NOTES ON NONCOMMUTATIVE GEOMETRY Igor Nikolaev E-mail: [email protected] arXiv:1503.05411v1 [math.OA] 17 Mar 2015 Foreword These notes are neither an introduction nor a survey (if only a brief) of noncommutative geometry – later NCG; rather, they strive to answer some naive but vital questions: What is the purpose of NCG and what is it good for? Why a number theorist or an algebraic geometer should care about the NCG? Can NCG solve open problems of classical geometry inaccessible otherwise? In other words, why does NCG matter? What is it anyway? Good answer means good examples. A sweetheart of NCG called non- commutative torus captures classical geometry of elliptic curves because such a torus is a coordinate ring for elliptic curves. In other words, one deals with a functor from algebraic geometry to the NCG; such functors are at the heart of our book. What is NCG anyway? It is a calculus of functors on the classical spaces (e.g. algebraic, geometric, topological, etc) with the values in NCG. Such an approach departs from the tradition of recasting geometry of the classi- cal space X in terms of the C∗-algebra C(X) of continuous complex-valued functions on X, see the monograph by [Connes 1994] [39]. Boston, March 2014 Igor Nikolaev i ii Introduction It is not easy to write an elementary introduction to the NCG because the simplest non-trivial examples are mind-blowing and involve the KK-groups, subfactors, Sklyanin algebras, etc. Such a material cannot be shrink-wrapped into a single graduate course being the result of a slow roasting of ideas from different (and distant) mathematical areas. But this universality of NCG is a thrill giving the reader a long lost sense of unity of mathematics. In writing these notes the author had in mind a graduate student in (say) number theory eager to learn something new, e.g. a noncommutative torus with real multiplication; it will soon transpire that such an object is linked to the K-rational points of elliptic curves and the Langlands program. The book has three parts. Part I is preparatory: Chapter 1 deals with the simplest examples of functors arising in algebraic geometry, number the- ory and topology; the functors take value in a category of the C∗-algebras known as noncommutative tori. Using these functors one gets a set of non- commutative invariants for elliptic curves and Anosov’s automorphisms of the two-dimensional torus. Chapter 2 is a brief introduction to the cate- gories, functors and natural transformations; they will be used throughout the book. Chapter 3 covers an essential information about the category of C∗-algebras and their K-theory; we introduce certain important classes of the C∗-algebras: the AF-algebras, the UHF-algebras and the Cuntz-Krieger algebras. Our choice of the C∗-algebras is motivated by their applications in Part II. Part II deals with the noncommutative invariants obtained from the func- tors acting on various classical spaces. Chapter 4 is devoted to such functors on the topological spaces with values in the category of the so-called station- ary AF-algebras; the noncommutative invariants are the Handelman triples (Λ, [I],K), where Λ is an order in a real algebraic number field K and [I] an equivalence class of the ideals of Λ. Chapter 5 deals with the examples iii iv of functors arising in projective algebraic geometry and their noncommuta- tive invariants. Finally, Chapter 6 covers functors in number theory and the corresponding invariants. Part III is a brief survey of the NCG; the survey is cursory yet an extensive guide to the literature has been compiled at the end of each chapter. We hope that the reader can instruct himself by looking at the original publications; we owe an apology to the authors whose works are not on the list. There exist several excellent textbooks on the NCG. The first and fore- most is the monograph by A. Connes “G´eom´etrie Non Commutative”, Paris, 1990 and its English edition “Noncommutative Geometry”, Academic Press, 1994. The books by J. Madore “An Introduction to Noncommutative Dif- ferential Geometry & its Applications”, Cambridge Univ. Press, 1995, by J. M. Gracia-Bondia, J. C. Varilly and H. Figueroa “Elements of Noncom- mutative Geometry”, Birkh¨auser, 2000 and by M. Khalkhali “Basic Non- commutative Geometry”, EMS Series of Lectures in Mathematics, 2007 treat particular aspects of Connes’ monograph. A different approach to the NCG is covered in a small but instructive book by Yu. I. Manin “Topics in Non- commutative Geometry”, Princeton Univ. Press, 1991. Finally, a more spe- cialized “Noncommutative Geometry, Quantum Fields and Motives”, AMS Colloquium Publications, 2008 by A. Connes and M. Marcolli is devoted to the links to physics and number theory. None of these books treat the NCG as a functor [115]. I thank the organizers, participants and sponsors of the Spring Institute on Noncommutative Geometry and Operator Algebras (NCGOA) held annu- ally at the Vanderbilt University in Nashville, Tennessee; these notes grew from efforts to find out what is going on there. (I still don’t have an an- swer.) I am grateful to folks who helped me with the project; among them are D. Anosov, P. Baum, D. Bisch, B. Blackadar, O. Bratteli, A. Connes, J. Cuntz, G. Elliott, K. Goodearl, D. Handelman, N. Higson, B. Hughes, V. F. R. Jones, M. Kapranov, M. Khalkhali, W. Krieger, Yu. Manin, V. Manuilov, M. Marcolli, V. Mathai, A. Mishchenko, S. Novikov, N. C. Phillips, M. Ri- effel, W. Thurston, V. Troitsky, G. Yu and others. Contents I BASICS 3 1 Model Examples 5 1.1 Noncommutativetorus . .. 