Summary: HOMOTOPY LIE ALGEBRAS AND POINCARŽE SERIES OF
ALGEBRAS WITH MONOMIAL RELATIONS
LUCHEZAR L. AVRAMOV
Dedicated to Jan-Erik Roos
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 PoincarŽe series of the residue field k
of S/I is rational by Backelin , and the homotopy Lie algebra of S/I is finitely
generated by Backelin and Roos . Constructions of Anick  and Roos ,
respectively, show that these properties may fail for general homogeneous ideals.
Recenly, Gasharov, Peeva, and Welker  showed that some homological prop-
erties 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 PoincarŽe 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 PoincarŽe series is finite, even when K ranges over all fields.