The multivariate Serre conjecture ring

TL;DR

This paper explores the properties of Serre conjecture rings in multivariate cases, aiming to generalize their Bézout domain structure to rational monomial orders, which supports the Gr"obner Ring Conjecture.

Contribution

It extends the known properties of Serre conjecture rings from lexicographic to arbitrary rational monomial orders, providing evidence for the Gr"obner Ring Conjecture.

Findings

Serre conjecture rings are Bézout domains of Krull dimension ≤ 1 when base ring has this property.

The paper proposes a generalization to rational monomial orders.

Supports the Gr"obner Ring Conjecture in the rational case.

Abstract

It is well-known that for any commutative unitary ring $\mathbf{R}$, the Serre conjecture ring $\mathbf{R}\langle X \rangle$, i.e., the localization of the univariate polynomial ring $\mathbf{R}[X]$ at monic polynomials, is a B\'ezout domain of Krull dimension $\leq 1$ if so is $\mathbf{R}$. Consequently, defining by induction $\mathbf{R}\langle X_1,\ldots,X_n \rangle:=(\mathbf{R}\langle X_1,\ldots,X_{n-1}\rangle)\langle X_n\rangle$, the ring $\mathbf{R}\langle X_1,\ldots,X_n \rangle$ is a B\'ezout domain of Krull dimension $\leq 1$ if so is $\mathbf{R}$. The fact that $\mathbf{R}\langle X_1,\ldots,X_n \rangle$ is a B\'ezout domain when $\mathbf{R}$ is a valuation domain of Krull dimension $\leq 1$ was the cornerstone of Brewer and Costa's theorem stating that if $\mathbf{R}$ is a one-dimensional arithmetical ring then finitely generated projective $\mathbf{R}[X_1,\dots,X_n]$-modules are extended. It is also the key of the proof of the Gr\"obner Ring Conjecture in the lexicographic order case, namely the fact that for any valuation domain $\mathbf{R}$ of Krull dimension $\leq 1$, any $n \in \mathbb{N}_{>0}$, and any finitely generated ideal $I$ of $\mathbf{R}[X_1, \dots, X_n]$, the ideal $\operatorname{LT}(I)$ generated by the leading terms of the elements of $I$ with respect to the lexicographic monomial order is finitely generated. Since the ring $\mathbf{R}\langle X_1,\ldots,X_n\rangle$ can also be defined directly as the localization of the multivariate polynomial ring $\mathbf{R}[X_1,\dots,X_n]$ at polynomials whose leading coefficients according to the lexicographic monomial order with $X_1<X_2<\cdots<X_n$ is $1$, we propose to generalize the fact that $\mathbf{R}\langle X_1,\ldots,X_n\rangle$ is a B\'ezout domain of Krull dimension $\leq 1$ if so is $\mathbf{R}$ to any rational monomial order, bolstering the evidence for the Gr\"obner Ring Conjecture in the rational case.