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Etale Homotopy Theory and Simplicial Sheaves Etale Homotopy Theory and Simplicial Sheaves (Thesis format: Monograph) by Michael D. Misamore Graduate Program in Mathematics A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy School of Graduate and Postdoctoral Studies The University of Western Ontario London, Ontario, Canada © Michael D. Misamore 2009 Library and Archives Bibliotheque et 1*1 Canada Archives Canada Published Heritage Direction du Branch Patrimoine de I'edition 395 Wellington Street 395, rue Wellington Ottawa ON K1A 0N4 Ottawa ON K1A 0N4 Canada Canada Your file Votre reference ISBN: 978-0-494-54316-0 Our file Notre r6f6rence ISBN: 978-0-494-54316-0 NOTICE: AVIS: The author has granted a non­ L'auteur a accorde une licence non exclusive exclusive license allowing Library and permettant a la Bibliotheque et Archives Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par I'lnternet, preter, telecommunication or on the Internet, distribuer et vendre des theses partout dans le loan, distribute and sell theses monde, a des fins commerciales ou autres, sur worldwide, for commercial or non­ support microforme, papier, electronique et/ou commercial purposes, in microform, autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in this et des droits moraux qui protege cette these. Ni thesis. Neither the thesis nor la these ni des extraits substantiels de celle-ci substantial extracts from it may be ne doivent etre imprimes ou autrement printed or otherwise reproduced reproduits sans son autorisation. without the author's permission. In compliance with the Canadian Conformement a la loi canadienne sur la Privacy Act some supporting forms protection de la vie privee, quelques may have been removed from this formulaires secondaires ont ete enleves de thesis. cette these. While these forms may be included Bien que ces formulaires aient inclus dans in the document page count, their la pagination, il n'y aura aucun contenu removal does not represent any loss manquant. of content from the thesis. 1+1 Canada THE UNIVERSITY OF WESTERN ONTARIO SCHOOL OF GRADUATE AND POSTDOCTORAL STUDIES CERTIFICATE OF EXAMINATION Supervisor Examiners Dr. J.F. Jardine Dr. Ajneet Dhillon Supervisory Committee Dr. Richard Kane Dr. Stephen Watt Dr. Ravi Vakil The thesis by Michael Misamore entitled: Etale Homotopy Theory and Simplicial Sheaves is accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Date: Chair of the Thesis Examination Board Bernd Frohinann n Abstract This thesis begins an examination of the foundations of etale homotopy theory via the homotopy theory of simplicial sheaves. Etale fundamental groups are studied by means of their relationships to torsors and pointed torsors on the corresponding etale sites, and short exact sequences associated to torsor trivializations are proven. New etale van Kampen theorems are proven by using known, results about the ho­ motopy theory of stacks and the characterization of etale fundamental groups by pointed torsors. A natural etale universal covering space construction is provided, whose etale homotopy pro-groups satisfy the expected properties and which works in both pointed and unpointed contexts. A new definition of etale homotopy type is given which applies to simplicial sheaves, and is shown to coincide with the classical etale homotopy type in the case of simplicial sheaves represented by schemes. This new etale homotopy type is invariant under local weak equivalences, and in particular any hypercover has the same etale homotopy type as its base. In the last chapter, calculational consequences of known results about classical etale homotopy types are worked out in some detail. Keywords: Etale homotopy theory, simplicial sheaves. 2000 Mathematics Subject Classification. Primary 18G30; Secondary 14F35. in Acknowledgements I would like to take this opportunity to thank my colleagues in the Mathematics Department at the University of Western Ontario for many stimulating discussions during my time here. Special thanks are due to my advisor Rick Jardine for his advice, patience, and guidance during the conception and execution of this research project. His careful reading of this document and his insightful comments on it have helped to clarify and improve it in many places; any remaining mistakes are of course mine alone. He has always been generous with his ideas, for which I also thank him, and his unique points of view continue to serve as inspiration for exciting new directions in my own research. I would also like to thank my family, and especially my parents, for the tremen­ dous amount of support they have given me over the past 26 years, and for their constant encouragement to