Linear syzygies of Stanley-Reisner ideals

Abstract

We give an elementary description of the maps in the linear strand of the minimal free resolution of a square-free monomial ideal, that is, the Stanley-Reisner ideal associated to a simplicial complex $\Delta$. The description is in terms of the homology of the canonical Alexander dual complex $\Delta^*$. As applications we are able to

• prove for monomial ideals and $j=1$ a conjecture of J. Herzog giving lower bounds on the number of $i$-syzygies in the linear strand of $j^{th}$-syzygy modules
• show that the maps in the linear strand can be written using only $\pm 1$ coefficients if $\Delta^*$ is a pseudomanifold
• exhibit an example where multigraded maps in the linear strand cannot be written using only $\pm 1$ coefficients
• compute the entire resolution explicitly when $\Delta^*$ is the complex of independent sets of a matroid