On the multigraded Hilbert and Poincaré–Betti series and the Golod property of monomial rings

Abstract

In this paper we study the multigraded Hilbert and Poincaré–Betti series of A = S / a , where S is the ring of polynomials in n indeterminates divided by the monomial ideal a . There is a conjecture about the multigraded Poincaré–Betti series by Charalambous and Reeves which they proved in the case where the Taylor resolution is minimal. We introduce a conjecture about the minimal A -free resolution of the residue class field and show that this conjecture implies the conjecture of Charalambous and Reeves and, in addition, gives a formula for the Hilbert series. Using Algebraic Discrete Morse theory, we prove that the homology of the Koszul complex of A with respect to x 1 , … , x n is isomorphic to a graded commutative ring of polynomials over certain sets in the Taylor resolution divided by an ideal r of relations. This leads to a proof of our conjecture for some classes of algebras A . We also give an approach for the proof of our conjecture via Algebraic Discrete Morse theory in the general case. The conjecture implies that A is Golod if and only if the product (i.e. the first Massey operation) on the Koszul homology is trivial. Under the assumption of the conjecture we finally prove that a very simple purely combinatorial condition on the minimal monomial generating system of a implies Golodness for A .