follow my dreams. IV To My Family V Table of Contents Certificate of Examination ii Abstract iii Acknowledgements iv Dedication v 1 Introduction 1 1.1 History of etale homotopy theory 1 1.2 The etale homotopy type of Artin-Mazur 4 1.3 Etale topological types 7 1.4 Introduction to the present work 11 1.4.1 Summary of results 13 2 Connected sites, connected components 19 2.1 Connected components of sheaves 19 2.2 Pointed sites 24 3 Fibred sites 26 3.1 Definition and basic properties 26 3.2 Degree restriction and its right adjoint 29 3.3 Locally constant sheaves 32 3.4 Etale cohomology on fibred sites 33 3.5 Morphisms of fibred sites 36 3.6 Example: fibred site of a Deligne-Mumford stack 37 4 Fundamental groups 41 4.1 Cofiltered categories of hypercovers 41 4.2 Torsors 44 4.3 Pro-homotopy characterization of 7r^al by Galois torsors 45 4.4 Pro-homotopy characterization of irf' by discrete group torsors .... 48 4.5 Pointed torsors and characterizations of fundamental groups 51 5 Torsors and descent 60 5.1 Base change for torsors 60 5.1.1 A fibre sequence for torsor base change 62 5.2 Short exact sequences associated to torsor trivializations 66 5.3 Van Kampen theorems 78 VI 6 Hypercovers and etale homotopy types 85 6.1 Definition and basic properties 85 6.2 Etale universal covering spaces 89 6.3 Artin-Mazur weak equivalences 95 7 Calculating Artin-Mazur etale homotopy pro-groups 105 7.1 The etale homotopy pro-groups of a field 105 7.2 Applications of Grothendieck-Riemann existence 107 7.3 Product formulas for n^ 112 Curriculum Vitae 121 vn 1 Chapter 1 Introduction This chapter consists of a brief introduction to this work, starting with the history of etale homotopy theory as it was classically formulated by Artin and Mazur (in [3]) and Friedlander (in [13]). For a summary of the primary results of this thesis, skip to subsection 1.4.1. 1.1 History of etale homotopy theory The history of etale homotopy theory may be traced back to its origins in Expose V of [2], where Grothendieck used an algebraic analogue of the theory of covering spaces in the context of the etale topology of a scheme X to define a kind of algebraic fundamental group of X, denoted here by 7Tj (X, x). As the notation indicates, this group is defined in terms of the Galois coverings of X, and recovers the absolute Galois group when X = Speck for a field k. This profinite group is defined as the group of automorphisms of a certain nonabelian fibre functor F^ on the finite etale site et(X) of the scheme X in question, where x is some fixed geometric point of X. Later, fibre functors would be used by Deligne in the theory of Tannakian categories to define various other algebraic types of fundamental groups, which in this context would be group schemes rather than profinite groups, and whose relation to the etale homotopy groups TT^ discussed here has remained mostly mysterious. In [2], Grothendieck also conjectured the existence of higher algebraic homotopy groups associated to any scheme X, along with long exact sequences of these profinite groups (with some variations) associated to geometric fibres of sufficiently good (say, 2 smooth and proper) maps of schemes. However, he could only prove six-term exact sequences as he had no definition of the higher homotopy groups. In [1], Verdier, a student of Grothendieck, set out the theory of hypercovers in arbitrary Grothendieck sites, and proved (in modern parlance) that the category of representable hypercovers together with simplicial homotopy classes of maps between them on an arbitrary small Grothendieck site is cofiltered (i.e., "left" filtered). Further, he proved that the hypercovers of a scheme X could be used to compute the etale cohomology WLiX, A) of X with coefficients in any locally constant sheaf of abelian groups A. After the work of J. Giraud, another student of Grothendieck, on nonabelian cohomology, it became clear that the set of isomorphism classes of torsors for any sheaf of groups H could also be computed by means of hypercovers. The next step was taken by Artin and Mazur in [3], where they showed that the category of hypercovers with simplicial homotbpy classes of maps between them, denoted HR(X), determined a pro-object Et(X) in the homotopy category Ho(sSet) of simplicial sets, and that these pro-objects could be used to recover ir^ (X) for sufficiently good schemes X (9.9, [3]) as well as Hn{X, A) for local coefficient sys­ tems of abelian groups A (4.3, [3]). Moreover, these pro-objects had naturally-defined higher etale homotopy pro-groups 7r^, and these were shown to satisfy at least some of the properties that one expects of homotopy groups by analogy with the topolog­ ical situation.
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