Componentwise linear ideals in Veronese rings

TL;DR

This paper characterizes when graded ideals in Veronese subrings of polynomial rings are componentwise linear, extending known results from polynomial rings to Veronese rings in characteristic zero.

Contribution

It provides a new characterization of componentwise linear ideals in Veronese subrings, generalizing previous results from polynomial rings.

Findings

Characterization of componentwise linear ideals in Veronese rings for char(K)=0

Extension of known polynomial ring results to Veronese subrings

Framework for analyzing graded ideals in Veronese contexts

Abstract

In this article, we study the componentwise linear ideals in the Veronese subrings of $R=K[x_1,\ldots,x_n]$. If char$(K)=0$, then we give a characterization for graded ideals in the $c^{th}$ Veronese ring $R^{(c)}$ to be componentwise linear. This characterization is an analogue of that over $R$ due to Aramova, Herzog and Hibi in \cite{ah00} and Conca, Herzog and Hibi in \cite{chh04}.