Gr\"obner bases of syzygies and Stanley depth
Abstract
Let F. be a any free resolution of a Zn-graded submodule of a free module over the polynomial ring K[x1, ..., xn]. We show that for a suitable term order on F., the initial module of the p'th syzygy module Zp is generated by terms miei where the mi are monomials in K[xp+1, ..., xn]. Also for a large class of free resolutions F., encompassing Eliahou-Kervaire resolutions, we show that a Gr\"obner basis for Zp is given by the boundaries of generators of Fp. We apply the above to give lower bounds for the Stanley depth of the syzygy modules Zp, in particular showing it is at least p+1. We also show that if I is any squarefree ideal in K[x1, ..., xn], the Stanley depth of I is at least of order the square root of 2n.
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