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On Stanley Depths of Certain Monomial Factor Algebras
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- Journal:
- Canadian Mathematical Bulletin / Volume 58 / Issue 2 / 01 June 2015
- Published online by Cambridge University Press:
- 20 November 2018, pp. 393-401
- Print publication:
- 01 June 2015
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- Article
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Let
$S\,=\,K\left[ {{x}_{1}},\,\ldots \,,\,{{x}_{n}} \right]$ be the polynomial ring in 
$n$-variables over a field 
$K$ and 
$I$ a monomial ideal of 
$S$. According to one standard primary decomposition of $I$, we get a Stanley decomposition of the monomial factor algebra 
$S/I$. Using this Stanley decomposition, one can estimate the Stanley depth of $S/I$. It is proved that 
$\text{sdept}{{\text{h}}_{s}}\left( S/I \right)\,\ge \,\text{siz}{{\text{e}}_{S}}\left( I \right)$. When $I$ is squarefree and 
$\text{bigsiz}{{\text{e}}_{S}}\left( I \right)\,\le \,2$, the Stanley conjecture holds for $S/I$, i.e., 
$\text{sdept}{{\text{h}}_{S}}\left( S/I \right)\ge \text{dept}{{\text{h}}_{S}}\left( S/I \right)$.
