Stanley depth of monomial ideals with small number of generators

Abstract

For a monomial ideal $I\subset S=K[x_1,...,x_n]$, we show that $\sdepth(S/I)\geq n-g(I)$, where $g(I)$ is the number of the minimal monomial generators of $I$. If $I=v{I}^{\prime }$, where $v\in S$ is a monomial, then we see that $\sdepth(S/I)=\sdepth(S/I')$. We prove that if $I$ is a monomial ideal $I\subset S$ minimally generated by three monomials, then $I$ and $S/I$ satisfy the Stanley conjecture. Given a saturated monomial ideal $I\subset K[x_1,x_2,x_3]$ we show that $\sdepth(I)=2$. As a consequence, $\sdepth(I)\geq \sdepth(K[x_1,x_2,x_3]/I)+1$ for any monomial ideal in $I\subset K[x_1,x_2,x_3]$.

7 pages. submitted to Central European Journal of Mathematics

References in corpus (6)

• Stanley Conjecture in small embedding dimension
• Stanley Decompositions, Pretty Clean Filtrations and Reductions Modulo Regular Elements
• Stanley decompositions and localization
• Sequentially Cohen-Macaulay monomial ideals of embedding dimension four
• Some remarks on the Stanley's depth for multigraded modules
• How to compute the Stanley depth of a monomial ideal

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