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[Li HUISHI] an Introduction to Commutative Algebra(Bookfi.Org) An Introduction to Commutative Algebra From the Viewpoint of Normalization This page intentionally left blank From the Viewpoint of Normalization I Jiaying University, China NEW JERSEY * LONDON * SINSAPORE - EElJlNG * SHANGHA' * HONG KONG * TAIPEI CHENNAI Published by World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202,1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library AN INTRODUCTION TO COMMUTATIVE ALGEBRA From the Viewpoint of Normalization Copyright 0 2004 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book, or parts thereof. may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rasewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 981-238-951-2 Printed in Singapore by World Scientific Printers (S)Pte Ltd For Pinpin and Chao This page intentionally left blank Preface Why normalization? Over the years I had been bothered by selecting material for teaching several one-semester courses. When I taught senior undergraduate stu- dents a first course in (algebraic) number theory, or when I taught first-year graduate students an introduction to algebraic geometry, students strongly felt the lack of some preliminaries on commutative algebra; while I taught first-year graduate students a course in commutative algebra by quoting some nontrivial examples from number theory and algebraic geometry, of- ten times I found my students having difficulty understanding. The problem is that in a short semester there is not enough time to go through the sig- nificant background material needed in my course (if you are an instructor of mathematics, can you ask your students to find and read those material themselves and do they usually follow?). Based on my lecture notes on (algebraic) number theory, algebraic ge- ometry, and commutative algebra used at Shaanxi Normal University and Bilkent University, I decided to fuse several things into one - the presenta- tion of this book. As a consequence, the text consists of five chapters that are designed for a (one-semester) common course taken by senior under- graduate students or by first-year graduate students in mathematics. The goal is to introduce to the students the concrete source of commutative vii ... Vlll Commutative Algebra algebra through the following diagram: DVR -+ PID UFD 1~DVR - + I= I I ED - DD - NND where DVR = Discrete valuation ring, PID = Principal ideal domain, UFD = Unique factorization domain, ED = Euclidean domain, DD = Dedekind domain, NND- = Normal Noetherian domain, Z[6]= The integral closure of 2[6](or equivalently the integral closure = - of Z) in the number field K Q(d), K[C]= The normalization (or integral closure) of the coordinate ring K[C]of an algebraic curve C in its field of rational functions K(C), - and K[V]= The normalization (or integral closure) of the coordinate ring K[V]of an irreducible algebraic set V in its field of rational func- tions K (V). Or more precisely, in terms of normal (normalized) structure the lectures demonstrate how-- to get to the center- of commutative algebra by recognizing the roles that 2[6],K[C] and K[V]play in number theory and algebraic geometry, so that, after reading this volume, interested readers may read a course in algebraic number theory at a higher level, or start an advanced course in algebraic geometry with a better background. I assume that the reader has taken the undergraduate courses including Linear Algebra and A First Course in Abstract Algebra (Galois theory may not be included). Moreover, very little about topological space is needed Preface ix to understand the Zariski topology in Chapter 5. Other than these prereq- uisites, the book is self-contained. Exercises are given at the end of each section. Though most of the exercises mainly test the understanding of the text in the usual way, the reader is involved in providing proofs and in working problems that have not been completely solved in the text; and furthermore, students are asked to extend some of the theory that is essential for the subsequent sections. Huishi Li This page intentionally left blank Contents Preface vii Chapter 1 Preliminaries 1 0. Conventional Review 1 1. Noetherian Rings 6 2. Factorization of Elements in a