Czechoslovak Mathematical Journal, Vol. 67, No. 3, pp. 767-778, 2017

# The cleanness of (symbolic) powers of Stanley-Reisner ideals

## Somayeh Bandari, Ali Soleyman Jahan

#### Received April 8, 2016.   First published August 8, 2017.

Somayeh Bandari, Department of Engineering Sciences and Physics, Buein Zahra Technical University, 3451745346, Buein Zahra, Iran, and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5746, Tehran, Iran, e-mail: somayeh.bandari@yahoo.com; Ali Soleyman Jahan, Department of Mathematics, University of Kurdistan, 66177-15175, Sanandaj, Iran, e-mail: solymanjahan@gmail.com

Abstract: Let $\Delta$ be a pure simplicial complex on the vertex set $[n]=\{1,\ldots,n\}$ and $I_\Delta$ its Stanley-Reisner ideal in the polynomial ring $S=K[x_1,\ldots,x_n]$. We show that $\Delta$ is a matroid (complete intersection) if and only if $S/I_\Delta^{(m)}$ ($S/I_\Delta^m$) is clean for all $m\in\mathbb{N}$ and this is equivalent to saying that $S/I_\Delta^{(m)}$ ($S/I_\Delta^m$, respectively) is Cohen-Macaulay for all $m\in\mathbb{N}$. By this result, we show that there exists a monomial ideal $I$ with (pretty) cleanness property while $S/I^m$ or $S/I^{(m)}$ is not (pretty) clean for all integer $m\geq3$. If $\dim(\Delta)=1$, we also prove that $S/I_\Delta^{(2)}$ ($S/I_\Delta^2$) is clean if and only if $S/I_\Delta^{(2)}$ ($S/I_\Delta^2$, respectively) is Cohen-Macaulay.

Keywords: clean; Cohen-Macaulay simplicial complex; complete intersection; matroid; symbolic power

Classification (MSC 2010): 13F20, 05E40, 13F55

DOI: 10.21136/CMJ.2017.0173-16

Full text available as PDF.

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