Gr\"obner bases of syzygies and Stanley depth

TL;DR

This paper investigates the structure of syzygy modules in free resolutions over polynomial rings, showing how Gr"obner bases can be characterized and applying these results to establish lower bounds on Stanley depth.

Contribution

It introduces a method to determine initial modules of syzygies and provides new bounds on Stanley depth for syzygy modules and squarefree ideals.

Findings

Initial modules of syzygies are generated by monomials in specific variables.

A Gr"obner basis for syzygies can be given by boundaries of generators.

Stanley depth of syzygies is at least p+1, and for squarefree ideals at least proportional to the square root of 2n.

Abstract

Let F. be a any free resolution of a Z^n-graded submodule of a free module over the polynomial ring K[x_1, ..., x_n]. We show that for a suitable term order on F., the initial module of the p'th syzygy module Z_p is generated by terms m_ie_i where the m_i are monomials in K[x_{p+1}, ..., x_n]. Also for a large class of free resolutions F., encompassing Eliahou-Kervaire resolutions, we show that a Gr\"obner basis for Z_p is given by the boundaries of generators of F_p. We apply the above to give lower bounds for the Stanley depth of the syzygy modules Z_p, in particular showing it is at least p+1. We also show that if I is any squarefree ideal in K[x_1, ..., x_n], the Stanley depth of I is at least of order the square root of 2n.