Generalized hilbert numerators

It is a well-known fact that if K is a field, then the Hilbert series of a quotient of the polynomial ring by a homogeneous ideal is of the form we call the polynomial q(t) the Hilbert numerator of the quotient algebra. We will generalize this concept to a class of non-finitely generated, graded, commuta-tive algebras, which are endowed with a surjective “co-filtration” of finitely generated algebras. Then, although the Hilbert series themselves can not be defined (since the sub-vector-spaces involved have infinite dimension), we get a sequence of Hilbert numerators qn (t), which we show converge to a power series in Z[[t]]. This power series we call the (generalized) Hilbert numerator of the non-finitely generated algebra. The question of determining when this power series is in fact a polynomial is the topic of the last part of this article. We show that quotients of the ring R' by homogeneous ideals that are generated by finitely many monomials have polynomial Hilbert numerator, as have quotients of R'...