The Gröbner ring conjecture in the lexicographic order case

Abstract

We prove that a valuation domain $\mathbf{V}$ has Krull dimension $\le$1 if and only if, for any $n$, fixing the lexicographic order as monomial order in $\mathbf{V}[X_1,\ldots ,X_n]$, for every finitely generated ideal $I$ of $\mathbf{V}[X_1,\ldots ,X_n]$, the ideal generated by the leading terms of the elements of $I$ is also finitely generated. This proves the Gröbner ring conjecture in the lexicographic order case. The proof we give is both simple and constructive. The same result is valid for Prüfer domains. As a “scoop”, contrary to the common idea that Gröbner bases can be computed exclusively on Noetherian ground, we prove that computing Gröbner bases over $\mathbf{R}[X_1,\ldots , X_n]$, where $\mathbf{R}$ is a Prüfer domain, has nothing to do with Noetherianity, it is only related to the fact that the Krull dimension of $\mathbf{R}$ is $\le$1.