapting the concept of a cone decomposition to . It is shown that these constants provide upper bounds for the degrees in which the local cohomology modules of are not zero. The results include an upper bound on the Castelnuovo-Mumford regularity of and a generalization of Gotzmann's Regularity Theorem from ideals to modules. As an application, an upper bound on the Castelnuovo-Mumford regularity of any coherent sheaf on projective space is established. The mentioned bounds are sharp even for cyclic modules. Furthermore, Macaulay constants are utilized to provide a characterization of Hilbert polynomials of finitely generated graded modules.
Paper Structure (4 sections, 1 theorem, 88 equations)

This paper contains 4 sections, 1 theorem, 88 equations.

Table of Contents

  1. Introduction
  2. Review of Cone Decompositions
  3. Macaulay Constants and Hilbert Polynomials
  4. Regularity Bounds

Key Result

Theorem 1

If $M \ne 0$ is any finitely generated graded $S$-module then one has Moreover, for any integers $0 \le t \le d \le \dim S$ and $b_t \ge b_{t+1}\ge \cdots \ge b_d > b_{d+1}$, there a is cyclic graded $S$-module $M$ satisfying $\dim M = d$, $b_i (M) = b_i$ for $k = t,t + 1,\ldots,d+1$ and

Theorems & Definitions (22)

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