Linear syzygies of Stanley-Reisner ideals

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.