Homotopy Lie algebras and Poincaré series of algebras with monomial relations

Abstract

To every homogeneous ideal of a polynomial ring $S$ over a field $K$, Macaulay assigned an ideal generated by monomials in the indeterminates and with the same Hilbert function. Thus, from the point of view of Hilbert series residue rings modulo monomial ideals display the most general behavior. The homological perspective reveals a very different picture. Two aspects are particularly relevant to this paper:

If $I$ is generated by monomials, then the Poincaré series of the residue field $k$ of $S/I$ is rational by Backelin [7], and the homotopy Lie algebra of $S/I$ is finitely generated by Backelin and Roos [8]. Constructions of Anick [1] Roos [15], respectively, show that these properties may fail for general homogeneous ideals.

Recently, Gasharov, Peeva, and Welker [12] showed that some homological properties of $S/I$, such as being Golod, depend only on combinatorial data gathered from a minimal set of monomial generators.

Here we prove that these data determine the Poincaré series of $k$ over $S/I$, along with most of its homotopy Lie algebra. As a consequence, we obtain the surprising result that if the number of generators of the ideal $I$ is fixed, then the number of such Poincaré series is finite, even when $K$ ranges over all fields.