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Simplicial Resolutions of Powers of Square-free Monomial Ideals

Susan M. Cooper, Sabine El Khoury, Sara Faridi, Sarah Mayes-Tang, Susan Morey, Liana M. Sega, Sandra Spiroff

Abstract

The Taylor resolution is almost never minimal for powers of monomial ideals, even in the square-free case. In this paper we introduce a smaller resolution for each power of any square-free monomial ideal, which depends only on the number of generators of the ideal. More precisely, for every pair of fixed integers $r$ and $q$, we construct a simplicial complex that supports a free resolution of the $r$-th power of any square-free monomial ideal with $q$ generators. The resulting resolution is significantly smaller than the Taylor resolution, and is minimal for special cases. Considering the relations on the generators of a fixed ideal allows us to further shrink these resolutions. We also introduce a class of ideals called "extremal ideals", and show that the Betti numbers of powers of all square-free monomial ideals are bounded by Betti numbers of powers of extremal ideals. Our results lead to upper bounds on Betti numbers of powers of any square-free monomial ideal that greatly improve the binomial bounds offered by the Taylor resolution.

Simplicial Resolutions of Powers of Square-free Monomial Ideals

Abstract

The Taylor resolution is almost never minimal for powers of monomial ideals, even in the square-free case. In this paper we introduce a smaller resolution for each power of any square-free monomial ideal, which depends only on the number of generators of the ideal. More precisely, for every pair of fixed integers and , we construct a simplicial complex that supports a free resolution of the -th power of any square-free monomial ideal with generators. The resulting resolution is significantly smaller than the Taylor resolution, and is minimal for special cases. Considering the relations on the generators of a fixed ideal allows us to further shrink these resolutions. We also introduce a class of ideals called "extremal ideals", and show that the Betti numbers of powers of all square-free monomial ideals are bounded by Betti numbers of powers of extremal ideals. Our results lead to upper bounds on Betti numbers of powers of any square-free monomial ideal that greatly improve the binomial bounds offered by the Taylor resolution.
Paper Structure (7 sections, 18 theorems, 137 equations, 3 figures)

This paper contains 7 sections, 18 theorems, 137 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Simplicial Resolutions
  3. Resolutions supported on quasi-trees
  4. The quasi-tree $\mathbb{L}^r_q$
  5. The quasi-tree $\mathbb{L}^r(I)$ supporting a free resolution of $I^r$
  6. Bounds on Betti numbers
  7. Extremal Ideals: When does $\mathbb{L}^r_q$ support a minimal resolution?

Key Result

Theorem 3.1

Let $\Delta$ be a simplicial complex whose vertices are labeled with a monomial generating set of a monomial ideal $I$ in a polynomial ring $S$ over a field. Then $\Delta$ supports a free resolution of $I$ over $S$ if and only if for every monomial $M$, the induced subcomplex $\Delta_M$ of $\Delta$

Figures (3)

  • Figure 1: An example of collapsing
  • Figure 2: A picture of $\mathbb{L}_3^1$, $\mathbb{L}_3^2$ and $\mathbb{L}_3^3$
  • Figure 3: A picture of $\mathbb{L}^6_3$

Theorems & Definitions (62)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Remark 2.6
  • Theorem 3.1: Criterion for a simplicial complex to support a free resolution
  • Remark 3.2
  • Proposition 3.3: Induced subcomplexes of quasi-forests are quasi-forests
  • proof
  • ...and 52 more