Combinatorics of multigraded Poincaré series for monomial rings

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, but until the recent work of Berglund there was no combinatorial formula for the denominator. Berglund's formula gives the denominator in terms of ranks of reduced homology groups of lower intervals in a certain lattice. We now express this lattice as the intersection lattice L A ( I ) of a subspace arrangement A ( I ) , use Crapo's Closure Lemma to drastically simplify the denominator in some cases (such as monomial ideals generated in degree two), and relate Golodness to the Cohen–Macaulay property for associated posets. In addition, we introduce a new class of finite lattices called complete lattices, prove that all geometric lattices are complete and provide a simple criterion for Golodness of monomial ideals whose lcm-lattices are complete.