Non-Archimedean Tame Topology and Stably Dominated Types (AM-192)
Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model theory has joined this work through the theory of o-minimality, providing finiteness and uniformity stat...
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| Médium: | E-kniha Kniha |
| Jazyk: | angličtina |
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Princeton University Press
2016
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| Vydání: | 1 |
| Edice: | Annals of Mathematics Studies |
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| ISBN: | 9781400881222, 1400881226, 9780691161686, 0691161682, 9780691161693, 0691161690 |
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| Abstract | Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model theory has joined this work through the theory of o-minimality, providing finiteness and uniformity statements and new structural tools.For non-archimedean fields, such as the p-adics, the Berkovich analytification provides a connected topology with many thoroughgoing analogies to the real topology on the set of complex points, and it has become an important tool in algebraic dynamics and many other areas of geometry.This book lays down model-theoretic foundations for non-archimedean geometry. The methods combine o-minimality and stability theory. Definable types play a central role, serving first to define the notion of a point and then properties such as definable compactness.Beyond the foundations, the main theorem constructs a deformation retraction from the full non-archimedean space of an algebraic variety to a rational polytope. This generalizes previous results of V. Berkovich, who used resolution of singularities methods.No previous knowledge of non-archimedean geometry is assumed. Model-theoretic prerequisites are reviewed in the first sections. |
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| AbstractList | Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model theory has joined this work through the theory of o-minimality, providing finiteness and uniformity statements and new structural tools.For non-archimedean fields, such as the p-adics, the Berkovich analytification provides a connected topology with many thoroughgoing analogies to the real topology on the set of complex points, and it has become an important tool in algebraic dynamics and many other areas of geometry.This book lays down model-theoretic foundations for non-archimedean geometry. The methods combine o-minimality and stability theory. Definable types play a central role, serving first to define the notion of a point and then properties such as definable compactness.Beyond the foundations, the main theorem constructs a deformation retraction from the full non-archimedean space of an algebraic variety to a rational polytope. This generalizes previous results of V. Berkovich, who used resolution of singularities methods.No previous knowledge of non-archimedean geometry is assumed. Model-theoretic prerequisites are reviewed in the first sections. No detailed description available for "Non-Archimedean Tame Topology and Stably Dominated Types (AM-192)". Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model theory has joined this work through the theory of o-minimality, providing finiteness and uniformity statements and new structural tools. For non-archimedean fields, such as thep-adics, the Berkovich analytification provides a connected topology with many thoroughgoing analogies to the real topology on the set of complex points, and it has become an important tool in algebraic dynamics and many other areas of geometry. This book lays down model-theoretic foundations for non-archimedean geometry. The methods combine o-minimality and stability theory. Definable types play a central role, serving first to define the notion of a point and then properties such as definable compactness. Beyond the foundations, the main theorem constructs a deformation retraction from the full non-archimedean space of an algebraic variety to a rational polytope. This generalizes previous results of V. Berkovich, who used resolution of singularities methods. No previous knowledge of non-archimedean geometry is assumed. Model-theoretic prerequisites are reviewed in the first sections. Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model theory has joined this work through the theory of o-minimality, providing finiteness and uniformity statements and new structural tools. For non-archimedean fields, such as the p-adics, the Berkovich analytification provides a connected topology with many thoroughgoing analogies to the real topology on the set of complex points, and it has become an important tool in algebraic dynamics and many other areas of geometry. This book lays down model-theoretic foundations for non-archimedean geometry. The methods combine o-minimality and stability theory. Definable types play a central role, serving first to define the notion of a point and then properties such as definable compactness. Beyond the foundations, the main theorem constructs a deformation retraction from the full non-archimedean space of an algebraic variety to a rational polytope. This generalizes previous results of V. Berkovich, who used resolution of singularities methods. No previous knowledge of non-archimedean geometry is assumed. Model-theoretic prerequisites are reviewed in the first sections. |
| Author | Loeser, François Hrushovski, Ehud |
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| Keywords | fundamental space stably dominated point imaginary base set v-continuity finite-dimensional vector space continuity criteria relatively compact set homotopy Abhyankar property Berkovich space path continuous map stability theory topological structure linear topology retraction definable topology Galois orbit algebraic variety connectedness algebraic geometry definable type Zariski topology Γ-internal set pseudo-Galois covering definable space stable domination model theory iso-definable set curve fibration definable compactness pro-definable bijection stably dominated type continuous definable map strong stability deformation retraction topological embedding ind-definable subset topology non-archimedean tame topology Zariski open subset germ orthogonality sequence natural functor definable topological space real numbers substructure o-minimality stably dominated stable completion Γ-internal space iterated place definably compact set Riemann-Roch definable set algebraically closed valued field inflation homotopy inflation ind-definable set transcendence degree analytic geometry pro-definable map g-continuity main theorem Γ-internal subset valued field o-minimal formulation pro-definable set definable function g-open set canonical extension inverse limit homotopy equivalence semi-lattice iso-definable subset definable subset forward-branching point definable homotopy type topological space morphism Zariski dense open set smooth case non-archimedean geometry g-continuous smoothness birational invariant finite simplicial complex iso-definability residue field extension good metric schematic distance pro-definable subset Polynomial Equivalence relation Parametrization Open set Bijection Subset Limit point Morphism of algebraic varieties Subgroup Substructure Projective variety Definable set Dense set Bounded set Mathematical induction Characterization (mathematics) Canonical map Quasi-projective variety Continuous function Topology Functor Cohomology Existential quantification Torsor (algebraic geometry) Transcendence degree Coset Direct limit Dimension (vector space) Codimension Homotopy Homeomorphism Topological space Saturated model Algebraically closed field Base change Abelian group Residue field Set (mathematics) Embedding Connected space Category of sets Finite morphism Pullback Equivalence of categories Linear topology Isolated point Theorem Yoneda lemma Valuation ring Closed set Irreducible component Constructible set (topology) Transitive relation Compact space Generic point Disjoint union Morphism Affine space Smoothness Union (set theory) Algebraic variety Finite set Parameter Galois extension Pullback (category theory) Surjective function Algebraic closure Irreducibility (mathematics) |
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| Notes | Includes bibliographical references (p. [207]-210) and index |
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| Snippet | Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its... No detailed description available for "Non-Archimedean Tame Topology and Stably Dominated Types (AM-192)". |
