Algebraic Cobordism
Following Quillen's approach to complex cobordism, the authors introduce the notion of oriented cohomology theory on the category of smooth varieties over a fixed field. They prove the existence of a universal such theory (in characteristic 0) called Alge
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M. Levine • F. Morel
Algebraic Cobordism
Marc Levine Department of Mathematics Northeastern University Boston, MA 02115, USA e-mail: [email protected] Fabien Morel Mathematisches Institut der LMU Theresienstr. 39 D - 80333 München, Germany e-mail: [email protected]
Library of Congress Control Number: 2006936486
Mathematics Subject Classification (2000): 14F43, 55N22, 14C15, 14C17, 14C40
ISSN 1439-7382 ISBN-10 3-540-36822-1 Springer Berlin Heidelberg New York ISBN-13 978-3-540-36822-9 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, 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. Typesetting by the authors and VTEX using a Springer LATEX macro package Cover design: Erich Kirchner, Heidelberg, Germany Printed on acid-free paper
SPIN: 11805953
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To Rebecca, Anna and Ute–M.L. To Juliette, Elise and Mymy–F.M.
Introduction
Motivation. This work grew out of our attempt to understand the analog in algebraic geometry of the fundamental paper of Quillen on the cobordism of differentiable manifolds [30]. In this paper, Quillen introduced the notion of a (complex) oriented cohomology theory on the category of differentiable manifolds, which basically means that the cohomology theory is endowed with suitable Gysin morphisms, and in particular gives the cohomology theory the additional structure of Chern classes for complex vector bundles. Quillen then observed that the complex cobordism theory X → M U ∗ (X) is the universal such cohomology theory. This new point of view allowed him to shed some new light on classical computations in cobordism theory. He made more precise the computation by Milnor and Novikov of the complex cobordism ring M U ∗ as a polynomial ring: it is in fact the Lazard ring L, the coefficient ring of the universal formal group law defined and studied in [16]. The isomorphism L∼ = MU∗ is obtained via the formal group law FM U (u, v) on M U ∗ defined as the expression of the Chern class c1 (L ⊗ M ) of a tensor product of line bundles as a power series in c1 (L) and c1 (M ) by the formula c1 (L ⊗ M ) = FM U (c1 (L), c1 (M )). This result of Quillen is in fact a particular case of
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