Monomial ideals under ideal operations

Abstract

In this paper, we show for a monomial ideal $I$ of $K[x_1,x_2,\dots ,x_n]$ that the integral closure $\ol{I}$ is a monomial ideal of Borel type (Borel-fixed, strongly stable, lexsegment, or universal lexsegment respectively), if $I$ has the same property. We also show that the $k^{th}$ symbolic power $I^{(k)}$ of $I$ preserves the properties of Borel type, Borel-fixed and strongly stable, and $I^{(k)}$ is lexsegment if $I$ is stably lexsegment. For a monomial ideal $I$ and a monomial prime ideal $P$, a new ideal $J(I,P)$ is studied, which also gives a clear description of the primary decomposition of $I^{(k)}$. Then a new simplicial complex ${}_J△$ of a monomial ideal $J$ is defined, and it is shown that $I_{{}_J{△}^{\vee }}=\sqrt{J}$. Finally, we show under an additional weak assumption that a monomial ideal is universal lexsegment if and only if its polarization is a squarefree strongly stable ideal.

18 pages

References in corpus (1)

• Some remarks on Borel type ideals