Hyperplane arrangements and free resolutions over the exterior algebra

Hal Schenck (Mathematics, Texas A&M University)

If X is the complement of a hyperplane arrangement, then the cohomology ring A of X has a beautiful and simple combinatorial description; algebraically it is a quotient of an exterior algebra. Of course, questions about the topology of X may be studied by studying A; for example from the fundamental group of X one can form a graded Lie algebra; the ranks of the graded pieces are the lower central series. These ranks can be computed as from a free resolution of the residue field over the cohomology ring. These ranks get large very fast, but by using the change of rings spectral sequence, we can relate them to the much smaller numbers which result from the free resolution of A over the exterior algebra. Our results have a particularly nice formulation in the case of graphic arrangements. (Joint work with Alex Suciu, Northeastern Univ.)