TANNAKIAN FORMALISM OVER FIELDS with OPERATORS 3 Construction Is Available When M Is Now a Formal Set, and S Is a (Nice) Scheme
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MAPPING STACKS and CATEGORICAL NOTIONS of PROPERNESS Contents 1. Introduction 2 1.1. Introduction to the Introduction 2 1.2
MAPPING STACKS AND CATEGORICAL NOTIONS OF PROPERNESS DANIEL HALPERN-LEISTNER AND ANATOLY PREYGEL Abstract. One fundamental consequence of a scheme being proper is that there is an algebraic space classifying maps from it to any other finite type scheme, and this result has been extended to proper stacks. We observe, however, that it also holds for many examples where the source is a geometric stack, such as a global quotient. In our investigation, we are lead naturally to certain properties of the derived category of a stack which guarantee that the mapping stack from it to any geometric finite type stack is algebraic. We develop methods for establishing these properties in a large class of examples. Along the way, we introduce a notion of projective morphism of algebraic stacks, and prove strong h-descent results which hold in the setting of derived algebraic geometry but not in classical algebraic geometry. Contents 1. Introduction 2 1.1. Introduction to the introduction2 1.2. Mapping out of stacks which are \proper enough"3 1.3. Techniques for establishing (GE) and (L)5 1.4. A long list of examples6 1.5. Comparison with previous results7 1.6. Notation and conventions8 1.7. Author's note 9 2. Artin's criteria for mapping stacks 10 2.1. Weil restriction of affine stacks 11 2.2. Deformation theory of the mapping stack 12 2.3. Integrability via the Tannakian formalism 16 2.4. Derived representability from classical representability 19 2.5. Application: (pGE) and the moduli of perfect complexes 21 3. Perfect Grothendieck existence 23 3.1. -
Notes on Tannakian Categories
NOTES ON TANNAKIAN CATEGORIES JOSE´ SIMENTAL Abstract. These are notes for an expository talk at the course “Differential Equations and Quantum Groups" given by Prof. Valerio Toledano Laredo at Northeastern University, Spring 2016. We give a quick introduction to the Tannakian formalism for algebraic groups. The purpose of these notes if to give a quick introduction to the Tannakian formalism for algebraic groups, as developed by Deligne in [D]. This is a very powerful technique. For example, it is a cornerstone in the proof of geometric Satake by Mirkovi´c-Vilonenthat gives an equivalence of tensor categories between the category of perverse sheaves on the affine Grassmannian of a reductive algebraic group G and the category of representations of the Langlands dual Gb, see [MV]. Our plan is as follows. First, in Section 1 we give a rather quick introduction to representations of al- gebraic groups, following [J]. The first main result of the theory says that we can completely recover an algebraic group by its category of representations, see Theorem 1.12. After that, in Section 2 we formalize some properties of the category of representations of an algebraic group G into the notion of a Tannakian category. The second main result says that every Tannakian category is equivalent to the category of repre- sentations of an appropriate algebraic group. Our first definition of a Tannakian category, however, involves the existence of a functor that may not be easy to construct. In Section 3 we give a result, due to Deligne, that gives a more intrinsic characterization of Tannakian categories in linear-algebraic terms. -
Homotopical and Higher Categorical Structures in Algebraic Geometry
