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Lecture Notes in Mathematics 2105 Siegfried Bosch Lectures on Formal and Rigid Geometry Lecture Notes in Mathematics 2105 Editors-in-Chief: J.-M. Morel, Cachan B. Teissier, Paris Advisory Board: Camillo De Lellis (Zürich) Mario di Bernardo (Bristol) Alessio Figalli (Austin) Davar Khoshnevisan (Salt Lake City) Ioannis Kontoyiannis (Athens) Gabor Lugosi (Barcelona) Mark Podolskij (Aarhus) Sylvia Serfaty (Paris and NY) Catharina Stroppel (Bonn) Anna Wienhard (Heidelberg) For further volumes: http://www.springer.com/series/304 Siegfried Bosch Lectures on Formal and Rigid Geometry 123 Siegfried Bosch Westfälische Wilhelms-Universität Mathematisches Institut Münster, Germany ISBN 978-3-319-04416-3 ISBN 978-3-319-04417-0 (eBook) DOI 10.1007/978-3-319-04417-0 Springer Cham Heidelberg New York Dordrecht London Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2014944738 Mathematics Subject Classification (2010): 14G22, 14D15 © Springer International Publishing Switzerland 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface This volume grew out of lectures that I gave on several occasions. First versions of the manuscript were prepared as handouts for students and later, in 2005, became a preprint of the Collaborative Research Center Geometrical Structures in Mathematics at the University of Münster. The present Lecture Notes Volume is a revised and slightly expanded version of the earlier preprint. Although I kept the lecture-style presentation, I added more motivation on basic ideas as well as some fundamental examples. To make the text virtually self-contained, the theory of completed tensor products was included in a separate appendix. It is a pleasure for me to express my gratitude to students, colleagues and, particularly, to M. Strauch for their valuable comments and suggestions. Also I would like to thank the referees for their constructive remarks which, finally, made the text more complete and easier to digest. Münster, Germany Siegfried Bosch February 2014 v Contents 1 Introduction: Analytic Functions over non-Archimedean Fields....... 1 Part I Classical Rigid Geometry 2 Tate Algebras ................................................................. 9 2.1 Topology Induced from a Non-Archimedean Absolute Value ........ 9 2.2 Restricted Power Series ................................................. 12 2.3 Ideals in Tate Algebras.................................................. 24 3 Affinoid Algebras and Their Associated Spaces .......................... 31 3.1 Affinoid Algebras ....................................................... 31 3.2 Affinoid Spaces ......................................................... 42 3.3 Affinoid Subdomains ................................................... 45 4 Affinoid Functions ........................................................... 65 4.1 Germs of Affinoid Functions ........................................... 65 4.2 Locally Closed Immersions of Affinoid Spaces ....................... 69 4.3 Tate’s Acyclicity Theorem.............................................. 82 5 Towards the Notion of Rigid Spaces ....................................... 93 5.1 Grothendieck Topologies ............................................... 93 5.2 Sheaves .................................................................. 99 5.3 Rigid Spaces............................................................. 103 5.4 The GAGA-Functor..................................................... 109 6 Coherent Sheaves on Rigid Spaces ......................................... 117 6.1 Coherent Modules....................................................... 117 6.2 Grothendieck Cohomology ............................................. 125 6.3 The Proper Mapping Theorem ......................................... 129 6.4 Proof of the Proper Mapping Theorem ................................ 133 vii viii Contents Part II Formal Geometry 7 Adic Rings and Their Associated Formal Schemes ...................... 151 7.1 Adic Rings............................................................... 151 7.2 Formal Schemes......................................................... 158 7.3 Algebras of Topologically Finite Type ................................. 162 7.4 Admissible Formal Schemes ........................................... 169 8 Raynaud’s View on Rigid Spaces........................................... 175 8.1 Coherent Modules....................................................... 175 8.2 Admissible Formal Blowing-Up ....................................... 179 8.3 Rig-Points in the Classical Rigid Setting .............................. 194 8.4 Rigid Spaces in Terms of Formal Models.............................. 202 9 More Advanced Stuff ........................................................ 215 9.1 Relative Rigid Spaces ................................................... 215 9.2 An Example: Raynaud’s Universal Tate Curve ........................ 217 9.3 The Zariski–Riemann Space............................................ 220 9.4 Further Results on Formal Models ..................................... 223 Appendix A: Classical Valuation Theory ...................................... 229 Appendix B: Completed Tensor Products ..................................... 237 References......................................................................... 249 Index ............................................................................... 251 Chapter 1 Introduction Analytic Functions over Non-Archimedean Fields Classical rigid geometry may be viewed as a theory of analytic functions over local fields or, more generally, over fields that are complete under a non-Archimedean absolute value; complete means that every Cauchy sequence is converging. For example, choosing a prime p, the field Qp of p-adic numbers is such a field. To construct it, we start out from the field Q of rational numbers and complete it with respect to its p-adic absolute value jjp, which is defined as follows: we set j0jp D 0, r Q r a Z − and jxjp D p for x 2 with x D p b where a; b; r 2 and p ab. Then jjp exhibits the usual properties of an absolute value, as it satisfies the following conditions: jxjp D 0 ” x D 0; jxyjp Djxjpjyjp; ˚ jx C yjp Ä max jxjp; jyjp : Furthermore, jjp extends to an absolute value on Qp with the same properties. The third condition above is called the non-Archimedean triangle inequality, it is a sharpening of the usual Archimedean triangle inequality jx C yjjxjCjyj. This way, the field Qp of p-adic numbers might be viewed as an analog of the field R of real numbers. There is also a p-adic analog Cp of the field C of complex numbers. Its construction is more complicated than in the Archimedean alg case. We first pass from Qp to its algebraic closure Qp . The theory of extensions of valuations and absolute values shows that there is a unique extension of jjp to alg this algebraic closure. However, as Qp is of infinite degree over Qp, we cannot alg alg conclude that Qp is complete again. In fact, it is not, and we have to pass from Qp to its completion. Fortunately, this completion remains algebraically closed; it is the field Cp we are looking for. S. Bosch, Lectures on Formal and Rigid Geometry, Lecture Notes 1 in Mathematics 2105, DOI 10.1007/978-3-319-04417-0__1, © Springer International Publishing Switzerland 2014 2 1 Introduction After the p-adic numbers had been discovered by Hensel in 1893, there were several attempts to develop a theory of analytic functions over p-adic fields. At the beginning, people were just curious about the question if there would exist a reasonable analog of classical function theory over C. However, later when algebraic geometry had progressed so that applications to number theory were possible, a good
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