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Positivity and nonstandard graded Betti numbers

Michael K. Brown, Daniel Erman

Abstract

A foundational principle in the study of modules over standard graded polynomial rings is that geometric positivity conditions imply vanishing of Betti numbers. The main goal of this paper is to determine the extent to which this principle extends to the nonstandard graded case. In this setting, the classical arguments break down, and the results become much more nuanced. We introduce a new notion of Castelnuovo-Mumford regularity and employ exterior algebra techniques to control the shapes of nonstandard graded minimal free resolutions. Our main result reveals a unique feature in the nonstandard graded case: the possible degrees of the syzygies of a graded module in this setting are controlled not only by its regularity, but also by its depth. As an application of our main result, we show that, given a simplicial projective toric variety and a module M over its coordinate ring, the multigraded Betti numbers of M are contained in a particular polytope when M satisfies an appropriate positivity condition.

Positivity and nonstandard graded Betti numbers

Abstract

A foundational principle in the study of modules over standard graded polynomial rings is that geometric positivity conditions imply vanishing of Betti numbers. The main goal of this paper is to determine the extent to which this principle extends to the nonstandard graded case. In this setting, the classical arguments break down, and the results become much more nuanced. We introduce a new notion of Castelnuovo-Mumford regularity and employ exterior algebra techniques to control the shapes of nonstandard graded minimal free resolutions. Our main result reveals a unique feature in the nonstandard graded case: the possible degrees of the syzygies of a graded module in this setting are controlled not only by its regularity, but also by its depth. As an application of our main result, we show that, given a simplicial projective toric variety and a module M over its coordinate ring, the multigraded Betti numbers of M are contained in a particular polytope when M satisfies an appropriate positivity condition.
Paper Structure (7 sections, 10 theorems, 13 equations)

This paper contains 7 sections, 10 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Regularity and related notions
  4. The multigraded BGG correspondence
  5. Tate resolutions on weighted projective stacks
  6. Proof of Theorem \ref{['introweighted']}
  7. An application to Betti numbers over Cox rings of toric varieties

Key Result

Theorem 1.6

Let $M$ be a finitely generated, graded $S$-module with Betti numbers $\beta_{i,j}(M) = \operatorname{Tor}_i^S(M, k)_j$. If $M$ is Koszul $r$-regular, then

Theorems & Definitions (29)

  • Definition 1.3
  • Definition 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Example 1.9
  • Definition 2.1: The standard graded case
  • Proposition 2.2: symonds2 Proposition 1.2
  • Remark 2.3
  • Example 2.4
  • ...and 19 more