On the Linear Syzygies Of

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. The description is in terms of the homology of the canonical Alexander dual complex. 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 coee-cients if is a pseudomanifold, exhibit an example where multigraded maps in the linear strand cannot be written using only 1 coeecients. compute the entire resolution explicitly when is the complex of independent sets of a matroid. 1. Introduction The goal of this paper is to describe in a simple topological fashion the maps in the linear strand of the minimal free resolution of a Stanley-Reisner ideal. It is an outgrowth of two recent trends in the theory of minimal free resolutions. The rst is a series of results 1, 2, 4, 5, 7, 20] giving explicit descriptions of the maps in the minimal free resolutions for various classes of ideals. The second is the realization