DOCSLIB.ORG

Invertible Objects in Motivic Homotopy Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Invertible Objects in Motivic Homotopy Theory Invertible Objects in Motivic Homotopy Theory Tom Bachmann M¨unchen2016 Invertible Objects in Motivic Homotopy Theory Tom Bachmann Dissertation an der Fakult¨atf¨urMathematik, Informatik und Statistik der Ludwig{Maximilians{Universit¨at M¨unchen vorgelegt von Tom Bachmann aus Chemnitz M¨unchen, den 19. Juli 2016 Erstgutachter: Prof. Dr. Fabien Morel Zweitgutachter: Prof. Dr. Marc Levine Drittgutachter: Prof. Dr. math. Oliver R¨ondigs Tag der m¨undlichen Pr¨ufung:18.11.2016 Eidesstattliche Versicherung Bachmann, Tom Name, Vorname Ich erkläre hiermit an Eides statt, dass ich die vorliegende Dissertation mit dem Thema Invertible Objects in Motivic Homotopy Theory selbständig verfasst, mich außer der angegebenen keiner weiteren Hilfsmittel bedient und alle Erkenntnisse, die aus dem Schrifttum ganz oder annähernd übernommen sind, als solche kenntlich gemacht und nach ihrer Herkunft unter Bezeichnung der Fundstelle einzeln nachgewiesen habe. Ich erkläre des Weiteren, dass die hier vorgelegte Dissertation nicht in gleicher oder in ähnlicher Form bei einer anderen Stelle zur Erlangung eines akademischen Grades eingereicht wurde. München, 1.12.2016 Ort, Datum Unterschrift Doktorandin/Doktorand Eidesstattliche Versicherung Stand: 31.01.2013 vi Abstract If X is a (reasonable) base scheme then there are the categories of interest in stable motivic homotopy theory SH(X) and DM(X), constructed by Morel-Voevodsky and others. These should be thought of as generalisations respectively of the stable homotopy category SH and the derived category of abelian groups D(Ab), which are studied in classical topology, to the \world of smooth schemes over X". Just like in topology, the categories SH(X); DM(X) are symmetric monoidal: there is a bifunctor (E; F ) 7! E ⊗ F satisfying certain properties; in particular there is a unit 1 satisfying E ⊗ 1 ' 1 ⊗ E ' E for all E. In any symmetric monoidal category C an object E is called invertible if there is an object F such that E ⊗ F ' 1. Modulo set theoretic problems (which do not occur in practice) the isomorphism classes of invertible objects of a symmetric monoidal category C form an abelian group P ic(C) called the Picard group of C. The aim of this work is to study P ic(SH(X)); P ic(DM(X)) and relations between these various groups. A complete computation seems out of reach at the moment. We can show that (in Q good cases) the natural homomorphism P ic(SH(X)) ! x2X P ic(SH(x)) coming from pull back to points has as kernel the locally trivial invertible spectra P ic0(SH(X)). (For DM, this homomorphism is injective.) Moreover if x = Spec(k) is a point, then (again in good cases) the homomorphism P ic(SH(x)) ! P ic(DM(x)) is injective. This reduces (in some sense) the study of P ic(SH(X)) to the study of the Picard groups of DM over fields, and the latter category is much better understood. We then show that, for example, the reduced motive of a smooth affine quadric is invertible in DM(k). By the previous results, it follows that affine quadric bundles over X yield (in good cases) invertible objects