Generalized hilbert numerators

Abstract

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 q_n (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' by ideals that are generated by two homogeneous elements. More generally, the Hilbert numerator is a polynomial whenever the ideal is generated by finitely many homogeneous elements, the images of which form a regular sequence under all but finitely many of the truncation homomorphisms ρ_n .

Keywords:

• Gröbner bases
• initial ideals
• Hilbert series
• non-noetherian commutative rings
• regular sequences