5 1.2 Elliptic curves . 10 1.3 Complex multiplication . 16 1.4 Anosovautomorphisms. 19 2 Categories and Functors 27 2.1 Categories............................. 27 2.2 Functors.............................. 31 2.3 Naturaltransformations . 34 3 C∗-Algebras 37 3.1 Basicdefinitions.......................... 37 3.2 Crossedproducts ......................... 41 3.3 K-theory of C∗-algebras ..................... 43 3.4 Noncommutativetori . 48 3.4.1 Generalcase........................ 48 3.4.2 Two-dimensionaltorus . 50 3.5 AF-algebras ............................ 52 3.5.1 Generalcase........................ 52 3.5.2 StationaryAF-algebras. 55 3.6 UHF-algebras ........................... 58 3.7 Cuntz-Kriegeralgebras . 59 v vi CONTENTS II NONCOMMUTATIVE INVARIANTS 63 4 Topology 65 4.1 Classificationofthesurfacemaps . 65 4.1.1 Pseudo-Anosovmapsofasurface . 66 4.1.2 Functorsandinvariants. 68 4.1.3 Jacobian of measured foliations . 70 4.1.4 Equivalent foliations . 73 4.1.5 Proofs ........................... 74 4.1.6 Anosovmapsofthetorus . 80 4.2 Torsioninthetorusbundles . 84 4.2.1 Cuntz-Kriegerfunctor . 84 4.2.2 Proofoftheorem4.2.1 . 85 4.2.3 Noncommutative invariants of torus bundles . 87 4.3 Obstruction theory for Anosov’s bundles . 89 4.3.1 FundamentalAF-algebra . 89 4.3.2 Proofs ........................... 94 4.3.3 Obstructiontheory . 99 5 Algebraic Geometry 107 5.1 Elliptic curves . 108 5.1.1 Noncommutative tori via Sklyanin algebras . 110 5.1.2 Noncommutative tori via measured foliations . 116 5.2 Highergenusalgebraiccurves . .122 5.2.1 ToricAF-algebras. .122 5.2.2 ProofofTheorem5.2.1. .124 5.3 Complexprojectivevarieties . .129 5.3.1 Serre C∗-algebras .....................130 5.3.2 Proofoftheorem5.3.3 . .133 5.3.3 Real multiplication revisited . 137 5.4 Application: Mapping class groups . 140 5.4.1 Harvey’sconjecture. .140 5.4.2 ProofofTheorem5.4.1. .141 6 Number Theory 147 6.1 Complex multiplication . 148 6.1.1 Functor on elliptic curves with complex multiplication . 150 6.1.2 ProofofTheorem6.1.1. .151 CONTENTS vii 6.1.3 ProofofTheorem6.1.2. .154 6.1.4 ProofofTheorem6.1.3. .156 6.2 Ranks of the K-rational elliptic curves . 161 6.2.1 Arithmetic complexity of noncommutative tori . 162 6.2.2 Q-curves ..........................164 6.2.3 ProofofTheorem6.2.2. .165 6.2.4 Numericalexamples . .168 6.3 Non-commutativereciprocity. 169 6.3.1 L-functionofnoncommutative tori . 170 6.3.2 ProofofTheorem6.3.1. .171 6.3.3 Supplement: Gr¨ossencharacters, units and π(n) ....178 6.4 Langlands program for noncommutative tori . 182 6.4.1 L( 2n ,s).........................182 ARM 6.4.2 ProofofTheorem6.4.1. .186 6.4.3 Supplement: Artin L-function . .189 6.5 Projective varieties over finite fields . 191 6.5.1 Traces of Frobenius endomorphisms . 191 6.5.2 ProofofTheorem6.5.1 . .194 6.5.3 Examples .........................199 6.6 Transcendentalnumbertheory . 204 6.6.1 Algebraic values of transcendental functions . 204 6.6.2 ProofofTheorem6.6.1. .205 III BRIEF SURVEY OF NCG 211 7 Finite Geometries 213 7.1 Axiomsofprojectivegeometry. .213 7.2 Projective spaces over skew fields . 215 7.3 DesarguesandPappusaxioms . .216 8 Continuous Geometries 219 8.1 W ∗-algebras............................219 8.2 VonNeumanngeometry . .222 9 Connes Geometries 225 9.1 ClassificationoftypeIIIfactors . .225 9.1.1 Tomita-TakesakiTheory . .225 viii CONTENTS 9.1.2 ConnesInvariants. .227 9.2 Noncommutative differentialgeometry . 228 9.2.1 Hochschild homology . 228 9.2.2 Cyclic homology . 229 9.2.3 Novikov Conjecture for hyperbolic groups . 230 9.3 Connes’IndexTheorem . .231 9.3.1 Atiyah-Singer Theorem for families of elliptic operators 231 9.3.2 Foliatedspaces ......................232 9.3.3 IndexTheoremforfoliatedspaces . 233 9.4 Bost-Connesdynamicalsystem . 234 9.4.1 Hecke C∗-algebra .....................235 9.4.2 Bost-ConnesTheorem . .235 10 Index Theory 237 10.1 Atiyah-SingerTheorem. .237 10.1.1 Fredholmoperators . .237 10.1.2 Elliptic operators on manifolds . 239 10.1.3 IndexTheorem . .240 10.2K-homology ............................241 10.2.1 TopologicalK-theory . .241 10.2.2 Atiayh’s realization of K-homology . 243 10.2.3 Brown-Douglas-Fillmore Theory . 244 10.3 Kasparov’sKK-theory .