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DFG-Forschergruppe Regensburg/Leipzig Algebraische Zykel Und L-Funktionen DFG-Forschergruppe Regensburg/Leipzig Algebraische Zykel und L-Funktionen On ¯bre sequences in motivic homotopy theory Matthias Wendt Preprint Nr. 10/2007 On Fibre Sequences in Motivic Homotopy Theory Der Fakultat¨ fur¨ Mathematik und Informatik der Universitat¨ Leipzig eingereichte Dissertation zur Erlangung des akademischen Grades Doctor rerum naturalium (Dr. rer. nat.) im Fachgebiet Mathematik vorgelegt von Diplom-Informatiker Matthias Wendt geboren am 22. Februar 1980 in Lutherstadt Wittenberg Leipzig, den 1. Juni 2007 ii Bibliographische Daten Wendt, Matthias On Fibre Sequences in Motivic Homotopy Theory Dissertation, Universit¨at Leipzig 2007 x+165 Seiten, 122 Literaturangaben Mathematics Subject Classification (2000): 14F35, (18G55, 55U35) Keywords: simplicial presheaves, Bousfield-Kan localization, A1-homotopy theory iii Wissenschaftlicher Werdegang • 1998 Abitur am Johann-Wolfgang-von-Goethe-Gymnasium in Chemnitz • 1999 – 2002 Studium an der TU Dresden im Studiengang Diplom Informatik • 2002 – 2004 Studium an der TU Dresden im Internationalen Studiengang Computational Logic • 2004 Diplom Informatik, Note 1.1 mit Auszeichnung • seit Juli 2004 Promotionsstudium an der Universit¨at Leipzig, gef¨ordert durch ein DFG-Stipendium im Graduiertenkolleg “Analysis, Geometrie und ihre Verbindung zu den Naturwissenschaften” iv Selbst¨andigkeitserkl¨arung Hiermit erkl¨are ich, die vorliegende Dissertation selbst¨andig und ohne unzul¨assige Hilfe angefertigt zu haben. Ich habe keine anderen als die angef¨uhrten Quellen und Hilfsmittel benutzt und s¨amtliche Textstellen, die w¨ortlich oder sinngem¨aß aus ver¨offentlichten oder unver¨offentlichten Schriften entnommen wurden, und alle Angaben, die auf m¨undlichen Ausk¨unften beruhen, als solche kenntlich gemacht. Ebenfalls sind alle von anderen Personen bereitgestellten Materialien oder er- brachten Dienstleistungen als solche gekennzeichnet. Leipzig, den 1. Juni 2007 (Matthias Wendt) v vi Danksagung Mein Dank gilt zuallererst meiner Betreuerin Annette Huber-Klawitter. F¨ur ihre Geduld, fortw¨ahrende Ermutigung, und vor allem daf¨ur, daß sie mir die M¨oglichkeit gegeben hat, mich ¨uberhaupt mit der Homotopietheorie von Schemata zu besch¨afti- gen. Ohne sie w¨are die vorliegende Arbeit nicht entstanden. Vielen Dank auch an die Menschen, die mir bei verschiedenen Gelegenheiten mit wertvollen mathematische Einsichten weitergeholfen haben: An Fabien Morel f¨ur seine Erl¨auterungen zu A1-Homotopietheorie und zu Homotopietypen von al- gebraischen Gruppen, an Jens Hornbostel f¨ur Gespr¨ache zur Lokalisierungstheo- rie, sowie an Marc Levine und Oliver R¨ondigs f¨ur ihre Erl¨auterungen zur Serre- Spektralsequenz f¨ur die motivische Kohomologie. Die mit Denis-Charles Cisinski gef¨uhrte Diskussion zu Fragen der Eigentlichkeit der lokalen Modellstruktur haben meine Darstellung dieser Problematik stark gepr¨agt. Meinem Wissen auf dem Gebiet der algebraischen Geometrie auf die Spr¨unge geholfen haben in Leipzig vor allem Shahram Biglari, Bernd Herzog, Sascha Orlik und Malte Witte. Bjørn M¨aurer, Hilke Reiter, Christoph Sachse, Cornelia Schneider und Malte Witte haben verschiedene Teile der Dissertation korrekturgelesen und zur Beseiti- gung vieler Fehler sowie zur Verbesserung der Darstellung beigetragen. Den gr¨oßten Teil der Zeit, die ich mit der Arbeit an dieser Dissertation ver- bracht habe, war ich Stipendiat im DFG-Graduiertenkolleg “Analysis, Geometrie und ihre Verbindung zu den Naturwissenschaften”, wodurch mir verschiedene Kon- ferenzbesuche erm¨oglicht wurden. In diesem Zusammenhang ebenso vielen Dank an Prof. Rademacher und Frau Leißner. Zus¨atzliche Finanzhilfen habe ich außer- dem vom Clay Mathematics Institute f¨ur den Besuch der Sommerschule “Arith- metic Geometry” im Jahr 2006, sowie von der DFG-Forschergruppe “Algebraische Zykel und L-Funktionen” f¨ur den Besuch des Workshops “Algebraic Cycles, Mo- tives and A1-Homotopy Theory over General Bases” im Februar 2007 erhalten, auf dem ich auch einen Teil meiner Ergebnisse vorstellen durfte. “Viel schon ist getan, mehr noch bleibt zu tun,” sprach der Wasserhahn zu dem Wasserhuhn. Robert Gernhardt vii viii Contents 1 Introduction 1 2 Preliminaries and Notation 9 2.1 HomotopicalAlgebra. .. .. .. .. .. .. .. 10 2.2 Sites for Motivic Homotopy Theory . 21 2.3 Remarks................................. 24 3 Homotopy Colimits and Fibrations 31 3.1 Homotopy Colimits of Simplicial Presheaves . 32 3.2 ClassifyingSpacesforFibreSequences . 43 3.3 ExamplesofFibreSequences . 65 3.4 Remarks on Cellular Inequalities . 75 3.5 RemarksontheSerreSpectralSequence . 77 4 Fibrewise Localization 87 4.1 Localization Functors for Simplicial Presheaves . ...... 88 4.2 Fibrewise Localization for Simplicial Presheaves . ....... 