Componentwise linear powers and the $x$-condition

TL;DR

This paper introduces the x-condition, a criterion based on Gr"obner bases, which ensures componentwise linearity of graded components in standard graded algebras, with applications to Rees rings, symmetric algebras, and graph theory.

Contribution

It provides a new Gr"obner basis-based condition, called the x-condition, to determine when graded components are componentwise linear, extending understanding of algebraic structures like Rees rings and symmetric algebras.

Findings

The x-condition guarantees linear quotients in graded components.

Applications to Rees rings and symmetric algebras demonstrate the criterion's usefulness.

The criterion helps analyze powers of vertex cover ideals in graph classes.

Abstract

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over a field and $A$ a standard graded $S$-algebra. In terms of the Gr\"obner basis of the defining ideal $J$ of $A$ we give a condition, called the x-condition, which implies that all graded components $A_k$ of $A$ have linear quotients and with additional assumptions are componentwise linear. A typical example of such an algebra is the Rees ring $R(I)$ of a graded ideal or the symmetric algebra $Sym(M)$ of a module $M$. We apply our criterion to study certain symmetric algebras and the powers of vertex cover ideals of certain classes of graphs.