Stanley depth of weakly polymatroidal ideals and squarefree monomial ideals (Q470866)

Let \(S=\mathbb{K}[x_1, \dots, x_n]\) be the polynomial ring over a field \(\mathbb{K}\). Let \(M\) be a \(\mathbb{Z}^n\)-graded \(S\)-module. Let \({\mathcal D}\) denote a direct sum decomposition \( M=\bigoplus_{i=1}^r u_i\mathbb{K}[Z]\) of \(M\) as \(\mathbb{K}\)-vector space, where \(u_i \in M\), \(Z \subset \{x_1, \dots, x_n\}\) such that \(u_i\mathbb{K}[Z]\) is a free \(\mathbb{K}[Z]\)-module. Setting \(\text{sdepth}({\mathcal D})=\text{min}_{i=1}^r|Z_i|\), the Stanley depth of \(M\) is defined to be the maximum of \(\text{sdepth}({\mathcal D})\) over all \({\mathcal D}\), denoted by \(\text{sdepth}(M)\). A monomial \({\mathbf x}^{\mathbf a}=x_1^{a_1}\cdots x_n^{a_n} \in S\) is squarefree if each \(a_i \in \{0, 1\}\), where \({\mathbf x}=(x_1, \dots, x_n)\) and \({\mathbf a}=(a_1, \dots, a_n)\), and an ideal \(I \subset S\) is squarefree if it can be minimally generated by a set of squrefree monomials. Let \(G(I)=\{{\mathbf x}^{\mathbf a_1}, \dots, {\mathbf x}^{a_n}\}\) be the set of minimal generators of \(I\). Then \(\text{rank}(I)\) is the cardinality of the largest subset of \(\{{\mathbf a_1}, \dots, {\mathbf a_n}\}\). In this paper, the author proves that for every squarefree monomial ideal \(I\) there hold NEWLINE\[NEWLINE\text{sdepth}(I) \geq n- \text{rank}(I) + 1 \quad\text{and} \quad \text{sdepth}(S/I) \geq n- \text{rank}(I).NEWLINE\]NEWLINE This proves a conjecture of the author in a special case [Collect. Math. 64, No. 3, 351--362 (2013; Zbl 1317.13024)]. Moreover, we have a similar result for a weakly polymatroidal ideal \(I\). In particular, when \(I\) is generated in a single degree, from the results described above we have that there hold \(\text{sdepth}(I) \geq n- \ell(I) + 1\) and \(\text{sdepth}(S/I) \geq n- \ell(I)\), regardless of the types of \(I\), where \(\ell(I)\) denotes the analytic spread of \(I\) (definition omitted).