Hilbert series and relations in algebras

Let A be a graded associative algebra over a field, I triangleleft A an ideal generated by a set α subset of A of homogeneous elements, and B=A/I. In this paper we get estimates relating the Hilbert series of the algebras A,B and the number of elements of α. As in the Golod-Shafarevich theorem, these estimates hold with equality exactly for strongly free sets α, which gives new characterizations of such sets. As a corollary, we prove that in the class of finitely generated algebras over a field of characteristic zero there is no algorithm to decide (from the given generators and relations of the algebra) whether the radius of convergence of the Hilbert series equals a given rational number, and there is no algorithm to decide whether the value of the Hilbert function at a given point is equal to a given number. We also introduce and study extremal graded algebras (such that taking any quotient strictly increases the radius of convergence of the Hilbert series). In particular, we prove that this class contains free products of two non-trivial algebras, quadratic algebras with one relation and at least three generators, and Artin-Shelter regular non-Noetherian algebras of global dimension 2