152 research outputs found
Ratliff-Rush Filtrations associated with ideals and modules over a Noetherian ring
Let be a commutative Noetherian ring, a finitely generated -module
and a proper ideal of . In this paper we introduce and analyze some
properties of , {\it the
Ratliff-Rush ideal associated with and }. When (or more generally
when is projective) then , the usual Ratliff-Rush
ideal associated with .
If is a regular ideal and \ann M=0 we show that is a stable -filtration. If M_{\p} is free for all
{\p}\in \spec R\setminus \mspec R, then under mild condition on we show
that for a regular ideal , is finite. Further
if A^*(I)\cap \mspec R =\emptyset (here is
the stable value of the sequence \Ass (R/{I^n})). Our generalization also
helps to better understand the usual Ratliff-Rush filtration. When is a
regular \m-primary ideal our techniques yield an easily computable bound for
such that for all .
For any ideal we show that \widetilde{I^nM}=I^nM+H^0_I(M)\quad\mbox{for
all} n\gg 0. This yields that is Noetherian if and only
if \depth M>0. Surprisingly if then is
always a Noetherian and a Cohen-Macaulay -module. Application to
Hilbert coefficients is also discussed.Comment: 27 pages. Many minor revisions made, including little changes in
title and abstract. Five additional refernces added. To appear in Journal of
algebr
Hilbert Functions of Filtered Modules
In this presentation we shall deal with some aspects of the theory of Hilbert
functions of modules over local rings, and we intend to guide the reader along
one of the possible routes through the last three decades of progress in this
area of dynamic mathematical activity. Motivated by the ever increasing
interest in this field, our goal is to gather together many new developments of
this theory into one place, and to present them using a unifying approach which
gives self-contained and easier proofs. In this text we shall discuss many
results by different authors, following essentially the direction typified by
the pioneering work of J. Sally. Our personal view of the subject is most
visibly expressed by the presentation of Chapters 1 and 2 in which we discuss
the use of the superficial elements and related devices. Basic techniques will
be stressed with the aim of reproving recent results by using a more elementary
approach. Over the past few years several papers have appeared which extend
classical results on the theory of Hilbert functions to the case of filtered
modules. The extension of the theory to the case of general filtrations on a
module has one more important motivation. Namely, we have interesting
applications to the study of graded algebras which are not associated to a
filtration, in particular the Fiber cone and the Sally-module. We show here
that each of these algebras fits into certain short exact sequences, together
with algebras associated to filtrations. Hence one can study the Hilbert
function and the depth of these algebras with the aid of the know-how we got in
the case of a filtration.Comment: 127 pages, revised version. Comments and remarks are welcom
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