Ideals with componentwise linear powers
Takayuki Hibi, Somayeh Moradi
TL;DR
The paper investigates when powers of ideals retain componentwise linearity by linking the $x$-condition on the defining ideal of the Rees algebra to linear quotients of graded components. It shows that a quadratic initial ideal $ ext{in}_{<}(J)$ guarantees linear quotients for each $A_k$, implying that $I^k$ is componentwise linear for all $k$ when $J$ is the Rees algebra of a graded ideal $I$. A graph-theoretic construction $G(H_1, frac{ o}{ o}H_n)$ is then used to transfer the $x$-condition to vertex cover ideals, proving that all powers of $I_{G(H_1, frac{ o}{ o}H_n)}$ are componentwise linear under suitable conditions. Specializing to graph families with quadratic Gröbner bases and cone graphs yields broad classes where cover ideals have componentwise linear powers, including that every power of the vertex cover ideal of a Cohen–Macaulay Cameron–Walker graph has a linear resolution. Overall, the work connects Gröbner-theoretic criteria, Rees-algebra structure, and combinatorial graph constructions to identify robust criteria for when powers preserve componentwise linearity.
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.