maximal depth property of bigraded modules

2020/07/11 by Rahimi, Ahad
#13C14 #13C15 #13D45 #13F20 #16W50 #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2007.05744

Abstract

Let S=K[x₁, …, x_m, y₁, …, y_n] be the standard bigraded polynomial ring over a field K. Let M be a finitely generated bigraded S-module and Q=(y₁, …, 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.

Related

• Maximal depth property of finitely generated modules
• Sequentially Cohen--Macaulayness of bigraded modules
• On associated graded modules of maximal Cohen-Macaulay modules over hypersurface rings
• Bigraded structures and the depth of blow-up algebras
• On the structure of sequentially Cohen--Macaulay bigraded modules