Introduction to Algebraic Geometry and Commutative Algebra 7725 tp.indd 1 4/15/10 2:33:41 PM This page intentionally left blank IISc Lecture Notes Series Introduction to Algebraic Geometry and Commutative Algebra Dilip P Patil Indian Institute of Science, India Uwe Storch Ruhr University, Germany World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI 7725 tp.indd 2 4/15/10 2:33:42 PM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. IISc Lecture Notes Series INTRODUCTION TO ALGEBRAIC GEOMETRY AND COMMUTATIVE ALGEBRA Copyright © 2010 by World Scientific Publishing Co. Pte. 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 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN-13 978-981-4304-56-6 ISBN-10 981-4304-56-5 Printed in Singapore. YeeSern - Intro to Algebraic Geometry.pmd 1 2/11/2010, 10:03 AM SERIES PREFACE World Scientific Publishing Company - Indian Institute of Science Collaboration IISc Press and WSPC are co-publishing books authored by world renowned scien- tists and engineers. This collaboration, started in 2008 during IISc’s centenary year under a Memorandum of Understanding between IISc and WSPC, has resulted in the establishment of three Series: IISc Centenary Lectures Series (ICLS), IISc Research Monographs Series (IRMS), and IISc Lecture Notes Series (ILNS). This pioneering collaboration will contribute significantly in disseminating current Indian scientific advancement worldwide. The "IISc Centenary Lectures Series" will comprise lectures by designated Cen- tenary Lecturers - eminent teachers and researchers from all over the world. The "IISc Research Monographs Series" will comprise state-of-the-art mono- graphs written by experts in specific areas. They will include, but not limited to, the authors’ own research work. The "IISc Lecture Notes Series" will consist of books that are reasonably self- contained and can be used either as textbooks or for self-study at the postgraduate level in science and engineering. The books will be based on material that has been class-tested for most part. Editorial Board for the IISc Lecture Notes Series (ILNS): Gadadhar Misra, Editor-in-Chief ([email protected]) Chandrashekar S Jog [email protected] Joy Kuri [email protected] K L Sebastian [email protected] Diptiman Sen [email protected] Sandhya Visweswariah [email protected] This page intentionally left blank PREFACE The present book is based on a course of lectures delivered by the second author at the Department of Mathematics, Indian Institute of Science, Bangalore during seven weeks in February/March 1998. The course met four hours weekly with tutorials of two hours in addition. The arrangement of chapters follows quite closely the sequence of these lectures and each chapter contains more or less the subject-matter of one week. In addition to the exercises covered in the tutorial sessions, further exercises are added at the appropriate places to enhance the understanding and to provide examples. We recommend to look at them while studying the text. To those exercises which are used at other places sufficient hints for straightforward solutions are given. Chapter 7 is an expanded version of the lectures given in the last week (and would at least need two weeks to deliver). The lecture notes [12] based on a series of lectures in 1971/72 and written by Dr. Michael Lippa constituted an important model. The objective of the lectures was to introduceAlgebraic Geometry and Commutative Algebra simultaneously and to show their interplay. This aspect was developed systematically and in full generality with all its consequences in the work of A. Grothendieck, cf. [4]. In Commutative Algebra we do not introduce and use the concept of completion. In geometry we start the language of sheaves and schemes from scratch, but we avoid sheaf cohomology completely. The Riemann–Roch theorem is formulated for arbitrary coherent sheaves on arbitrary projective curves over an arbitrary field. Its proof we reduce to the case of the projective line. Instead of (first) cohomology it uses the dualizing sheaf. Since the uniqueness of this sheaf is not so important for the understanding of the Riemann–Roch theorem, its proof which uses some homological algebra is postponed to the end. We have added a lot of illustrative examples and related concepts to draw many consequences, especially about the genus of a projective curve. We start with basic Commutative Algebra and emphasize on normalization. As geometric counterpart we then introduce the K-spectrum of a finitely generated algebra over a field K. We extend these concepts to prime spectra of arbitrary commutative rings and develop the dimension theory for arbitrary