The multivariate Serre conjecture ring

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.