in SH(X). This is related to a conjecture of Po Hu. viii Zusammenfassung Ist X ein (nicht zu exotisches) Basis-Schema dann sind die Kategorien SH(X) and DM(X), welche fur¨ die stabile motivische Homotopietheorie von interesse sind, von Morel-Voevodsky und anderen konstruiert worden. Man sollte sich diese Kategorien respektive als Verallgemeinerungen der stabilen Homotopiekategorie SH und der derivierten Kategorie abelscher Gruppen D(Ab) vorstellen, welche in der klassischen Topologie studiert werden. Genau wie in der Topologie sind die Kategorien SH(X); DM(X) symmetrisch monoidal: es gibt einen Bifunktor (E; F ) 7! E ⊗F der gewisse Bedingungen erfullt;¨ insbesondere gibt es eine Einheit mit der Eigenschaft, dass E⊗1 ' 1⊗E ' E fur¨ alle E. In einer beliebigen symmetrisch monoidalen Kategorie heißt ein Objekt E invertierbar wenn es ein Objekt F gibt, so dass E ⊗ F ' 1. Unter Vernachl¨assigung mengentheoretischer Probleme (die in der Praxis nicht auftreten) bilden die Isomorphieklassen der invertierbaren Objekte einer symmetrisch moidalen Kategorie eine abelsche Gruppe P ic(C), genannt die Picard-Gruppe von C. Das Ziel dieser Arbeit ist es, die Gruppen P ic(SH(X)) und P ic(DM(X)) zu studieren. Eine vollst¨andige Berechnung erscheint derzeit nicht durchfuhrbar.¨ Wir k¨onnen zeigen, dass der Kern Q des naturlichen¨ Homomorphismus P ic(SH(X)) ! x2X P ic(SH(x)) (in guten F¨allen) genau aus den lokal trivialen invertierbaren Spektra besteht. (Im Falle von DM ist dieser Homomorphismus injektiv.) Daruber¨ hinaus zeigen wir, dass wenn x = Spec(k), dann ist der Homomorphismus P ic(SH(x)) ! P ic(DM(x)) (wiederum in guten F¨allen) injektiv. Das reduziert (in gewissem Sinne) das Studium von P ic(SH(X)) auf das studium der Picard-Gruppen von DM uber¨ K¨orpern, und die letztere Kategorie ist sehr viel besser verstanden. Wir zeigen dann, zum Beispiel, dass das reduzierte Motiv einer glatten, affinen Quadrik in DM(k) invertierbar ist. Es folgt aus den vorherigen Ergebnissen, dass Bundel¨ von affinen Qua- driken uber¨ X (in guten F¨allen) Beispiele von invertierbare Objekte in SH(X) liefern. Dies h¨angt zusammen mit einer Vermutung von Po Hu. x Contents 1 Introduction 1 1.1 Background . .1 1.1.1 Stable Homotopy Theory . .1 1.1.2 Symmetric Monoidal Categories and P ic ....................2 1.1.3 Motives and Motivic Homotopy Theory . .4 1.2 Prior and Analogous Work . .5 1.2.1 Picard Groups of Derived Categories . .5 1.2.2 Picard Groups in Equivariant Stable Homotopy Theory . .6 1.2.3 Hu's Conjectures on Pfister Quadrics . .7 1.3 Overview of the Main Results . .8 1.3.1 Invertible Motives over a Field . .8 1.3.2 Invertible Motivic Spectra over General Bases . 11 1.3.3 Bridging the Gap . 12 1.3.4 Organisation of this Thesis . 14 1.4 Notations and Conventions . 15 1.5 Acknowledgements . 15 2 Classifying Spaces of Invertible Objects 17 2.1 Introduction to Chapter Two . 17 2.1.1 Results in Concrete Terms . 17 2.1.2 The Language of Quillen Sheaves . 18 2.1.3 Overview of the Chapter . 18 2.2 The τ-Quillen Sheaves of Interest . 19 2.3 The Classifying Spaces of Interest . 22 2.3.1 Monoids of Homotopy Automorphisms . 22 2.3.2 Nerves of Categories of Locally Trivial Objects . 