Homotopical and Higher Categorical Structures in Algebraic Geometry Bertrand To¨en CNRS UMR 6621 Universit´ede Nice, Math´ematiques Parc Valrose 06108 Nice Cedex 2 France arXiv:math/0312262v2 [math.AG] 31 Dec 2003 M´emoire de Synth`ese en vue de l’obtention de l’ Habilitation a Diriger les Recherches 1 Je d´edie ce m´emoire `aBeussa la mignonne. 2 Pour ce qui est des choses humaines, ne pas rire, ne pas pleurer, ne pas s’indigner, mais comprendre. Spinoza Ce dont on ne peut parler, il faut le taire. Wittgenstein Ce que ces gens l`asont cryptiques ! Simpson 3 Avertissements La pr´esent m´emoire est une version ´etendue du m´emoire de synth`ese d’habilitation originellement r´edig´epour la soutenance du 16 Mai 2003. Le chapitre §5 n’apparaissait pas dans la version originale, bien que son contenu ´etait pr´esent´e sous forme forte- ment r´esum´e. 4 Contents 0 Introduction 7 1 Homotopy theories 11 2 Segal categories 16 2.1 Segalcategoriesandmodelcategories . 16 2.2 K-Theory ................................. 21 2.3 StableSegalcategories . .. .. 23 2.4 Hochschild cohomology of Segal categories . 24 2.5 Segalcategoriesandd´erivateurs . 27 3 Segal categories, stacks and homotopy theory 29 3.1 Segaltopoi................................. 29 3.2 HomotopytypeofSegaltopoi . 32 4 Homotopical algebraic geometry 35 4.1 HAG:Thegeneraltheory. 38 4.2 DAG:Derivedalgebraicgeometry . 40 4.3 UDAG: Unbounded derived algebraic geometry . 43 4.4 BNAG:Bravenewalgebraicgeometry . 46 5 Tannakian duality for Segal categories 47 5.1 TensorSegalcategories . 49 5.2 Stacksoffiberfunctors . .. .. 55 5.3 TheTannakianduality .......................... 58 5.4 Comparison with usual Tannakian duality . -
Modified Mixed Realizations, New Additive Invariants, and Periods Of
MODIFIED MIXED REALIZATIONS, NEW ADDITIVE INVARIANTS, AND PERIODS OF DG CATEGORIES GONC¸ALO TABUADA Abstract. To every scheme, not necessarily smooth neither proper, we can associate its different mixed realizations (de Rham, Betti, ´etale, Hodge, etc) as well as its ring of periods. In this note, following an insight of Kontsevich, we prove that, after suitable modifications, these classical constructions can be extended from schemes to the broad setting of dg categories. This leads to new additive invariants of dg categories, which we compute in the case of differential operators, as well as to a theory of periods of dg categories. Among other applications, we prove that the ring of periods of a scheme is invariant under projective homological duality. Along the way, we explicitly describe the modified mixed realizations using the Tannakian formalism. 1. Modified mixed realizations Given a perfect field k and a commutative Q-algebra R, Voevodsky introduced in [49, §2] the category of geometric mixed motives DMgm(k; R). By construction, this R-linear rigid symmetric monoidal triangulated category comes equipped with a ⊗-functor M(−)R : Sm(k) → DMgm(k; R), defined on smooth k-schemes of finite type, and with a ⊗-invertible object T := R(1)[2] called the Tate motive. Moreover, when k is of characteristic zero, the preceding functor can be extended from Sm(k) to the category Sch(k) of all k-schemes of finite type. Recall also the construction of Voevodsky’s big category of mixed motives DM(k; R). This R-linear symmet- ric monoidal triangulated category admits arbitrary direct sums and DMgm(k; R) identifies with its full triangulated subcategory of compact objects. -
Introduction
Chapter 1 Introduction In the late 1940s Weil posed the challenge to create a cohomology theory for algebraic varieties X over an arbitrary field k with coefficients in Z. Weil had a deep arithmetic application in mind, namely to prove a conjecture of his concerning the -function attached to a variety over a finite field: Its first part asserts the rationality of this - function, the second claims the existence of a functional equation, and the third predicts the locations of its zeros and poles. Assuming a well-developed cohomology theory with coefficients in Z, possessing in particular a Lefschetz trace formula, he indicated a proof of these conjectures, cf. [48]. As we now know, for fields k of positive characteristic one cannot hope to con- struct meaningful cohomology groups H i .X; Z/. Nevertheless Weil’s quest provided an important stimulus. Starting with Serre and then with the main driving force of Grothendieck, several good substitutes for such a theory have been constructed. Many later developments in arithmetic algebraic geometry hinge crucially on these theories. Among the most important cohomology theories for algebraic varieties over a field k are the following: If k is of characteristic zero, one may pass from k to the field of complex numbers C and consider the singular cohomology of the associated complex analytic space X.C/. Other invariants in the same situation are given by the algebraic de Rham cohomology of X. For general k and a prime ` different from the characteristic of k, there is the cohomology theory of étale `-adic sheaves due to Grothendieck et al. -
