On Characteristic Poset and Stanley Decomposition of S/I

Let K be a field and S = K[x1,…,xn] be the polynomial ring in n variables. Let I ⊂ S be a monomial ideal such that S/I is Cohen-Macaulay. By associating a finite poset [Formula: see text] to S/I, we show that if S/I is a Stanley ideal then T/Ĩ is also a Stanley ideal, where T = K[x11,…,x1a1,…,xn1,…,xnan] and Ĩ is the polarization of I.

On Stanley Depths of Certain Monomial Factor Algebras

Let S = K[x1 , . . . , xn] be the polynomial ring in n-variables over a ûeld 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 sdepthS(S/I) ≤ sizeS(I). When I is squarefree and bigsizeS(I) ≤ 2, the Stanley conjecture holds for S/I, i.e., sdepthS(S/I) ≥ depthS(S/I).

On Characteristic Poset and Stanley Decomposition

Let J ⊂ I be two monomial ideals such that I/J is Cohen Macaulay. By associating a finite posets $P_{I/J}^g$ to I/J, we show that if I/J is a Stanley ideal then $\widetilde{I/J}$ is also a Stanley ideal, where $\widetilde{I/J}$ is the polarization of I/J. We also give relations between sdepth and fdepth of I/J and $\widetilde{I/J}$