24 2.4 Relations Between Classifying Spaces . 27 2.4.1 Comparing BAuth and LOC .......................... 27 2.4.2 Comparing Stable and Unstable Fibrations . 28 2.4.3 Comparing Pointed and Unpointed Fibrations . 32 2.4.4 Examples . 34 2.5 Comparison of cdh-locally Trivial Objects and Nisnevich-locally Trivial Objects . 35 3 Recollement and Reduction to Fields 39 3.1 Gluing and Picard Groups . 40 3.2 A Long Exact Sequence . 44 3.3 Recollections about the Six Functors and Continuity . 46 3.4 Recollections about Resolution of Singularities . 47 3.5 Application 1: Pointwise Trivial Motivic Spectra . 48 3.6 Application 2: Pointwise Trivial Motives . 51 xii CONTENTS 4 The Motivic Hurewicz Theorem 53 4.1 The Abstract Hurewicz Theorem . 53 4.2 The Motivic Hurewicz Theorem . 55 4.3 The Slice Filtration and the Conservativity Theorem . 58 4.4 Applications . 61 4.5 Extensions . 62 5 Computations 65 5.1 Change of Coefficients and Base for DM ........................ 66 5.1.1 Recollections about DM ............................. 66 5.1.2 Base Change . 67 5.1.3 Change of Coefficients . 69 5.2 Weight Structures and Fixed Point Functors . 72 5.2.1 Generalities about Weight Structures . 72 5.2.2 More About Chow Motives . 75 5.2.3 The Abstract Fixed Point Functors Theorem . 78 5.3 Quadrics . 81 5.3.1 The Geometric Fixed Point Functors Theorem for Motives of Quadrics . 81 5.3.2 The Invertibility of Affine Quadrics . 83 5.3.3 Pfister Quadrics and the Conjectures of Po Hu . 85 5.4 Artin Motives . 87 5.4.1 The Category of Derived Artin Motives . 88 5.4.2 Geometric Fixed Point Functors . 89 5.4.3 Borel-Smith Conditions . 91 5.4.4 Relationship to Equivariant Stable Homotopy Theory . 97 5.4.5 Artin-Tate Motives . 100 5.5 Tate Spectra . 101 5.6 Rational Coefficients, the Standard Conjectures, and the Etale´ Topology . 104 A Model Categories of Simplicial Presheaves 109 A.1 Review of Model Categories . 110 A.1.1 Basic Definitions; Properness . 110 A.1.2 Adjunctions in Two Variables; Simplicial and Monoidal Model Categories . 112 A.1.3 Bousfield Localization . 113 A.1.4 Mapping Spaces and Nerves . 114 A.1.5 Homotopy Limits and Colimits . 115 A.2 The Four Model Category Structures . 117 A.3 Review of Hypercovers . 120 A.4 Almost Finitely Generated Model Categories . 123 A.5 Descent Spectral Sequences and t-structures . 124 A.6 Pseudofunctors and Fibred Categories . 127 A.6.1 Pseudofunctors and Strictification . 127 A.6.2 Grothendieck Construction, Fibred Categories and Homotopy Colimits . 128 A.6.3 Homotopy Limits and Further Comments . 130 A.7 Homotopy Limits of Model Categories and their Nerves . 132 A.7.1 Quillen (Pseudo-) Presheaves and Homotopy Limits of Model Categories . 132 A.7.2 Changing the Index Category . 135 A.7.3 Decomposition along Pushouts . 139 A.7.4 Commutation of Homotopy Limits and Nerves . 142 A.8 Descent in τ-Quillen Presheaves . 145 A.9 Constructions with τ-Quillen Sheaves . 149 A.9.1 Pointing . 149 A.9.2 Localisation . 153 A.9.3 Stabilisation . 154 Chapter 1 Introduction In Section 1.1, we set the stage: we introduce symmetric monoidal categories and their Picard groups, the stable homotopy category SH, the motivic stable homotopy category SH(X) and the triangulated categories of motives.
