Homotopy theory and arithmetic geometry : motivic and diophantine aspects, LMS-CMI Research School, London, July 2018 / / edited by Frank Neumann and Ambrus Pál

• Pubblicazione: Cham, Switzerland : , : Springer, , [2021]; ©2021
• Descrizione fisica: 1 online resource (223 pages)
• Disciplina: 514.24
• Soggetto topico: Arithmetical algebraic geometry; Homotopy theory; Teoria de l'homotopia; Geometria algebraica aritmètica
• Soggetto genere / forma: Congressos; Llibres electrònics
• Persona (resp. second.): NeumannFrank (Mathematician); PálAmbrus
• Note generali: Includes index.
• Nota di contenuto: Intro -- Preface -- Contents -- 1 Homotopy Theory and Arithmetic Geometry-Motivic and Diophantine Aspects: An Introduction -- 1.1 Overview of Themes -- 1.2 Summaries of Individual Contributions -- References -- 2 An Introduction to A1-Enumerative Geometry -- 2.1 Introduction -- 2.2 Preliminaries -- 2.2.1 Enriching the Topological Degree -- 2.2.2 The Grothendieck-Witt Ring -- 2.2.3 Lannes' Formula -- 2.2.4 The Unstable Motivic Homotopy Category -- 2.2.5 Colimits -- 2.2.6 Purity -- 2.3 A1-enumerative Geometry -- 2.3.1 The Eisenbud-Khimshiashvili-Levine Signature Formula -- 2.3.2 Sketch of Proof for Theorem 4 -- 2.3.3 A1-Milnor Numbers -- 2.3.4 An Arithmetic Count of the Lines on a Smooth Cubic Surface -- 2.3.5 An Arithmetic Count of the Lines Meeting 4Lines in Space -- Notation Guide -- References -- 3 Cohomological Methods in Intersection Theory -- 3.1 Introduction -- 3.2 Étale Motives -- 3.2.1 The h-topology -- 3.2.2 Construction of Motives, After Voevodsky -- 3.2.3 Functoriality -- 3.2.4 Representability Theorems -- 3.3 Finiteness and Euler Characteristic -- 3.3.1 Locally Constructible Motives -- 3.3.2 Integrality of Traces and Rationality of ζ-Functions -- 3.3.3 Grothendieck-Verdier Duality -- 3.3.4 Generic Base Change: A Motivic Variation on Deligne's Proof -- 3.4 Characteristic Classes -- 3.4.1 Künneth Formula -- 3.4.2 Grothendieck-Lefschetz Formula -- References -- 4 Étale Homotopy and Obstructions to Rational Points -- 4.1 Introduction -- 4.2 ∞-Categories -- 4.2.1 Motivation -- 4.2.2 Quasi-Categories -- 4.2.3 ∞-Groupoids and the Homotopy Hypothesis -- 4.2.4 Quasi-Categories from Topological Categories -- 4.2.5 ∞-Category Theory -- 4.2.6 The Homotopy Category -- 4.2.7 ∞-Categories and Homological Algebra -- 4.2.8 Stable ∞-Categories -- 4.2.9 Localization -- 4.3 ∞-Topoi -- 4.3.1 Definitions -- 4.3.2 The Shape of an ∞-Topos. ; 4.4 Obstruction Theory -- 4.4.1 Obstruction Theory for Homotopy Types -- 4.4.2 For ∞-Topoi and Linear(ized) Versions -- 4.5 Étale Homotopy and Rational Points -- 4.5.1 The étale ∞-Topos -- 4.5.2 Rational Points -- 4.5.3 The Local-to-Global Principle -- 4.6 Galois Theory and Embedding Problems -- 4.6.1 Topoi and Embedding Problems -- References -- 5 A1-homotopy Theory and Contractible Varieties: A Survey -- 5.1 Introduction: Topological and Algebro-Geometric Motivations -- 5.1.1 Open Contractible Manifolds -- 5.1.2 Contractible Algebraic Varieties -- 5.2 A User's Guide to A1-homotopy Theory -- 5.2.1 Brief Topological Motivation -- 5.2.2 Homotopy Functors in Algebraic Geometry -- 5.2.3 The Unstable A1-homotopy Category: Construction -- Spaces -- Nisnevich and cdh Distinguished Squares -- Localization -- 5.2.4 The Unstable A1-homotopy Category: Basic Properties -- Motivic Spheres -- Representability Statements -- Representability of Chow Groups -- The Purity Isomorphism -- Comparison of Nisnevich and cdh-local A1-weak Equivalences -- 5.2.5 A Snapshot of the Stable Motivic Homotopy Category -- Stable Representablity of Algebraic K-theory -- Milnor-Witt K-theory -- 5.3 Concrete A1-weak Equivalences -- 5.3.1 Constructing A1-weak Equivalences of Smooth Schemes -- 5.3.2 A1-weak Equivalences vs. Weak Equivalences -- 5.3.3 Cancellation Questions and A1-weak Equivalences -- 5.3.4 Danielewski Surfaces and Generalizations -- 5.3.5 Building Quasi-Affine A1-contractible Varieties -- Unipotent Quotients -- Other Quasi-Affine A1-contractible Varieties -- 5.4 Further Computations in A1-homotopy Theory -- 5.4.1 A1-homotopy Sheaves -- Basic Definitions -- A1-rigid Varieties Embed into H(k) -- 5.4.2 A1-connectedness and Geometry -- A1-connectedness and Rationality Properties -- 5.4.3 A1-homotopy Sheaves Spheres and Brouwer Degree -- 5.4.4 A1-homotopy at Infinity. ; One-point Compactifications -- Stable End Spaces -- 5.5 Cancellation Questions and A1-contractibility -- 5.5.1 The Biregular Cancellation Problem -- 5.5.2 A1-contractibility vs Topological Contractibility -- Affine Lines on Topologically Contractible Surfaces -- Chow Groups and Vector Bundles on Topologically Contractible Surfaces -- 5.5.3 Cancellation Problems and the Russell Cubic -- The Russell Cubic and Equivariant K-theory -- Higher Chow Groups and Stable A1-contractibility -- 5.5.4 A1-contractibility of the Koras-Russell Threefold -- 5.5.5 Koras-Russell Fiber Bundles -- References -- Index.
• Titolo autorizzato: Homotopy theory and arithmetic geometry
• ISBN: 3-030-78977-2
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 996466408503316
• Lo trovi qui: Univ. di Salerno
• Serie: Lecture Notes in Mathematics