maximal depth property of bigraded modules

Let $S=K[x_1, \dots, x_m, y_1, \dots, y_n]$ be the standard bigraded polynomial ring over a field $K$. Let $M$ be a finitely generated bigraded $S$-module and $Q=(y_1, \dots, y_n)$. We say $M$ has maximal depth with respect to $Q$ if there is an associated prime $\pp$ of $M$ such that $\grade(Q, M)=\cd(Q, S/\pp)$. In this paper, we study finitely generated bigraded modules with maximal depth with respect to $Q$. It is shown that sequentially Cohen--Macaulay modules with respect to $Q$ have maximal depth with respect to $Q$. In fact, maximal depth property generalizes the concept of sequentially Cohen--Macaulayness. Next, we show that if $M$ has maximal depth with respect to $Q$ with $\grade(Q, M)>0$, then $H^{\grade(Q, M)}_{Q}(M)$ is not finitely generated. As a consequence, "generalized Cohen--Macaulay modules with respect to $Q$" having "maximal depth with respect to $Q$" are Cohen--Macaulay with respect to $Q$. All hypersurface rings that have maximal depth with respect to $Q$ are classified.

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