Load more

Recommended publications
  • Cohomological Descent on the Overconvergent Site
    COHOMOLOGICAL DESCENT ON THE OVERCONVERGENT SITE DAVID ZUREICK-BROWN Abstract. We prove that cohomological descent holds for finitely presented crystals on the overconvergent site with respect to proper or fppf hypercovers. 1. Introduction Cohomological descent is a robust computational and theoretical tool, central to p-adic cohomology and its applications. On one hand, it facilitates explicit calculations (analo- gous to the computation of coherent cohomology in scheme theory via Cechˇ cohomology); on another, it allows one to deduce results about singular schemes (e.g., finiteness of the cohomology of overconvergent isocrystals on singular schemes [Ked06]) from results about smooth schemes, and, in a pinch, sometimes allows one to bootstrap global definitions from local ones (for example, for a scheme X which fails to embed into a formal scheme smooth near X, one actually defines rigid cohomology via cohomological descent; see [lS07, comment after Proposition 8.2.17]). The main result of the series of papers [CT03,Tsu03,Tsu04] is that cohomological descent for the rigid cohomology of overconvergent isocrystals holds with respect to both flat and proper hypercovers. The barrage of choices in the definition of rigid cohomology is burden- some and makes their proofs of cohomological descent very difficult, totaling to over 200 pages. Even after the main cohomological descent theorems [CT03, Theorems 7.3.1 and 7.4.1] are proved one still has to work a bit to get a spectral sequence [CT03, Theorem 11.7.1]. Actually, even to state what one means by cohomological descent (without a site) is subtle. The situation is now more favorable.
    [Show full text]
  • THE SIX OPERATIONS for SHEAVES on ARTIN STACKS II: ADIC COEFFICIENTS ? by YVES LASZLO and MARTIN OLSSON
    THE SIX OPERATIONS FOR SHEAVES ON ARTIN STACKS II: ADIC COEFFICIENTS ? by YVES LASZLO and MARTIN OLSSON ABSTRACT In this paper we develop a theory of Grothendieck’s six operations for adic constructible sheaves on Artin stacks continuing the study of the finite coefficients case in [14]. 1. Introduction In this paper we continue the study of Grothendieck’s six operations for sheaves on Artin stacks begun in [14]. Our aim in this paper is to extend the theory of finite coefficients of loc. cit. to a theory for adic sheaves. In a subsequent paper [15] we will use this theory to study perverse sheaves on Artin stacks. Throughout we work over an affine excellent finite-dimensional scheme S. Let ` be a prime invertible in S, and such that for any S-scheme X of finite type we have cd`(X) < ∞ (see [14], 1.0.1 for more discussion of this assumption). In what follows, all stacks considered will be algebraic locally of finite type over S. Let Λ be a complete discrete valuation ring with maximal ideal m and n with residue characteristic `, and for every n let Λn denote the quotient Λ/m so that Λ = lim Λ . We then define for any stack X a triangulated category ←− n Dc(X ,Λ) which we call the derived category of constructible Λ–modules on X (of course as in the classical case this is abusive terminology). The cat- egory Dc(X ,Λ) is obtained from the derived category of projective systems {Fn} of Λn–modules by localizing along the full subcategory of complexes whose cohomology sheaves are AR-null (see 2.1 for the meaning of this).
    [Show full text]
  • Partial Mal'tsevness and Partial Protomodularity
    PARTIAL MAL’TSEVNESS AND PARTIAL PROTOMODULARITY DOMINIQUE BOURN Abstract. We introduce the notion of Mal’tsev reflection which allows us to set up a partial notion of Mal’tsevness with respect to a class Σ of split epimorphisms stable under pullback and containing the isomorphisms, and we investigate what is remaining of the properties of the global Mal’tsev context. We introduce also the notion of partial protomodularity in the non-pointed context. Introduction A Mal’tsev category is a category in which any reflexive relation is an equivalence relation, see [9] and [10]. The categories Gp of groups and K-Lie of Lie K-algebras are major examples of Mal’tsev categories. The terminology comes from the pi- oneering work of Mal’tsev in the varietal context [16] which was later on widely developped in [18]. In [3], Mal’tsev categories were characterized in terms of split epimorphisms: a finitely complete category D is a Mal’tsev one if and only if any pullback of split epimorphisms in D: t¯ X′ o X O g¯ / O ′ ′ f s f s t Y ′ o Y g / is such that the pair (s′, t¯) is jointly extremally epic. More recently the same kind of property was observed, but only for a certain class Σ of split epimorphisms (f,s) which is stable under pullback and contains arXiv:1507.02886v1 [math.CT] 10 Jul 2015 isomorphisms. By the classes of Schreier or homogeneous split epimorphisms in the categories Mon of monoids and SRng of semi-rings [8], by the classes of puncturing or acupuncturing split epimorphims in the category of quandles [6], by the class of split epimorphic functors with fibrant splittings in the category CatY of categories with a fixed set of objects Y [5].
