Ideals with componentwise linear powers

TL;DR

This paper establishes conditions under which the powers of certain ideals in polynomial rings are componentwise linear, with applications to graph theory and Cohen-Macaulay properties.

Contribution

It proves that quadratic initial ideals imply componentwise linearity of all graded components and provides constructions for graphs with ideals having this property.

Findings

Quadratic initial ideals lead to componentwise linear powers.

Cohen-Macaulay Cameron-Walker graphs have all powers with linear resolutions.

Graph constructions can produce ideals with componentwise linear powers.

Abstract

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over a field $K$, and let $A$ be a finitely generated standard graded $S$-algebra. We show that if the defining ideal of $A$ has a quadratic initial ideal, then all the graded components of $A$ are componentwise linear. Applying this result to the Rees ring $\mathcal{R}(I)$ of a graded ideal $I$ gives a criterion on $I$ to have componentwise linear powers. Moreover, for any given graph $G$, a construction on $G$ is presented which produces graphs whose cover ideals $I_G$ have componentwise linear powers. This in particular implies that for any Cohen-Macaulay Cameron-Walker graph $G$ all powers of $I_G$ have linear resolutions. Moreover, forming a cone on special graphs like unmixed chordal graphs, path graphs and Cohen-Macaulay bipartite graphs produces cover ideals with componentwise linear powers.