## Nagoya Mathematical Journal

### Componentwise linear ideals

#### Abstract

A componentwise linear ideal is a graded ideal $I$ of a polynomial ring such that, for each degree $q$, the ideal generated by all homogeneous polynomials of degree $q$ belonging to $I$ has a linear resolution. Examples of componentwise linear ideals include stable monomial ideals and Gotzmann ideals. The graded Betti numbers of a componentwise linear ideal can be determined by the graded Betti numbers of its components. Combinatorics on squarefree componentwise linear ideals will be especially studied. It turns out that the Stanley-Reisner ideal $I_{\Delta}$ arising from a simplicial complex $\Delta$ is componentwise linear if and only if the Alexander dual of $\Delta$ is sequentially Cohen-Macaulay. This result generalizes the theorem by Eagon and Reiner which says that the Stanley-Reisner ideal of a simplicial complex has a linear resolution if and only if its Alexander dual is Cohen-Macaulay.

#### Article information

Source
Nagoya Math. J., Volume 153 (1999), 141-153.

Dates
First available in Project Euclid: 27 April 2005

https://projecteuclid.org/euclid.nmj/1114630823

Mathematical Reviews number (MathSciNet)
MR1684555

Zentralblatt MATH identifier
0930.13018

#### Citation

Herzog, Jürgen; Hibi, Takayuki. Componentwise linear ideals. Nagoya Math. J. 153 (1999), 141--153. https://projecteuclid.org/euclid.nmj/1114630823

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