Stanley conjecture on monomial ideals of mixed products (Q2259542)

Let \(I'\subseteq I\) be monomial ideals in a polynomial ring \(S_1=k[x_1,\ldots,x_m]\), and \(J' \subseteq J\) be monomial ideals in a polynomial ring \(S_2=k[y_1,\ldots,y_n]\). The ideal \(I\) generated by \(I'J+IJ'\) in \(S=S_1\otimes_k S_2\) is called by the authors a \textit{generalized mixed product} ideal. Let \(I_q\) denotes the ideal generated by squarefree monomials of degree \(q\) of \(S_1\). Define similarly the ideal \(J_q\) in \(S_2\). If \(I'=I_q, I=I_r\) where \(q\geq r\), and \(J'=J_s, J=J_t\) where \(s\geq t\), then the ideal \(I=(I'J+IJ')S\) is called a \textit{mixed product} ideal. In this paper, the authors prove that if \(I\) is a mixed product ideal, then the Stanley conjecture holds for \(I\) and \(S/I\). The proof is based on a careful analysis of the depth and Stanley depth of the multigraded \(S\)-modules of type \(IS\cap JS, IS+JS, S/(IS\cap JS), S/(IS+JS)\).