Componentwise linearity, Fröberg's analogue, and classification of linear support-two monomial ideals

Abstract

In this paper, we investigate the componentwise linear property of support-two monomial ideals. Our first main result shows that if $I$ is a support-two monomial ideal, then its underlying simple graph $G_I$ is co-chordal; equivalently, by Fröberg's theorem, $\sqrt{I}$ admits a linear resolution. This phenomenon is quite rare for general monomial ideals. In fact, there exist monomial ideals with linear resolutions whose radicals fail to have linear resolutions, even when the radical is the edge ideal of a graph. For any monomial ideal $I$, one always has $μ(I)\geq μ(\sqrt{I})$. In the literature, support-two monomial ideals satisfying $μ(I)=μ(\sqrt{I})$ are of special interest, as they include edge ideals of simple graphs, weighted oriented graphs, edge-weighted graphs, and vertex-weighted graphs. We refer to such ideals as minimal support-two monomial ideals. We explicitly characterize all minimal support-two monomial ideals, as well as their powers, that admit linear resolutions. Next, we classify the linearity of non-minimal ones. Consequently, we obtain a complete classification of linear support-two monomial ideals, which shows that the property of being linear does not depend on the characteristics of the base field for support-two monomial ideals.