Gröbner bases of syzygies and Stanley depth

Let F
• be any free resolution of a finitely generated Zn-graded module over the polynomial ring K[x1,…,xn]. We show that for a suitable term order on F
• , then for 0⩽p• , encompassing Eliahou–Kervaire resolutions, we show that a Gröbner basis for Zp is given by the boundaries of generators of Fp+1.We apply the above to give lower bounds for the Stanley depth of the p+1'th syzygy module Zp, in particular showing it is greater or equal to 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 .