    [Show full text]
  • Etale Homotopy Theory of Non-Archimedean Analytic Spaces
    Etale homotopy theory of non-archimedean analytic spaces Joe Berner August 15, 2017 Abstract We review the shape theory of ∞-topoi, and relate it with the usual cohomology of locally constant sheaves. Additionally, a new localization of profinite spaces is defined which allows us to extend the ´etale realization functor of Isaksen. We apply these ideas to define an ´etale homotopy type functor ´et(X ) for Berkovich’s non-archimedean analytic spaces X over a complete non-archimedean field K and prove some properties of the construction. We compare the ´etale homotopy types coming from Tate’s rigid spaces, Huber’s adic spaces, and rigid models when they are all defined. 0 Introduction The ´etale homotopy type of an algebraic variety is one of many tools developed in the last century to use topological machinery to attack algebraic questions. For example sheaves and their cohomology, topoi, spectral sequences, Galois categories, algebraic K-theory, and Chow groups. The ´etale homotopy type enriches ´etale cohomology and torsor data from a collection of groups into a pro-space. This allows us to almost directly apply results from algebraic topol- ogy, with the book-keeping primarily coming from the difference between spaces and pro-spaces. In this paper, we apply this in the context of non-archimedean geometry. The major results are two comparison theorems for the ´etale homo- topy type of non-archimedean analytic spaces, and a theorem indentifying the homotopy fiber of a smooth proper map with the ´etale homotopy type of any geometric fiber. The first comparison theorem concerns analytifications of va- rieties, and states that after profinite completion away from the characteristic arXiv:1708.03657v1 [math.AT] 11 Aug 2017 of the field that the ´etale homotopy type of a variety agrees with that of its analytification.
    [Show full text]
  • Universality of Multiplicative Infinite Loop Space Machines
    UNIVERSALITY OF MULTIPLICATIVE INFINITE LOOP SPACE MACHINES DAVID GEPNER, MORITZ GROTH AND THOMAS NIKOLAUS Abstract. We establish a canonical and unique tensor product for commutative monoids and groups in an ∞-category C which generalizes the ordinary tensor product of abelian groups. Using this tensor product we show that En-(semi)ring objects in C give rise to En-ring spectrum objects in C. In the case that C is the ∞-category of spaces this produces a multiplicative infinite loop space machine which can be applied to the algebraic K-theory of rings and ring spectra. The main tool we use to establish these results is the theory of smashing localizations of presentable ∞-categories. In particular, we identify preadditive and additive ∞-categories as the local objects for certain smashing localizations. A central theme is the stability of algebraic structures under basechange; for example, we show Ring(D ⊗ C) ≃ Ring(D) ⊗ C. Lastly, we also consider these algebraic structures from the perspective of Lawvere algebraic theories in ∞-categories. Contents 0. Introduction 1 1. ∞-categories of commutative monoids and groups 4 2. Preadditive and additive ∞-categories 6 3. Smashing localizations 8 4. Commutative monoids and groups as smashing localizations 11 5. Canonical symmetric monoidal structures 13 6. More functoriality 15 7. ∞-categories of semirings and rings 17 8. Multiplicative infinite loop space theory 19 Appendix A. Comonoids 23 Appendix B. Algebraic theories and monadic functors 23 References 26 0. Introduction The Grothendieck group K0(M) of a commutative monoid M, also known as the group completion, is the universal abelian group which receives a monoid map from M.
