Local Cohomology and complete intersections of rank 1

Abstract

The number of equations needed to cut out a variety given by an ideal is called the arithmetic rank (of the ideal). It was shown in [8] that the notion of arithmetic rank is strongly related to the concept of regular sequences on the Matlis duals of certain local cohomology modules and thus is related to the set of associated primes of such Matlis duals. While in [8] the general situation with no restrictions on arithmetic rank and cohomological dimension was investigated, this work concentrates on the case ``cohomological dimension of the given ideal is (at most) one'' and also on the interplay between non-graded local and graded situations. The main results are new characterizations for the case $\ara \leq 1$ (theorem 1 and theorem 2), the fact that given a graded ring $R$ and a homogenous ideal $I$ of cohomological dimension at most one the inclusion $$\{x\in I| x\hbox {non-homogenous}\} \subseteq \bigcup_{\goth p\in \Ass_R(D(\LCMo ^1_I(R)))}\goth p$$ holds (theorem 4, $D$ is a Matlis dual functor) and a (somewhat technical) statement on regular sequences on Matlis duals of certain local cohomology modules (theorem 5), which shows that regular sequences on such Matlis duals behave well in some sense although these modules are not finite in general.