## Nagoya Mathematical Journal

### Alternative polarizations of Borel fixed ideals

Kohji Yanagawa

#### Abstract

For a monomial ideal $I$ of a polynomial ring $S$, a polarization of $I$ is a square-free monomial ideal $J$ of a larger polynomial ring $\widetilde {S}$ such that $S/I$ is a quotient of $\widetilde {S}/J$ by a (linear) regular sequence. We show that a Borel fixed ideal admits a nonstandard polarization. For example, while the usual polarization sends $xy^{2}\in S$ to $x_{1}y_{1}y_{2}\in \widetilde {S}$, ours sends it to $x_{1}y_{2}y_{3}$. Using this idea, we recover/refine the results on square-free operation in the shifting theory of simplicial complexes. The present paper generalizes a result of Nagel and Reiner, although our approach is very different.

#### Article information

Source
Nagoya Math. J., Volume 207 (2012), 79-93.

Dates
First available in Project Euclid: 26 July 2012

https://projecteuclid.org/euclid.nmj/1343309819

Digital Object Identifier
doi:10.1215/00277630-1630032

Mathematical Reviews number (MathSciNet)
MR2957143

Zentralblatt MATH identifier
1256.13005

#### Citation

Yanagawa, Kohji. Alternative polarizations of Borel fixed ideals. Nagoya Math. J. 207 (2012), 79--93. doi:10.1215/00277630-1630032. https://projecteuclid.org/euclid.nmj/1343309819

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