    [Show full text]
  • Monads of Effective Descent Type and Comonadicity
    Theory and Applications of Categories, Vol. 16, No. 1, 2006, pp. 1–45. MONADS OF EFFECTIVE DESCENT TYPE AND COMONADICITY BACHUKI MESABLISHVILI Abstract. We show, for an arbitrary adjunction F U : B→Awith B Cauchy complete, that the functor F is comonadic if and only if the monad T on A induced by the adjunction is of effective descent type, meaning that the free T-algebra functor F T : A→AT is comonadic. This result is applied to several situations: In Section 4 to give a sufficient condition for an exponential functor on a cartesian closed category to be monadic, in Sections 5 and 6 to settle the question of the comonadicity of those functors whose domain is Set,orSet, or the category of modules over a semisimple ring, in Section 7 to study the effectiveness of (co)monads on module categories. Our final application is a descent theorem for noncommutative rings from which we deduce an important result of A. Joyal and M. Tierney and of J.-P. Olivier, asserting that the effective descent morphisms in the opposite of the category of commutative unital rings are precisely the pure monomorphisms. 1. Introduction Let A and B be two (not necessarily commutative) rings related by a ring homomorphism i : B → A. The problem of Grothendieck’s descent theory for modules with respect to i : B → A is concerned with the characterization of those (right) A-modules Y for which there is X ∈ ModB andanisomorphismY X⊗BA of right A-modules. Because of a fun- damental connection between descent and monads discovered by Beck (unpublished) and B´enabou and Roubaud [6], this problem is equivalent to the problem of the comonadicity of the extension-of-scalars functor −⊗BA :ModB → ModA.
    [Show full text]
  • HODGE FILTERED COMPLEX BORDISM 11 Given by the Constant Presheaf with Value E
    HODGE FILTERED COMPLEX BORDISM MICHAEL J. HOPKINS AND GEREON QUICK Abstract. We construct Hodge filtered cohomology groups for complex man- ifolds that combine the topological information of generalized cohomology the- ories with geometric data of Hodge filtered holomorphic forms. This theory provides a natural generalization of Deligne cohomology. For smooth complex algebraic varieties, we show that the theory satisfies a projective bundle for- mula and A1-homotopy invariance. Moreover, we obtain transfer maps along projective morphisms. 1. Introduction For a complex manifold, Deligne cohomology is an elegant way to combine the topological information of integral cohomology with the geometric data of holomor- n phic forms. For a given p, the Deligne cohomology group HD(X; Z(p)) of a complex manifold X is defined to be the nth hypercohomology of the Deligne complex ZD(p) p (2πi) 1 d d p−1 0 → Z −−−−→OX → ΩX −→ ... −→ ΩX → 0 k of sheaves over X, where ΩX denotes the sheaf of holomorphic k-forms. Deligne cohomology is an important tool for many fundamental questions in al- gebraic, arithmetic and complex geometry. Let us just highlight one of the algebraic geometric applications of Deligne cohomology. Let X be a smooth projective com- plex variety. The associated complex manifold is an example of a compact K¨ahler manifold. Let CHpX be the Chow group of codimension p algebraic cycles modulo rational equivalence. There is the classical cycle map p 2p clH : CH X → H (X; Z) from Chow groups to the integral cohomology of the associated complex manifold of X. This map sends an irreducible subvariety Z ⊂ X of codimension p in X to arXiv:1212.2173v3 [math.AT] 27 Aug 2014 the Poincar´edual of the fundamental class of a resolution of singularities of Z.
