Combinatorics of Poincaré series for monomial rings

Professor Patricia Hersh
Indiana University

Friday, April 8, 2005
4 pm, 845 Patterson Office Tower

Abstract:

Backelin proved that the multigraded Poincaré series for resolving a residue field over a polynomial ring modulo a monomial ideal is a rational function. The numerator is simple and well-understood, but recently Alex Berglund found a formula for the denominator, expressed in terms of ranks of homology groups of certain simplicial complexes. I'll discuss recent joint work with Jonah Blasiak in which we express the denominator in terms of lower intervals in intersection lattices of subspace arrangements that in many cases have been studied before. Crapo's Closure Lemma and related combinatorics allow us to simplify the denominator substantially in some cases (such as monomial ideals generated in degree two), make it more explicit in other cases, and relate Golodness to the Cohen-Macaulay property for associated posets. I'll briefly review the notions of minimal free resolution, Golodness, etc. along the way.