A counterexample to the Gröbner ring conjecture
This paper contributes to the generalization of Gröbner basis theory to general commutative rings, in this case considering a question related to valuation domains. The question first arose during the author's development of dynamical Gröbner bases [\textit{I. Yengui}, J. Algebra 301, No. 2, 447--458 (2006; Zbl 1113.13013)]. The ``Gröbner ring conjecture'' posits an equivalence between the following statements: \begin{itemize} \item It is possible to compute in a finite number of steps a Gröbner basis for each finitely generated nonzero ideal of a ring over a valuation ring \(\mathbf V\) by a generalized Buchberger algorithm. \item \(V\)'s Krull dimension is at most 1. \item \(V\) is a ``Gröbner ring''; that is, for every finitely generated ideal of \(\mathbf V\), the ideal of its leading terms is also finitely generated. \end{itemize} (Although it appears in the original formulation, no mention is made in this paper of the first, algorithmic characterization.) The conjecture rests on several parameters, one of which is the monomial ordering, and the counterexample to this case rests on a valuation domain with valuation group \((\mathbb Z+\mathbb Z[\sqrt2],+)\) and a monomial ordering defined by the vector \((1 \ \sqrt2)\). In this context the author describes a finitely-generated ideal whose ideal of leading terms is not finitely generated. The paper is relatively short (10 pages) and largely self-contained, so that the reader familiar with the basic algebraic concepts need not be familiar with the previous investigations to understand this result. The author concludes with an updated conjecture that restricts the claim to a \textit{rational} monomial order.
scientific article; zbMATH DE number 7385172
scientific article