    [Show full text]
  • Protomodular Aspect of the Dual of a Topos
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector ARTICLE IN PRESS Advances in Mathematics 187 (2004) 240–255 http://www.elsevier.com/locate/aim Protomodular aspect of the dual of a topos Dominique Bourn Centre Universitaire de la Mi-voix, Lab. d’Analyse, Ge´ome´trie et Alge`bre, Universite´ du Littoral, 50 rue Ferdinand Buisson, BP699, 62228 Calais Cedex, France Received 3 March 2003; accepted 15 September 2003 Communicated by Ross Street Abstract The structure of the dual of a topos is investigated under its aspect of a Barr exact and protomodular category. In particular the normal monomorphisms in the fibres of the fibration of pointed objects are characterized, and the change of base functors with respect to this same fibration are shown to reflect those normal monomorphisms. r 2003 Elsevier Inc. All rights reserved. MSC: 18B25; 18D30; 08B10 Keywords: Topos; Fibrations; Mal’cev and protomodular categories 0. Introduction It is well known that, given a topos E; the subobject classifier O is endowed with an internal structure of Heyting Algebra, and that consequently the contravariant ‘‘power-set’’ functor OðÞ : Eop-E takes its values in the category HeytE of internal Heyting algebras in E: But the category Heyt of Heyting Algebras shares with the categories Gp of groups or Rg of rings the property of being protomodular. This property [3] says that the fibration of pointed objects p : PtC-C; whose fibre at X is the category PtX C of split epimorphisms with codomain X; has its change of base functors conservative (i.e.
    [Show full text]
  • EQUIVARIANT MOTIVIC HOMOTOPY THEORY Table of Contents 1
    EQUIVARIANT MOTIVIC HOMOTOPY THEORY GUNNAR CARLSSON AND ROY JOSHUA Abstract. In this paper, we develop the theory of equivariant motivic homotopy theory, both unstable and stable. While our main interest is the case when the group is pro-finite, we discuss our results in a more general setting so as to be applicable to other contexts, for example when the group is in fact a smooth group scheme. We also discuss 1 how A -localization behaves with respect to ring and module spectra and also with respect to mod-l-completion, where l is a fixed prime. In forthcoming papers, we apply the theory developed here to produce a theory of Spanier-Whitehead duality and Becker-Gottlieb transfer in this framework, and explore various applications of the transfer. Table of contents 1. Introduction 2. The basic framework of equivariant motivic homotopy theory: the unstable theory 3. The basic framework of equivariant motivic homotopy theory: the stable theory 4. Effect of A1-localization on ring and module structures and on mod-l completions 5. References. The second author was supported by the IHES, the MPI (Bonn) and a grant from the NSA. 1 2 Gunnar Carlsson and Roy Joshua 1. Introduction The purpose of this paper is to set up a suitable framework for a series of papers exploring the notions of Spanier-Whitehead duality and Becker-Gottlieb transfer in the setting of stable motivic homotopy theory and their applications. Some of the notable applications we have in mind are the conjectures due to the first author on equivariant algebraic K-theory, equivariant with respect to the action of a Galois group, see [Carl1].
    [Show full text]
  • Triangulated Categories of Motives Over Fs Log Schemes
    Triangulated Categories of Motives over fs Log Schemes by Doosung Park A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Martin Olsson, Chair Professor Kam-Biu Luk Professor Constantin Teleman Professor Xinyi Yuan Fall 2016 Triangulated Categories of Motives over fs Log Schemes Copyright 2016 by Doosung Park Abstract Triangulated Categories of Motives over fs Log Schemes by Doosung Park Doctor of Philosophy in Mathematics University of California, Berkeley Professor Martin Olsson, Chair In this thesis, we construct triangulated categories of motives over fs log schemes with rational coefficients and formulate its six operations formalism. For these, we introduce pw- topology and log-weak equivalences to study the homotopy equivalences of fs log schemes. We also introduce equivariant cd-structures to deal with descent theory of motives more systematically. 1 Contents Contents i Introduction v 1 Construction 1 1.1 Premotivic categories . .1 1.2 Equivariant cd-structures . .4 1.3 Descents . 13 1.4 Compactness . 16 1.5 Localizing subcategories . 17 1.6 Bousfield localization . 22 1.7 log-localization . 27 2 Properties of premotivic triangulated categories 33 2.1 Elementary properties . 34 2.2 Localization property . 35 2.3 Support property . 36 2.4 Homotopy properties . 39 2.5 Purity . 40 2.6 Base change property . 42 2.7 Projection formula . 45 2.8 Orientation . 46 2.9 Log motivic categories . 46 3 Some results on log geometry and motives 48 3.1 Charts of log smooth morphisms .
