B.I. Dundas M. Levine P.A. Østvær O. Röndigs. Motivic Homotopy Theory. Lectures at a Summer School in Nordfjordeid, Norway, August 2002 ABC

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2 B.I. Dundas M. Levine P.A. Østvær O. Röndigs V. Voevodsky Motivic Homotopy Theory Lectures at a Summer School in Nordfjordeid, Norway, August 2002 ABC

3 Bjørn Ian Dundas Department of Mathematics University of Oslo PO Box 1053, Blindern 0316 Oslo Norway Marc Levine Northeastern University Department of Mathematics 360 Huntington Avenue Boston, MA USA Paul Arne Østvær Department of Mathematics University of Oslo PO Box 1053, Blindern 0316 Oslo Norway Oliver Röndigs Fakultät für Mathematik Universität Bielefeld Postfach Bielefeld Germany Vladimir Voevodsky School of Mathematics Princeton University Princeton, NJ USA Editor: Bjørn Jahren Department of Mathematics University of Oslo Box 1053 Blindern 0316 Oslo Norway Mathematics Subject Classification (2000): 14-xx, 18-xx, 19-xx, 55-xx Library of Congress Control Number: ISBN-10 ISBN Springer Berlin Heidelberg New York 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 techbooks using a Springer TEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper SPIN: /techbooks

4 Preface This book is based on lectures given at a summer school held in Nordfjordeid on the Norwegian west coast in August In the little town with the spectacular surroundings where Sophus Lie was born in 1842, the municipality, in collaboration with the mathematics departments at the universities, has established the Sophus Lie conference center. The purpose is to help organizing conferences and summer schools at a local boarding school during its summer vacation, and the algebraists and algebraic geometers in Norway had already organized such summer schools for a number of years. In 2002 a joint project with the algebraic topologists was proposed, and a natural choice of topic was Motivic homotopy theory, which depends heavily on both algebraic topology and algebraic geometry and has had deep impact in both fields. The organizing committee consisted of Bjørn Jahren and Kristian Ranestad, Oslo, Alexei Rudakov, Trondheim and Stein Arild Strømme, Bergen, and the summer school was partly funded by NorFA Nordisk Forskerutdanningsakademi. It was primarily intended for Norwegian graduate students, but it attracted students from a number of other countries as well. These summer schools traditionally go on for one week, with three series of lectures given by internationally known experts. Motivic homotopy theory was an obvious choice for one of the series, and, especially considering the diverse background of the participants, the two remaining series were chosen to cover necessary background material from algebraic topology and model categories, and from algebraic geometry. The background lectures were given by Bjørn I. Dundas and Marc Levine, both of whom have done important work in their respective areas in connection with the main topic of the school. Motivic homotopy theory was taught by one of the founders of the subject and certainly its most prominent figure: Vladimir Voevodsky. We were very happy to have such great and inspiring experts come and share their knowledge and insight with a new generation of students. After the summer school, Dundas and Levine agreed to write up their lecture series for publication, and Voevodsky agreed to let Oliver Röndigs and Paul Arne Østvær write up his. Röndigs and Østvær have also added an

5 VI Preface extensive appendix with a more detailed discussion of the homotopy theory and model structures involved. In this volume the contributions of Dundas and Levine are presented first, since they contain the prerequisites for Voevodsky s lectures. They are basically independent and can be read in any order, or just referred to while reading the third part, depending on the background of the reader. Finally, we would like to thank Springer Verlag for offering to publish this book. We apologize that this has taken longer than expected, but now that the lectures are available, our hope is that many students will find it useful and convenient to find both an introduction to the fascinating subject of motivic homotopy theory and the background material in one place. Oslo, August 2006 Bjørn Jahren

6 Contents Prerequisites in Algebraic Topology the Nordfjordeid Summer School on Motivic Homotopy Theory Bjørn Ian Dundas... 1 Preface... 3 I Basic Properties and Examples TopologicalSpaces SingularHomology WeakEquivalences MappingSpaces SimplicialSets The Category SimplicialSetsvs.TopologicalSpaces WeakEquivalences Some Constructions in S SimplicialAbelianGroups Simplicial Abelian Groups vs. Chain Complexes TheNormalizedChainComplex ThePointedCase Spectra Introduction RelationtoSimplicialSets StableEquivalences HomologyTheories RelationtoChainComplexes II Deeper Structure: Simplicial Sets Realization as an Extension Through Presheaves (Co)fibrations SimplicialSetsareBuiltOutofSimplices... 30

7 VIII Contents 1.2 LiftingPropertiesandFactorizations SmallObjects Fibrations Combinatorial Homotopy Groups Homotopies and Fibrant Objects III Model Categories Liftings TheAxioms SimpleConsequences ProperModelCategories Quillen Functors FunctorCategories:TheProjectiveStructure CofibrantlyGeneratedModelCategories SimplicialModelCategories Spectra PointwiseStructure StableStructure IV Motivic Spaces and Spectra MotivicSpaces The A 1 -Structure MotivicFunctors TwoQuestions AlgebraicStructure TheMotivicEilenberg-MacLaneSpectrum Wanted Model Structures of Motivic Functors and Relation to Spectra The Homotopy Functor Model Structure MotivicSpectra The Connection F S Spt S References Index Background from Algebraic Geometry Marc Levine I Elementary Algebraic Geometry The Spectrum of a Commutative Ring IdealsandSpec TheZariskiTopology FunctorialProperties Naive Algebraic Geometry and Hilbert s Nullstellensatz

8 Contents IX 1.5 Krull Dimension, Height One Primes and the UFD Property Open Subsets and Localization RingedSpaces PresheavesandSheavesonaSpace The Sheaf of Regular Functions on Spec A LocalRingsandStalks TheCategoryofSchemes ObjectsandMorphisms GluingConstructions Open and Closed Subschemes FiberProducts SchemesandMorphisms NoetherianSchemes Irreducible Schemes, Reduced Schemes and Generic Points SeparatedSchemesandMorphisms FiniteTypeMorphisms Proper,FiniteandQuasi-FiniteMorphisms FlatMorphisms ValuativeCriteria The Category Sch k R-ValuedPoints Group-Schemes and Bundles Dimension FlatnessandDimension SmoothMorphismsandétaleMorphisms TheJacobianCriterion ProjectiveSchemesandMorphisms The Functor Proj Properness ProjectiveandQuasi-ProjectiveMorphisms Globalization Blowing Up a Subscheme II Sheaves for a Grothendieck Topology Limits Definitions FunctorialityofLimits Representability and Exactness Cofinality Presheaves LimitsandExactness Functoriality and Generators for Presheaves Generators for Presheaves PreShv Ab (C) asanabeliancategory...121

9 X Contents 9 Sheaves Grothendieck Pre-Topologies and Topologies SheavesonaSite References Index Voevodsky s Nordfjordeid Lectures: Motivic Homotopy Theory Vladimir Voevodsky, Oliver Röndigs, Paul Arne Østvær Introduction Motivic Stable Homotopy Theory Spaces The Motivic s-stable Homotopy Category SH A1 s (k) The Motivic Stable Homotopy Category SH(k) CohomologyTheories The Motivic Eilenberg-MacLane Spectrum HZ The Algebraic K-Theory Spectrum KGL The Algebraic Cobordism Spectrum MGL The Slice Filtration Appendix TheNisnevichTopology ModelStructuresforSpaces ModelStructuresforSpectraofSpaces References Index...221