Lectures on the asymptotic theory of ideals / D. Rees.

Format:

• Book

Author/Creator:

• Rees, D., 1918-2013, author.

Series:

• London Mathematical Society lecture note series ; 113.
• London Mathematical Society lecture note series ; 113

Language:

English

Subjects (All):

• Ideals (Algebra)--Asymptotic theory.

Physical Description:

1 online resource (201 pages) : digital, PDF file(s).

Place of Publication:

Cambridge : Cambridge University Press, 1988.

Language Note:

English

Summary:

In this book Professor Rees introduces and proves some of the main results of the asymptotic theory of ideals. The author's aim is to prove his Valuation Theorem, Strong Valuation Theorem, and Degree Formula, and to develop their consequences. The last part of the book is devoted to mixed multiplicities. Here the author develops his theory of general elements of ideals and gives a proof of a generalised degree formula. The reader is assumed to be familiar with basic commutative algebra, as covered in the standard texts, but the presentation is suitable for advanced graduate students. The work is an expansion of lectures given at Nagoya University.

Contents:

• Cover; Half-title; Title; Copyright; Dedication; Contents; Preface; Introduction; Graded Rings and Modules; 1. Definitions and Samuel's theorem.; 2. Rappel on Koszul complexes.; 3. Additive functions on modules.; 4. The Hilbert series of a graded module.; Filtrations and Noether Filtrations; 1. Generalities on nitrations.; 2. Integer-valued nitrations.; 3. Noether nitrations.; 4. Miscellaneous results.; The Theorems of Matijevic and Mori-Nagata; 1. Matijevic's Theorem.; 2. The Mori-Nagata Theorem.; The Valuation Theorem; 1. The Valuation Theorem.; 2. Miscellaneous results.
• The Strong Valuation Theorem1. Preliminaries.; 2. Completions, the Cohen Structure Theorems, and Nagata's Theorem.; 3. The Strong Valuation Theorem.; 4. A criterion for analytic unramification.; Ideal Valuations (1); 1. Introduction.; 2. The ideal valuations of a local domain.; Ideal Valuations (2); 1. Introduction.; 2. Ideal valuations of finitely generated extensions.; 3. Applications.; 4. More on the rings Qn.; The Multiplicity Function associated with a Filtration; 1. Filtrations on a module.; 2. The multiplicity function of m-primary filtrations.
• The Degree Function of a Noether Filtration1. Definition and elementary properties.; 2. The degree formula: generalities.; 3. The degree formula: preliminary form.; 4. The degree formula: final version.; The General Extension of a Local Ring; 1. Introduction.; 2. Prime ideals of Qg.; 3. Valuations on general extensions.; General Elements; 1. Introduction.; 2. The ideal generated by a set of general elements.; 3. Some invariants of sets of ideals of a local ring.; Generalised Degree Formula; 1. Multiplicities again.; 2. Mixed multiplicities.; 3. The generalised degree formula.
• 4. A final illustration.Bibliography; Index; Index of Symbols

Notes:

• Title from publisher's bibliographic system (viewed on 05 Oct 2015).
• Includes bibliographical references and indexes.

ISBN:

• 1-139-88169-8
• 1-107-36616-X
• 1-107-37089-2
• 1-107-36125-7
• 1-107-36960-6
• 1-299-40396-4
• 1-107-36370-5
• 0-511-52595-8