论文标题
大型模块的最大深度特性
maximal depth property of bigraded modules
论文作者
论文摘要
令$ s = k [x_1，\ dots，x_m，y_1，\ dots，y_n] $是字段$ k $上的标准大型多项式环。令$ m $为有限生成的bigraded $ s $ module，$ q =（y_1，\ dots，y_n）$。我们说，如果有关联的prime $ \ pp $ $ m $，则$ m $相对于$ q $具有最大深度，以便$ \级（q，m）= \ cd（q，s/\ pp）$。在本文中，我们研究了有限生成的大型模块，相对于$ Q $，最大程度地深度为。结果表明，关于$ Q $，相对于$ q $，依次cohen-麦克劳莱模块具有最大的深度。实际上，最大深度属性概括了依次的cohen--macaulayness。接下来，我们表明，如果$ m $与$ \级（q，m）> 0 $相对于$ q $具有最大深度，则$ h^{\等级（q，m）} _ {q}（q}（m）$不是有限生成的。结果，“广义cohen- -macaulay模块相对于$ q $”具有“相对于$ q $的最大深度”是Cohen-与$ Q $相对于$ Q $。所有相对于$ Q $具有最大深度的Hypersurface环均分类。
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.