    [Show full text]
  • Grothendieck and the Six Operations
    The Fifth International Conference on History of Modern Mathematics Xi’an, China August 18–24, 2019 Grothendieck and the six operations Luc Illusie Université Paris-Sud Plan 1. Serre’s duality theorem 2. Derived categories: Grothendieck’s revolution 3. The f ! functor: duality in the coherent setting 4. Duality in étale cohomology and the six operations 5. Further developments 1. Serre’s duality theorem Theorem 1 (ICM Amsterdam, 1954) k algebraically closed, X =k smooth, projective, irreducible, of dimension m, _ V a vector bundle on X , V = Hom(V ; OX ), i i 1 Ω := Λ ΩX =k . Then: m m (a) dimk H (X ; Ω ) = 1; (b) For all q 2 Z, the pairing Hq(X ; V ) ⊗ Hm−q(X ; V _ ⊗ Ωm) ! Hm(X ; Ωm)(!∼ k) is perfect. Remarks (a) Serre had previously proved (FAC) that, for any coherent sheaf q q F on X , and all q, dimk H (X ; F) < 1 and H (X ; F) = 0 for q > m. (b) Serre doesn’t exhibit a distinguished basis of Hm(X ; Ωm). Proof by induction on m, his vanishing theorems on Hq(X ; F(n)) for q > 0 and n large play a key role. Construction of a distinguished basis crucial in further work by Grothendieck et al. (c) Serre proved analogue for X =C smooth, compact analytic, V a vector bundle on X (Comm. Helv., 1955). Quite different techniques. Th. 1 revisited by Grothendieck: 1955-56, Sém. Bourbaki 149, May 1957 X =k smooth, projective, irreducible, dimension m as above. Theorem 2 (Grothendieck, loc. cit., Th.
    [Show full text]
  • Abstract Homotopy Theory, Generalized Sheaf Cohomology, Homotopical Algebra, Sheaf of Spectra, Homotopy Category, Derived Functor
    TRANSACTIONSOF THE AMERICANMATHEMATICAL SOCIETY Volume 186, December 1973 ABSTRACTHOMOTOPY THEORY AND GENERALIZED SHEAF COHOMOLOGY BY KENNETHS. BROWN0) ABSTRACT. Cohomology groups Ha(X, E) are defined, where X is a topological space and £ is a sheaf on X with values in Kan's category of spectra. These groups generalize the ordinary cohomology groups of X with coefficients in an abelian sheaf, as well as the generalized cohomology of X in the usual sense. The groups are defined by means of the "homotopical algebra" of Quillen applied to suitable categories of sheaves. The study of the homotopy category of sheaves of spectra requires an abstract homotopy theory more general than Quillen's, and this is developed in Part I of the paper. Finally, the basic cohomological properties are proved, including a spectral sequence which generalizes the Atiyah-Hirzebruch spectral sequence (in gen- eralized cohomology theory) and the "local to global" spectral sequence (in sheaf cohomology theory). Introduction. In this paper we will study the homotopy theory of sheaves of simplicial sets and sheaves of spectra. This homotopy theory will be used to give a derived functor definition of generalized sheaf cohomology groups H^iX, E), where X is a topological space and E is a sheaf of spectra on X, subject to certain finiteness conditions. These groups include as special cases the usual generalized cohomology of X defined by a spectrum [22] and the cohomology of X with coefficients in a complex of abelian sheaves. The cohomology groups have all the properties one would expect, the most important one being a spectral sequence Ep2q= HpiX, n_ E) =■»Hp+*iX, E), which generalizes the Atiyah-Hirzebruch spectral sequence (in generalized coho- mology) and the "local to global" spectral sequence (in sheaf cohomology), and Received by the editors June 8, 1972 and, in revised form, October 15, 1972.
    [Show full text]