commutative Noetherian rings and their spectra. After introducing the language of sheaves we develop the theory of schemes, in particular, projective schemes. The main theo- rem of elimination and the mapping theorem of Chevalley are proved. Regularity, normality and smoothness are discussed in detail including the theory of K¨ahler dif- ferentials. We give a self-contained treatment of the module of K¨ahler differentials and use the sheaf of K¨ahler differentials as a fundamental example of a coherent and quasi-coherent module on a scheme. Before we prove the Riemann–Roch theorem we describe the coherent and quasi-coherent modules on projective schemes with the help of graded modules. viii Preface With very few exceptions full proofs are given under the assumption that the reader has some experience with the basic concepts of algebra, as groups, rings, fields, vector spaces, modules etc. It should be emphasized that, for a reader who has these prerequisites at his or her fingertips, this book is largely self-contained. This work would have been impossible without the financial support from Deut- scherAkademischerAustauschdienst (DAAD). Both authors have got opportunities for visiting the Ruhr University Bochum and the Indian Institute of Science in Ban- galore respectively and thank DAAD for the generous support and the encouraging cooperation. The second author was partially supported by the GARP Funds, Indian Institute of Science and Part II B-UGC-SAP grant of Department of Mathematics Phase IV-Visiting Fellows, and he would like to express his gratitude for the kind hospitality during his stays in 1998 and 2008. A first draft of the first five chapters was written by Dr. Indranath Sengupta. Dr. Abhijit Das further pushed for the finer draft, especially for the Chapters 5 and 6, during his stay in Bochum. Both were also supported by DAAD. We express our special thanks for their interest and competent work. Dr. Hartmut Wiebe from Ruhr University Bochum has helped us in many ways. He gave us technical support and steady encouragement to come to an end. We thank him wholeheartedly. Bangalore and Bochum, April 2008 Dilip Patil and Uwe Storch [email protected] [email protected] CONTENTS SERIES PREFACE .......................v PREFACE .......................... vii CHAPTER 1 : Finitely Generated Algebras 1.A Algebras over a Ring ..................1 1.B Factorization in Rings ..................2 1.C Noetherian Rings and Modules ..............4 1.D Graded Rings and Modules ................7 1.E Integral Extensions ...................8 1.F Noether’s Normalization Lemma and Its Consequences . 12 CHAPTER 2 : The K-Spectrum and the Zariski Topology 2.A The K-Spectrum of a K-Algebra ............ 19 2.B Affine Algebraic Sets ................. 21 2.C Strong Topology ................... 32 CHAPTER 3 : Prime Spectra and Dimension 3.A The Prime Spectrum of a Commutative Ring ........ 41 3.B Dimension ...................... 48 CHAPTER 4 : Schemes 4.A Sheaves of Rings ................... 61 4.B Schemes ...................... 68 4.C Finiteness Conditions on Schemes ............ 75 4.D Product of Schemes .................. 77 4.E Affine Morphisms ................... 83 CHAPTER 5 : Projective Schemes 5.A Projective Schemes .................. 87 5.B Main Theorem of Elimination ..............102 5.C Mapping Theorem of Chevalley .............107 x Contents CHAPTER 6 : Regular, Normal and Smooth Points 6.A Regular Local Rings .................111 6.B Normal Domains ...................118 6.C Normalization of a Scheme ...............125 6.D The Module of K¨ahler Differentials ...........128 6.E Quasi-coherent Sheaves and the Sheaf of K¨ahler Differentials . 139 CHAPTER 7 : Riemann–Roch Theorem 7.A Coherent Modules on Projective Schemes .........153 7.B Projective Curves ...................158 7.C The Projective Line ..................163 7.D Riemann–Roch Theorem for General Curves ........167 7.E Genus of a Projective Curve ..............175 References ..........................199 List of Symbols .......................201 Index ............................203 Biography of Authors .....................209 CHAPTER 1 : Finitely Generated Algebras Throughout this book a ring will always mean a commutative ring with identity if not stated otherwise. The letter K will always denote a field and the letters A, B, C, R will be generally used for rings. As usual we use Z, Q, R and C to denote the ring of integers, the fields of rational, real and complex numbers respectively. 1.A. Algebras over a Ring Let A be a ring. An A-algebra is a pair (B, ϕ) where B is a ring and ϕ : A → B is a ring homomorphism called the structure homomorphism oftheA-algebra (B, ϕ).