Linear syzygies of Stanley-Reisner ideals - Reiner, V; Welker, V (MATHEMATICA SCANDINAVICA, 2001)
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 $Δ$. The description is in terms of the homology of the canonical Alexander dual complex $Δ^*$. 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 $± 1$ coefficients if $Δ^*$ is a pseudomanifold exhibit an example where multigraded maps in the linear strand cannot be written using only $± 1$ coefficients compute the entire resolution explicitly when $Δ^*$ is the complex of independent sets of a matroid