Tannakian Formalism Applied to the Category of Mixed Hodge Structures
Tannakian Formalism Applied to the Category of Mixed Hodge Structures Licentiate Thesis Submitted to the Department of Mathematics and Statistics in Partial Fulfillment of the Requirements for the degree Licentiate of Philosophy Field of Mathematics By Antti Veilahti University of Helsinki Faculty of Science June 2008 c Copyright by Antti Veilahti 2008 All Rights Reserved ii Abstract Tannakian Formalism Applied to the Category of Mixed Hodge Structures Antti Veilahti Many problems in analysis have been solved using the theory of Hodge struc- tures. P. Deligne started to treat these structures in a categorical way. Following him, we introduce the categories of mixed real and complex Hodge structures. We describe and apply the Tannakian formalism to treat the category of mixed Hodge structures as a category of representations of a certain affine group scheme. Using this approach we give an explicit formula and method for calculating the Ext1-groups in the category and show that the higher Ext-groups vanish. Also, we consider some examples illustrating the usefulness of these calculations in analysis and geometry. iii Acknowledgements During my graduate studies I have received a lot of support from Professor Kari Vilonen, who made it possible for me to study in Northwestern University with all the wonderful mathematicians. With his excellent knowledge in many different areas he kept guiding me and helped me to find paths to the very core of modern mathematics. Especially, I would like to thank him for the discussions on Hodge theory that ultimately led me to this thesis. I would also like to thank Professor Kalevi Suominen, for reading the manu- script and his comments. -
Twisted Differential Operators and Q-Crystals Michel Gros, Bernard Le Stum, Adolfo Quirós
Twisted differential operators and q-crystals Michel Gros, Bernard Le Stum, Adolfo Quirós To cite this version: Michel Gros, Bernard Le Stum, Adolfo Quirós. Twisted differential operators and q-crystals. 2020. hal-02559467 HAL Id: hal-02559467 https://hal.archives-ouvertes.fr/hal-02559467 Preprint submitted on 30 Apr 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Twisted differential operators and q-crystals Michel Gros, Bernard Le Stum & Adolfo Quirós∗ Version of April 29, 2020 Abstract We describe explicitly the q-PD-envelopes considered by Bhatt and Scholze in their recent theory of q-crystalline cohomology and explain the relation with our notion of a divided polynomial twisted algebra. Together with an interpretation of crystals on the q-crystalline site, that we call q-crystals, as modules endowed with some kind of stratification, it allows us to associate a module on the ring of twisted differential operators to any q-crystal. Contents Introduction 2 1 δ-structures 3 2 δ-rings and twisted divided powers 5 3 q-divided powers and twisted divided powers 7 4 Complete q-PD-envelopes 14 5 Complete q-PD-envelope of a diagonal embedding 17 6 Hyper q-stratifications 19 7 q-crystals 22 8 Appendix: 1-crystals vs usual crystals 24 References 25 ∗Supported by grant PGC2018-095392-B-I00 (MCIU/AEI/FEDER, UE). -
Grothendieck Toposes As Unifying ‘Bridges’ in Mathematics
Grothendieck toposes as unifying ‘bridges’ in Mathematics Mémoire pour l’obtention de l’habilitation à diriger des recherches* Olivia Caramello *Successfully defended on 14 December 2016 in front of a Jury composed by Alain Connes (Collège de France) - President, Thierry Coquand (University of Gothenburg), Jamshid Derakhshan (University of Oxford), Ivan Fesenko (University of Nottingham), Anatole Khelif (Universitè de Paris 7), Frédéric Patras (Universitè de Nice - Sophia Antipolis), Enrico Vitale (Universitè Catholique de Louvain) and Boban Velickovic (Universitè de Paris 7). 1 Contents 1 Introduction 2 2 General theory 5 2.1 The ‘bridge-building’ technique . .6 2.1.1 Decks of ‘bridges’: Morita-equivalences . .7 2.1.2 Arches of ‘bridges’: Site characterizations . 