On Gr\"obner Bases and Krull Dimension of Residue Class Rings of Polynomial Rings over Integral Domains

TL;DR

This paper develops a novel method using Gröbner bases to compute the Krull dimension of residue class rings over Noetherian integral domains, extending classical results from fields to more general rings.

Contribution

It introduces the notion of combinatorial dimension and relates it to Krull dimension for residue class rings over Noetherian integral domains, providing a uniform computational approach.

Findings

Defines combinatorial dimension for residue class rings over Noetherian domains.

Establishes a relation between Krull and combinatorial dimensions.

Shows Gröbner basis methods can compute Hilbert functions, series, and polynomials.

Abstract

Given an ideal $\mathfrak{a}$ in $A[x_1, \ldots, x_n]$, where $A$ is a Noetherian integral domain, we propose an approach to compute the Krull dimension of $A[x_1,\ldots,x_n]/\mathfrak{a}$, when the residue class polynomial ring is a free $A$-module. When $A$ is a field, the Krull dimension of $A[x_1,\ldots,x_n]/\mathfrak{a}$ has several equivalent algorithmic definitions by which it can be computed. But this is not true in the case of arbitrary Noetherian rings. For a Noetherian integral domain, $A$ we introduce the notion of combinatorial dimension of $A[x_1, \ldots,x_n]/\mathfrak{a}$ and give a Gr\"obner basis method to compute it for residue class polynomial rings that have a free $A$-module representation w.r.t. a lexicographic ordering. For such $A$-algebras, we derive a relation between Krull dimension and combinatorial dimension of $A[x_1, \ldots, x_n]/\mathfrak{a}$. An immediate application of this relation is that it gives a uniform method, the first of its kind, to compute the dimension of $A[x_1, \ldots, x_n]/\mathfrak{a}$ without having to consider individual properties of the ideal. For $A$-algebras that have a free $A$-module representation w.r.t. degree compatible monomial orderings, we introduce the concepts of Hilbert function, Hilbert series and Hilbert polynomials and show that Gr\"obner basis methods can be used to compute these quantities. We then proceed to show that the combinatorial dimension of such $A$-algebras is equal to the degree of the Hilbert polynomial. This enables us to extend the relation between Krull dimension and combinatorial dimension to $A$-algebras with a free $A$-module representation w.r.t. a degree compatible ordering as well.