The index of reducibility of parameter ideals

Mark Wayne Rogers, Purdue University

Abstract

Let A be a Noetherian local ring with maximal ideal [special characters omitted], and let M be a finitely generated A-module of dimension d. Every submodule N of M may be written as a finite intersection of irreducible submodules of M. If this intersection is irredundant, then the number of irreducible submodules depends only on N. This number is called the index of reducibility of N; it equals one precisely when N is irreducible. When M/N has finite length, the index of reducibility of N equals the socle dimension of the quotient M/N. If M is Cohen-Macaulay, it is well known that the index of reducibility of [special characters omitted]M, where [special characters omitted] is a parameter ideal for M, depends only on M and not on [special characters omitted]. Without the assumption that M is Cohen-Macaulay, the index of reducibility of parameter ideals on M is not so well understood. We consider the case where M has finite local cohomologies; that is, when the local cohomology modules [special characters omitted] (M) have finite length except for i = d. Results are presented concerning the following question: If M has finite local cohomologies, does there exist an integer ℓ such that every parameter ideal for M contained in [special characters omitted] has the same index of reducibility? We show that the answer is yes if d = 1, or if M has positive depth and [special characters omitted] (M) = 0 for 2 ≤ i ≤ d − 1. We present an example showing the necessity of having finite local cohomologies. Along the way we develop results concerning the behavior of the unmixed components of parts of systems of parameters for a module having finite local cohomologies, provided the system is in a high power of the maximal ideal. When d = 1, we provide information on ℓ in terms of the reduction number of [special characters omitted].

Degree

Ph.D.

Advisors

Heinzer, Purdue University.

Subject Area

Mathematics

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