Tight closure, joint reductions, and mixed multiplicities

Irena Swanson, Purdue University

Abstract

In Chapter 3 we extend Rees' Multiplicity theorem to mixed multiplicities and joint reductions. The main theorem of Chapter 3 also generalizes Boger's Multiplicity theorem for certain pairs of ideals not necessarily primary to the maximal ideal. By using Hochster and Huneke's theory of tight closure, we prove in Chapter 4 several Briancon-Skoda-type theorems, namely theorems of the type $\overline{I\sp{n}}\subseteq(I\sp{n-k})\sp\#$ for some ideal I, some integer k, and all integers $n\ge k,$ where $\sp\#$ sometimes represents the identity operator, sometimes tight and sometimes plus closure. We prove several versions of the corresponding statements for several ideals and joint reductions. In most cases we can choose k (or the equivalent for several ideals) to be independent of I (resp. of the several ideals). We generalize Hochster and Huneke's version of the Briancon-Skoda theorem for several ideals. In Chapter 5 we prove some asymptotic properties of primary decompositions of powers of an ideal and of ideals generated by different powers of elements in some regular sequence. The main result is that for any ideal I in a Noetherian ring there exists an integer k such that for every prime $P\in\ \cup\sb{n}{\rm Ass}(R/I\sp{n})$ of height at most 1 over I, and for every integer n, there exists an irredundant primary decomposition $q\sb1\ \cap\cdots\cap\ q\sb{m}=I\sp{n}$ such that if $\sqrt{q\sb{i}}=P,$ then $P\sp{kn}\subseteq q\sb{i}.$ In particular, if R is a local ring with maximal ideal m and I is a prime ideal of dimension 1, then $m\sp{kn}I\sp{(n)}\subseteq I\sp{n}.$

Degree

Ph.D.

Advisors

Huneke, Purdue University.

Subject Area

Mathematics

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