Hilbert-Samuel Polynomials for Algebras with Special Filtrations

TL;DR

This paper develops an axiomatic framework for multiplicity in noncommutative algebras, establishing the existence of Hilbert-Samuel polynomials and applying these concepts to various algebraic structures including Cherednik algebras.

Contribution

It introduces the concept of very nice and modest algebras, generalizes multiplicity results beyond Ore domains, and applies the theory to min-holonomic modules and rational Cherednik algebras.

Findings

Hilbert-Samuel polynomials imply a notion of multiplicity in abstract algebraic settings.

Generalization of multiplicity results to prime algebras beyond Ore domains.

Rational Cherednik algebras admit a well-defined notion of multiplicity.

Abstract

The notion of multiplicity of a module first arose as consequence of Hilbert's work on commutative algebra, relating the dimension of rings with the degree of certain polynomials. For noncommutative rings, the notion of multiplicity first appeared in the context of modules for the Weyl algebra in Bernstein's solution of the problem of analytic continuation posed by I. Gelfand. The notion was shown to be useful to many more noncommutative rings, especially enveloping algebras, rings of differential operators, and quantum groups. In all these cases, the existence of multiplicity is related to the existence of Hilbert-Samuel polynomials. In this work we give an axiomatic definition of algebras with a notion of multiplicity, which we call very nice and modest algebras. We show, in an abstract setting, how the existence of Hilbert-Samuel polynomials implies the existence of a notion of multiplicity. We apply our results for the category of min-holonomic modules -- a notion which coincides with holonomic modules for simple algebras -- and that shares many similarities with it. In particular, we generalize the usual results in the literature that are stated for Ore domains, in the more general context of prime algebras, and we show that rational Cherednik algebras admit a notion of multiplicity.