11 2.1.3 The flexibility of Logic: a two-level view . 13 2.1.4 The duality between real and imaginary . 18 2.1.5 A theory of ‘structural translations’ . 20 2.2 Theories, Sites, Toposes ....................... 21 2.3 Universal models and classifying toposes . 30 3 Theory and applications of toposes as ‘bridges’ 35 3.1 Characterization of topos-theoretic invariants . 35 3.2 De Morgan and Boolean toposes . 39 3.3 Atomic two-valued toposes . 40 3.3.1 Fraïssé’s construction from a topos-theoretic persective . 40 3.3.2 Topological Galois Theory . 45 3.3.3 Motivic toposes . 50 4 Dualities, bi-interpretations and Morita-equivalences 53 4.1 Dualities, equivalences and adjunctions for preordered structures and topological spaces . 54 4.1.1 The topos-theoretic construction of Stone-type dualities and adjunctions . 54 4.1.2 A general method for building reflections . -
Higher Tannaka and Beyond
Higher Tannaka and Beyond Renaud Gauthier ∗ September 17, 2018 Abstract We consider the origins of Higher Tannaka duality, as well as it consequences. In a first time we review the work of J. Wallbridge ([W]) on that subject, which shows in particular that Hopf algebras are essential to generating Tannakian ∞-categories. While doing so we provide an analysis of what it means for Higher Tannaka to be a reconstruction program for stacks. Putting Wallbridge’s work in per- spective paves the way for causal models in Higher Category Theory. We introduce two interesting concepts that we deem to be instrumen- tal within this framework; that of blow-ups of categories as well as that of fractal ∞-categories. arXiv:1303.3076v1 [math.AG] 13 Mar 2013 ∗[email protected] 1 1 Introduction The origin of Tannaka duality can be found in [C], and from the perspective of a reconstruction program, the duality theorems of Tannaka and Krein were the first to show that some algebraic objects such as compact groups could be recovered from the collection of their representations. Tannaka ([Ta]) showed that a compact group can be recovered from its category of repre- sentations. Krein ([K]) worked on those categories that arise as the category of representations of such groups. A nice account of Tannaka duality can be found in [JS], with a formal development of the basis for a modern form of Tannaka duality given in [DM]. It is worth giving the main result of that latter paper to give a flavor of the Tannaka formalism. Deligne and Milne considered a rigid abelian tensor category (C, ⊗) for which k = End(1) and let ω : C → Veck be an exact faithful k-linear tensor functor. -
HOMOTOPY SPECTRA and DIOPHANTINE EQUATIONS Yuri I. Manin, Matilde Marcolli to Xenia and Paolo, from Yuri and Matilde, with All O
HOMOTOPY SPECTRA AND DIOPHANTINE EQUATIONS Yuri I. Manin, Matilde Marcolli To Xenia and Paolo, from Yuri and Matilde, with all our love and gratitude. ABSTRACT. Arguably, the first bridge between vast, ancient, but disjoint do- mains of mathematical knowledge, { topology and number theory, { was built only during the last fifty years. This bridge is the theory of spectra in the stable homotopy theory. In particular, it connects Z, the initial object in the theory of commutative rings, with the sphere spectrum S: see [Sc01] for one of versions of it. This con- nection poses the challenge: discover a new information in number theory using the developed independently machinery of homotopy theory. (Notice that a pas- sage in reverse direction has already generated results about computability in the homotopy theory: see [FMa20] and references therein.) In this combined research/survey paper we suggest to apply homotopy spectra to the problem of distribution of rational points upon algebraic manifolds. Subjects: Algebraic Geometry (math.AG); Number Theory (math.NT); Topology (math.AT) Comments: 72 pages. MSC-classes: 16E35, 11G50, 14G40, 55P43, 16E20, 18F30. CONTENTS 0. Introduction and summary 1. Homotopy spectra: a brief presentation 2. Diophantine equations: distribution of rational points on algebraic varieties 3. Rational points, sieves, and assemblers 4. Anticanonical heights and points count 5. Sieves \beyond heights" ? 6. Obstructions and sieves 7. Assemblers and spectra for Grothendieck rings with exponentials References 1 2 0. Introduction and summary 0.0. A brief history. For a long stretch of time in the history of mathematics, Number Theory and Topology formed vast, but disjoint domains of mathematical knowledge. -
