Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- Chapter 1 Graded Rings and Modules
- Chapter 2 Filtrations and Noether Filtrations
- Chapter 3 The Theorems of Matijevic and Mori-Nagata
- Chapter 4 The Valuation Theorem
- Chapter 5 The Strong Valuation Theorem
- Chapter 6 Ideal Valuations (1)
- Chapter 7 Ideal Valuations (2)
- Chapter 8 The Multiplicity Function associated with a Filtration
- Chapter 9 The Degree Function of a Noether Filtration
- Chapter 10 The General Extension of a Local Ring
- Chapter 11 General Elements
- Chapter 12 Mixed Multiplicities and the Generalised Degree Formula
- Bibliography
- Index
- Index of Symbols
Chapter 7 - Ideal Valuations (2)
Published online by Cambridge University Press: 17 September 2009
- Frontmatter
- Contents
- Preface
- Introduction
- Chapter 1 Graded Rings and Modules
- Chapter 2 Filtrations and Noether Filtrations
- Chapter 3 The Theorems of Matijevic and Mori-Nagata
- Chapter 4 The Valuation Theorem
- Chapter 5 The Strong Valuation Theorem
- Chapter 6 Ideal Valuations (1)
- Chapter 7 Ideal Valuations (2)
- Chapter 8 The Multiplicity Function associated with a Filtration
- Chapter 9 The Degree Function of a Noether Filtration
- Chapter 10 The General Extension of a Local Ring
- Chapter 11 General Elements
- Chapter 12 Mixed Multiplicities and the Generalised Degree Formula
- Bibliography
- Index
- Index of Symbols
Summary
Introduction.
The purpose of this chapter is to relate the set of ideal valuations of a finitely generated domain B over a noetherian domain A to the ideal valuations of A itself. For this purpose the description of the ideal valuations of A in terms of the completions of the localisations of A is not convenient and we therefore use a somewhat different one, namely that a valuation ≥0 on A is an ideal valuation of A if and only if there is a finitely generated extension B of A with the same field of fractions, such that v(x) ≥ 0 on B and the centre of v on B has height 1. Note that this implies that v is a Krull valuation of B by Theorem 3.24, and we could weaken the above condition by simply requiring that v be a Krull valuation of B. The proof of this criterion is obtained by putting together Theorem 4.24 and the Corollary to Lemma 6.11. This is done in the proof of the following theorem.
THEOREM 7.11. Let A be a noetherian domain, v be a valuation on the field of fractions F of A such that v(x) ≥ 0 on A. Then v is an ideal valuation of A if and only if there exists a finitely generated extension B of A, contained in F, such that v(x) ≥ 0 on B and the centrepof v on B has height 1.
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- Lectures on the Asymptotic Theory of Ideals , pp. 103 - 123Publisher: Cambridge University PressPrint publication year: 1988