Maximal depth property of bigraded modules (Q2047469)

Let \(K\) be a field and \(S = K[x_1, \ldots , x_m, y_1, \ldots, y_n]\) be the standard bigraded polynomial ring over \(K\). Set the bigraded irrelevant ideals \(P = (x_1,\ldots, x_m)\) and \(Q = (y_1, \ldots, y_n).\) Let \(M\) be a finitely generated bigraded \(S\)-module. Denote \(\mathrm{mgrade}(Q, M) =: \min{\mathrm{cd}(Q, S/\mathfrak{p}) \mid \mathfrak{p} \in \mathrm{Ass}(M)}.\) The \(S\)-module \(M\) has maximal depth with respect to \(Q\) if \(\mathrm{grade}(Q, M) = \mathrm{mgrade}(Q, M).\) The author shows some properties of \(\mathrm{mgrade}(Q, M).\) The paper provided a result on the non-vanishing of top local cohomology module \(H^{\mathrm{grade}(Q, M)}_Q(M).\) The author is interested in classifying all hypersurface rings that have maximal depth with respect to \(Q.\) Main results: (i) Assume \(M\) has maximal depth with respect to \(Q\) with \(\mathrm{grade}(Q, M) > 0\) and \( |K| = \infty.\) Then \(H^{\mathrm{grade}(Q, M)}_Q(M)\) is not finitely generated. (ii) Let \(I \subseteq S\) be a monomial ideal. Then \(\mathrm{mgrade}(Q, S/I) = n - d\) where \(d\) is the maximal height of an associated prime of \(I\) in \(Q.\) Moreover, if \(S/I\) is Cohen-Macaulay, then \(S/I\) has maximal depth with respect to \(P\) and \(Q.\)