Stanley depth of monomial ideals with small number of generators

Details Stanley depth of monomial ideals with small number of generators Stanley depth of monomial ideals with small number of generators

2009-06-05

arxiv.org/abs/0906.1105v1

Mathematics

Commutative Algebra

7 pages. submitted to Central European Journal of Mathematics

Scientific paper

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 $I=vI'$, 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 $I\subset K[x_1,x_2,x_3]$.

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