Graded ideals of K\"onig type

TL;DR

This paper introduces graded ideals of K"onig type, showing their Cohen--Macaulay property is characteristic-independent and exploring their applications to edge ideals, binomial edge ideals, and Hibi rings.

Contribution

It defines graded ideals of K"onig type and analyzes their properties, especially regarding Cohen--Macaulayness and canonical modules, with applications to various algebraic structures.

Findings

Cohen--Macaulay property of initial ideals is characteristic-independent for K"onig type ideals.

A sequence of linear forms acts as a system of parameters for these ideals.

Explicit descriptions of canonical modules for edge ideals, binomial edge ideals, and Hibi rings.

Abstract

Inspired by the notion of K\"onig graphs we introduce graded ideals of K\"onig type with respect to a monomial order $<$. It is shown that if $I$ is of K\"onig type, then the Cohen--Macaulay property of $\ini_<(I)$ does not depend on the characteristic of the base field. This happens to be the case also for $I$ itself when $I$ is a binomial edge ideal. Attached to an ideal of K\"onig type is a sequence of linear forms, whose elements are variables or differences of variables. This sequence is a system of parameters for $\ini_<(I)$, and is a potential system of parameters for $I$ itself. We study in detail the ideals of K\"onig type among the edge ideals, binomial edge ideals and the toric ideal of a Hibi ring and use the K\"onig property to determine explicitly their canonical module.