Noncommutative Motives in Positive Characteristic and Their Applications
Noncommutative motives in positive characteristic and their applications The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Tabuada, Gonçalo. “Noncommutative motives in positive characteristic and their applications.” Advances in mathematics, vol. 349, 2019, pp. 648-681 © 2019 The Author As Published 10.1016/J.AIM.2019.04.020 Publisher Elsevier BV Version Original manuscript Citable link https://hdl.handle.net/1721.1/126181 Terms of Use Creative Commons Attribution-NonCommercial-NoDerivs License Detailed Terms http://creativecommons.org/licenses/by-nc-nd/4.0/ NONCOMMUTATIVE MOTIVES IN POSITIVE CHARACTERISTIC AND THEIR APPLICATIONS GONC¸ALO TABUADA Abstract. Let k be a base field of positive characteristic. Making use of topo- logical periodic cyclic homology, we start by proving that the category of non- commutative numerical motives over k is abelian semi-simple, as conjectured by Kontsevich. Then, we establish a far-reaching noncommutative generaliza- tion of the Weil conjectures, originally proved by Dwork and Grothendieck. In the same vein, we establish a far-reaching noncommutative generalization of the cohomological interpretations of the Hasse-Weil zeta function, originally proven by Hesselholt. As a third main result, we prove that the numerical Grothendieck group of every smooth proper dg category is a finitely generated free abelian group, as claimed (without proof) by Kuznetsov. Then, we in- troduce the noncommutative motivic Galois (super-)groups and, following an insight of Kontsevich, relate them to their classical commutative counterparts. Finally, we explain how the motivic measure induced by Berthelot’s rigid co- homology can be recovered from the theory of noncommutative motives. -
Tannakian Categories of Perverse Sheaves On
INAUGURAL-DISSERTATION zur Erlangung der Doktorwurde¨ an der Naturwissenschaftlich-Mathematischen Gesamtfakultat¨ der Ruprecht-Karls-Universitat¨ Heidelberg TANNAKIAN CATEGORIES OF PERVERSE SHEAVES ON ABELIAN VARIETIES vorgelegt von Dipl.-Math. Thomas Kramer¨ aus Frankenberg (Eder) Tag der mundlichen¨ Prufung:¨ Betreuer: Prof. Dr. Rainer Weissauer Zusammenfassung. Die vorliegende Dissertation beschaftigt¨ sich mit Tannaka-Kategorien, welche durch Faltung perverser Garben auf abelschen Varietaten¨ entstehen. Die Konstruktion dieser Kategorien ist eng verbunden mit einem kohomologischen Verschwindungssatz, der ein Analogon zum Satz von Artin-Grothendieck darstellt und als Spezialfall die generischen Verschwindungssatze¨ von Green und Lazarsfeld enthalt.¨ Die erhaltenen Tannaka-Gruppen bilden interessante geometrische Invarianten, welche in vielen Situationen eine Rolle spielen — nachdem die Grundlagen fur¨ das Studium dieser Gruppen bereitgestellt sind, wird als ein wichtiges Beispiel der Thetadivisor einer prinzipal polarisierten komplexen abelschen Varietat¨ behandelt. Durch Entartungsargumente wird die Tannaka-Gruppe fur¨ eine generische abelsche Varietat¨ bestimmt, welche wenigstens in Dimension 4 eine neue Antwort auf das Schottky-Problem liefert. Das Faltungsquadrat des Thetadivisors in Dimension 4 hangt¨ eng zusammen mit einer Familie glatter Flachen¨ vom allgemeinen Typ, und ein Studium dieser Familie fuhrt¨ auf eine Variation von Hodge-Strukturen mit Monodromiegruppe W(E6), die mit der Prym-Abbildung verbunden ist. Die Dissertation schließt ab mit einer Rekursionsformel fur¨ den generischen Rang der durch Faltungen von Kurven entstehenden Brill-Noether-Garben auf Jacobischen Varietaten.¨ Abstract. We study Tannakian categories that arise from convolutions of perverse sheaves on abelian varieties. The construction of these categories is closely intertwined with a cohomological vanishing theorem which is an analog of the Artin-Grothendieck theorem and contains as a special case the generic vanishing theorems of Green and Lazarsfeld.