A1-Algebraic Topology over a Field
This text deals with A1-homotopy theory over a base field, i.e., with the natural homotopy theory associated to the category of smooth varieties over a field in which the affine line is imposed to be contractible. It is a natural sequel to the foundationa
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2052
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Fabien Morel
A1-Algebraic Topology over a Field
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Fabien Morel Mathematisches Institut der LMU München, Germany
ISBN 978-3-642-29513-3 ISBN 978-3-642-29514-0 (eBook) DOI 10.1007/978-3-642-29514-0 Springer Heidelberg New York Dordrecht London Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2012942079 Mathematics Subject Classification (2010): 14-XX, 19-XX, 55-XX c Springer-Verlag Berlin Heidelberg 2012 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 work should be considered as a natural sequel to the foundational paper [59] where the A1 -homotopy category of smooth schemes over a base scheme was defined and its first properties studied. In this text the base scheme will always be the spectrum of a perfect field k. One of our first motivations is to emphasize that, contrary to the first impression, the relationship between the A1 -homotopy theory over k and the category Smk of smooth k-schemes is of the same nature as the relationship between the classical homotopy theory and the category of differentiable manifolds. This explains the title of this work; we hope to convince the reader in this matter.
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