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arXiv · 1609.00089

Quadratic Generated Normal Domains From Graphs

Drew J. Lipman·Michael A. Burr
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Abstract

Determining whether an arbitrary subring $R$ of $k[x_1^{\pm 1},\dots, x_n^{\pm 1}]$ is a normal domain is, in general, a nontrivial problem, even in the special case of a monomial generated domain. In this paper, we provide a complete characterization of the normality and normalizations of quadratic-monomial generated domains. For a quadratic-monomial generated domain $R$, we develop a combinatorial structure that assigns, to each quadratic monomial of the ring, an edge in a mixed signed, directed graph $G$, i.e., a graph with signed edges and directed edges. We classify the normality and the normalizations of such rings in terms of a generalization of the combinatorial odd cycle condition on $G$.

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Drew J. Lipman, Michael A. Burr. 2016-09-01. Quadratic Generated Normal Domains From Graphs. https://arxiv.org/abs/1609.00089

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