92 4.3 Nullification and Local Fibre Sequences . 102 4.4 ApplicationsandExamples . 116 5 Geometry and Fibre Sequences 135 5.1 Patching and the Affine Brown-Gersten Property . 137 5.2 Patching and Local-Global Principles . 141 5.3 Example: Chevalley Groups and Torsors . 144 5.4 MoreExamplesofFibreSequences . 150 Bibliography 157 ix x CONTENTS Chapter 1 Introduction In this thesis, we will examine the motivic or A1-homotopy theory of schemes, a very recent field of algebraic geometry which applies methods and constructions of algebraic topology and the theory of model categories to schemes. Since the early days of Grothendieck-style algebraic geometry, intuition and methods from topology have always played an important role in the investigation of algebraic varieties and their structure. To give just one example, the study of algebraic K-theory and vector bundles on schemes, which has led to deep insights into the nature of linear algebra over general rings, would never have taken the shape it nowadays has without an existing theory of vector bundles on manifolds, and the use of elaborate methods from algebraic topology. The motivation for developing a homotopy theory of schemes is the analogy between smooth schemes and manifolds, and homotopy theoretic arguments are ubiquituous in the theory of manifolds: In order to decide if there exist vector bundles with certain properties on a manifold, one translates this geometric prob- lem into a homotopy theory problem, which then in turn is solved by computing some homotopy groups. It is reasonable to hope that a similar procedure can be used in algebraic geometry: Take a geometric problem, translate it into a question about A1-homotopy theory, and then solve it by computing some homotopy groups. However, all of the steps seem to be more difficult than the corresponding ones in differential topology, since the main difference between schemes and manifolds is, that the arithmetic of the ring over which the scheme is defined intervenes in basically every question concerning the structure of schemes. Both the above abstract motivation and a couple of already existing computa- tions of A1-homotopy groups make it seem worthwhile to study motivic homotopy theory. This is where we enter the scope of the thesis: Already in classical al- gebraic topology, there are only very few spaces whose homotopy groups can be computed directly; for all further computations, one has to use fibre sequences, which are sequences of spaces F → E → B which induce long exact sequences for the corresponding homotopy groups. The main geometric examples of such fibre sequences are fibre bundles like SO(n)-bundles or projective space bundles. Hence, the goal of this thesis is an investigation of the theory of fibre sequences in A1-homotopy theory. The definition of A1-homotopy theory uses Quillen’s ho- motopical algebra and model categories, which is an abstract framework to do algebraic topology and homotopy theory. This framework provides a notion of 1 2 CHAPTER 1. INTRODUCTION fibre sequences, and it also follows from the model category machinery that every morphism of schemes p : E → B sits in a fibre sequence F → E → B for some F . However, the homotopy fibre F of p need not be the same as the point-set fibre p−1(b) for a base point b ∈ B. Moreover, the description of the space F is a very abstract one, and what happens in the course of its construction is hard to keep track of. In the thesis, we will prove general criteria to decide when the homotopy fibre really can be identified with the point-set fibre. These general criteria allow to construct interesting fibre sequences which can be used to compute some A1- homotopy groups. As an example, the following is a fibre sequence in A1-homotopy theory: P1 → P∞ ∨ P∞ → P∞. The space P1 is the projective line, say over a field k, and P∞ is the infinite projective space. The space P∞ is the classifying space for line bundles, and therefore, its homotopy groups of P∞ are known. The fibre sequence can then A1 P1 be used to give a presentation of the motivic fundamental group π1 ( ). Both the methods of computation and the presentation obtained are different from the A1 P1 methods that Morel has used to describe π1 ( ). Other examples of fibre sequences that we can construct using our general criteria are provided by fibre bundles whose structure groups are the general linear groups GLn or the symplectic groups Sp2n. In particular, any projective space bundle E → B induces a fibre sequence A1 Pn A1 A1 Sing• ( ) → Sing• (E) → Sing• (B), A1 where Sing• (X) is an analogue of the singular complex of a topological space, using the affine spaces An as standard simplices. In contrast to topology, it is not clear that any fibre bundle also yields a fibre sequence. To obtain the above fibre sequence for a projective space bundle,
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