Maximal depth property of bigraded modules

Abstract

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 $${\mathfrak {p}}$$ of M such that $${\text {grade}}(Q, M)={\text {cd}}(Q, S/{\mathfrak {p}})$$ . 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 $${\text {grade}}(Q, M)>0$$ , then $$H^{{\text {grade}}(Q, M)}_{Q}(M)$$ 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.