Growth and Relations in Graded Rings

Abstract

Suppose $A$ is a graded associative algebra over a field, $I$ is its ideal generated by a set $\alpha$ of homogeneous elements, and B = A/I. In this note, some inequalities between Hilbert series of algebras $A,B$ and the number of elements of the set $\alpha$ are announced. As in the Golod--Shafarevich inequality as in our case the equality in every estimate is exact iff the set $\alpha$ is strongly free: so we obtain some new characterizations of such sets. As a consequence it is proved that over a field of zero characteristic for the class of finitely defined graded algebras there is no algorithm to answer the following question: for an algebra $A$ and a rational number $R$, is the convergence radius of the Hilbert series of $A$ equal to $R$?