# Algebra Seminar

Title: Minimal resolutions of monomial ideals

Abstract: There are two general constructions for minimal free resolutions of arbitrary monomial ideals. The first construction by Eagon is obtained via spectral sequences associated to certain Koszul complexes, and the second construction by Yuzvinsky is obtained from the Taylor resolution and utilizes the lcm lattice of monomial generators. In order to obtain a canonical, minimal free resolution for an arbitrary monomial ideal with a closed-form, combinatorial description of the differential, a combinatorial formula for the Moore--Penrose pseudoinverse is combined with these two constructions. The result is two differentials given as sums over lattice paths, one in the lcm lattice and one in N^n, of weights associated to higher-dimensional analogues of spanning trees. This is based on joint work with John Eagon and Ezra Miller.