Introductory Notes on Valuation Rings and Function Fields in One Variable
The book deals with the (elementary and introductory) theory of valuation rings. As explained in the introduction, this represents a useful and important viewpoint in algebraic geometry, especially concerning the theory of algebraic curves and their funct
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Introductory Notes on Valuation Rings and Function Fields in One Variable
EDIZIONI DELLA NORMALE
14
APPUNTI LECTURE NOTES
Renata Scognamillo and Umberto Zannier Scuola Normale Superiore Piazza dei Cavalieri, 7 56126 Pisa Introductory Notes on Valuation Rings and Function Fields in One Variable
Renata Scognamillo and Umberto Zannier
Introductory Notes on Valuation Rings and Function Fields in One Variable
c 2014 Scuola Normale Superiore Pisa ISBN 978-88-7642-500-4 ISBN 978-88-7642-501-1 (eBook)
Contents
Preface
vii
Introduction Generalities on algebraic functions of one variabile 1 A Urst viewpoint for algebraic functions: Riemann surfaces 2 A second viewpoint: geometry of curves . . . . . . . . . 3 A third viewpoint: Uelds . . . . . . . . . . . . . . . . . 1 Basic notions on function Delds of one variable 1.1 Function Uelds of one variable, rational Uelds . . . . 1.1.1 Unirational and rational function Uelds . . . 1.2 Function Uelds (of one variable) deUne (plane) curves 1.3 Algebraic varieties, rings of regular functions . . . . 1.4 Field inclusions and rational maps . . . . . . . . . .
. . . . .
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1 3 4 5 7 7 11 16 23 25
2 Valuation rings 2.1 Valuation rings and places . . . . . . . . . . . . . . . . 2.1.1 Some signiUcant examples . . . . . . . . . . . . 2.2 Existence and extensions of valuation rings . . . . . . . 2.2.1 Some applications of Theorem 2.2.1 . . . . . . . 2.3 Discrete valuation rings . . . . . . . . . . . . . . . . . . 2.4 Simultaneous approximations with several discrete valuation rings . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Extension of discrete valuation rings . . . . . . . . . . . 2.6 Completions of discrete valuation rings . . . . . . . . . 2.6.1 On valuation rings and geometric points again . . 2.7 Notes to Chapter 2 . . . . . . . . . . . . . . . . . . . .
31 31 34 36 38 42 49 50 61 70 83
A Hilbert’s Nullstellensatz A.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . .
85 85
vi Renata Scognamillo and Umberto Zannier
A.1.1 Preliminaries on algebraic sets . . . . . . . . . . A.2 Proof of the double implication ‘weak form’ ⇔ ‘strong form’ . . . . . . . . . . . . . . . . . . . . . . . . . . .
86 88
B Puiseux series B.1 Field of deUnition, convergence and Eisenstein Theorem
95 95
C Discrete valuation rings and Dedekind domains C.1 Dedekind domains . . . . . . . . . . . . . . . . . . . . C.2 Factorization . . . . . . . . . . . . . . . . . . . . . . . C.3 Notes to appendices . . . . . . . . . . . . . . . . . . . .
107 107 109 111
References
113
Index
115
Preface
The present notes arise from courses delivered at the Scuola Normale in the years 2004/2006. These courses were intended to give to students of III and IV at Scuola Normale year some background on function Uelds of one variable, from the viewpoint of valuations, these topics not being usually touched in university courses in Pisa. The initial purpose was to limit the treatment to basic theory. The notes follow very similar principles, and are addressed to a public with
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