Domain 9 3. Field Extensions 18 4. Symmetric Polynomials 28 5. Trace and Norm 33 6. Free Abelian Groups of Finite Rank 40 7. Noetherian Modules 45 Chapter 2 Local Rings, DVRs, and Localization 53 1. SpecR, m-SpecR, and Radicals 53 2. Local Rings and DVRs 58 3. The Ring of Fractions and Localization 67 4. The Module of Fractions 73 Chapter 3 Integral Extensions and Normalization 79 1. Integral Extensions 79 2. Noether Normalization 85 3. Normal Domains and Normalization 90 4. Normal Domains and DVRs 94 xi xii Contents Chapter 4 The Ring $2~in K = Q(t9) 99 1. dK is Normal and Free of %Rank [K : Q] 102 2. a(&) and Q(u) 108 3. Factorization of Elements in dK 112 4. From dK to Dedekind Domains 117 Chapter 5 Algebraic Geometry 127 1. Finite Field Extension and Nullstellensatz 127 2. Irreducible V and the Prime I(V) 134 3. Point P and the Local Ring Op,v 146 4. Nonsingular Points and DVRs 151 5. Normalization of Algebraic Curves 157 6. Parametrize a Rational Curve via Normalization 162 7. Rational Curves and Diophantine Equations 167 References 171 Index 173 Chapter 1 Preliminaries This introductory chapter concentrates on basic notions and properties con- cerning Noetherian rings, factorization of elements in a domain, field exten- sions, symmetric polynomials, trace and norm, free abelian groups of finite rank, and Noetherian modules, which might not be familiar to some readers. So we include most of necessary proofs for the reader’s convenience. 0. Conventional Review In this book all rings are commutative associative rings with identity 1, and throughout the text, N = the set of nonnegative integers, Z = the set of integers (ring of integers), Zf = the set of positive integers, Q = the set of rational numbers (field of rational numbers), R = the set of real numbers (field of real numbers), C = the set of complex numbers (field of complex numbers). Let R be a ring and A a subring of R. Then we insist that A has identity 1~ and And we write RX = R - (0). If {Ui}iE~and {Vl,..., Vm}are collections of nonempty subsets of R, 1 2 Commutative Algebra then the sum of {Ui}i,=~and the product of {Vl,..., Vm}are defined as vl...vm={cvl...v, I}ViEV, , where the sums involved in both C Ui and VI . V, are finite sums. So one understands that 0 the sum of given ideals is an ideal; the product of finitely many given subrings (ideals) is a subring (an ideal); and 0 for subrings (ideals) I, J and K, I(J + K)= IJ + IK = JI + KI = (J+ K)I. Let S be a nonempty subset of R and A a subring of R. We set Z[S]= the subring of R generated by S = {~sz...s:; I si, E S, m E z+,aj EN} A[S]= the subring of R generated by S over A = { ~a(,,~)s~...s:; 1 a(,,j) E A, si, E S, m E z+,aj E N} (S) = the ideal of R generated by S = {crisi I ri E R, si E s}, where the sums involved in Z[S],A[S] and (S) are finite sums. If S = {sl, ..., s,} is finite, we write Z[S]= Z[sl,..., sn],A[S] = A[SI,..., s,], (S)= (s1, ..., sn), and call them a finitely generated subring, a finitely generated subring over A, and a finitely generated ideal of R, respectively. Clearly we n can also write (~1,..., Sn) = C,=lRsi. Let RLR' be a ring homomorphism. Then we insist that cp is not the zero-homomorphism and cp(1~)= l~,, and we write Kercp, Imcp for the kernel and image of cp, respectively. A very useful consequence of the first isomorphism theorem on ring homomorphism states that Preliminaries 3 if R-%A and R-B* are ring homomorphisms, cp is surjective, and Kercp C Ker$, then there is a ring homomorphism A-%B defined by p(a) = $(r),where cp(r) = a, such that the following diagram commutes: Kercp L) R A *I JP pocp=$ B if furthermore 1c, is surjective, then p is surjective as well. Let R be a ring with identity 1 = 1~.If R has no divisors of zero, i.e., a, b E R and ab = 0 implies a = 0 or b = 0, then R is called an integral domain, or simply a domain. If R is a domain, then so is the polynomial ring R[zl, ..., zn]in variables z1 , ..., zn over R. If a, b E R and ab = 1, then a (hence b) is called a unit of R. If every nonzero a E R is a unit, then R is called a field. 0.1. Proposition Every finite domain is a field. Proof Exercise. 17 Thus, if p E Z is a prime number, then the ring Z/(p)of integers modulo p, usually denoted Z,, is a field. Let K be a field.
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