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| SubjectTerms | Abelian group Affine space Algebraic Algebraic closure Algebraic variety Algebraically closed field Analytic Base change Berkovich space Bijection Bounded set Canonical map Category of sets Characterization (mathematics) Closed set Codimension Cohomology Compact space Connected space Constructible set (topology) Continuous function Coset Definable set Dense set Dimension (vector space) Direct limit Disjoint union Embedding Equivalence of categories Equivalence relation Existential quantification Finite morphism Finite set Functor Galois extension General Topics for Engineers Generic point Geometry Geometry, Algebraic Homeomorphism Homotopy Irreducibility (mathematics) Irreducible component Isolated point Limit point Linear topology Mathematical induction MATHEMATICS MATHEMATICS / Geometry / General Morphism Morphism of algebraic varieties Open set Parameter Parametrization Polynomial Projective variety Pullback Pullback (category theory) Quasi-projective variety Residue field Saturated model Set (mathematics) Smoothness Subgroup Subset Substructure Surjective function Tame algebras Theorem Topological space Topology Torsor (algebraic geometry) Transcendence degree Transitive relation Union (set theory) Valuation ring Yoneda lemma Zariski topology |
| SubjectTermsDisplay | Algebraic Analytic Geometry Mathematics Topology |
| TableOfContents | Non-archimedean tame topology and stably dominated types -- Contents -- Chapter One: Introduction -- Chapter Two: Preliminaries -- Chapter Three: The space V of stably dominated types -- Chapter Four: Definable compactness -- Chapter Five: A closer look at the stable completion -- Chapter Six: Γ-internal spaces -- Chapter Seven: Curves -- Chapter Eight: Strongly stably dominated points -- Chapter Nine: Specializations and ACV2F -- Chapter Ten: Continuity of homotopies -- Chapter Eleven: The main theorem -- Chapter Twelve: The smooth case -- Chapter Thirteen: An equivalence of categories -- Chapter Fourteen: Applications to the topology of Berkovich spaces -- Bibliography -- Index -- List of notations. Front Matter Table of Contents Chapter One: Introduction Chapter Two: Preliminaries Chapter Three: The space \widehat{V} of stably dominated types Chapter Four: Definable compactness Chapter Five: A closer look at the stable completion Chapter Six: Γ-internal spaces Chapter Seven: Curves Chapter Eight: Strongly stably dominated points Chapter Nine: Specializations and ACV²F Chapter Ten: Continuity of homotopies Chapter Eleven: The main theorem Chapter Twelve: The smooth case Chapter Thirteen: An equivalence of categories Chapter Fourteen: Applications to the topology of Berkovich spaces Bibliography Index List of notations 9.6 g-continuity criterion -- 9.7 Some applications of the continuity criteria -- 9.8 The v-criterion on V̂ -- 9.9 Definability of v- and g-criteria -- 10 Continuity of homotopies -- 10.1 Preliminaries -- 10.2 Continuity on relative ℙ^1 -- 10.3 The inflation homotopy -- 10.4 Connectedness and the Zariski topology -- 11 The main theorem -- 11.1 Statement -- 11.2 Proof of Theorem 11.1.1: Preparation -- 11.3 Construction of a relative curve homotopy -- 11.4 The base homotopy -- 11.5 The tropical homotopy -- 11.6 End of the proof -- 11.7 Variation in families -- 12 The smooth case -- 12.1 Statement -- 12.2 Proof and remarks -- 13 An equivalence of categories -- 13.1 Statement of the equivalence of categories -- 13.2 Proof of the equivalence of categories -- 13.3 Remarks on homotopies over imaginary base sets -- 14 Applications to the topology of Berkovich spaces -- 14.1 Berkovich spaces -- 14.2 Retractions to skeleta -- 14.3 Finitely many homotopy types -- 14.4 More tame topological properties -- 14.5 The lattice completion -- 14.6 Berkovich points as Galois orbits -- Bibliography -- Index -- List of notations Cover -- Title -- Copyright -- Contents -- 1 Introduction -- 2 Preliminaries -- 2.1 Definable sets -- 2.2 Pro-definable and ind-definable sets -- 2.3 Definable types -- 2.4 Stable embeddedness -- 2.5 Orthogonality to a definable set -- 2.6 Stable domination -- 2.7 Review of ACVF -- 2.8 Г-internal sets -- 2.9 Orthogonality to Г -- 2.10 V̂ for stable definable V -- 2.11 Decomposition of definable types -- 2.12 Pseudo-Galois coverings -- 3 The space V̂ of stably dominated types -- 3.1 V̂ as a pro-definable set -- 3.2 Some examples -- 3.3 The notion of a definable topological space -- 3.4 V̂ as a topological space -- 3.5 The affine case -- 3.6 Simple points -- 3.7 v-open and g-open subsets, v+g-continuity -- 3.8 Canonical extensions -- 3.9 Paths and homotopies -- 3.10 Good metrics -- 3.11 Zariski topology -- 3.12 Schematic distance -- 4 Definable compactness -- 4.1 Definition of definable compactness -- 4.2 Characterization of definable compactness -- 5 A closer look at the stable completion -- 5.1 A^n and spaces of semi-lattices -- 5.2 A representation of ℙ^n -- 5.3 Relative compactness -- 6 Г-internal spaces -- 6.1 Preliminary remarks -- 6.2 Topological structure of Г-internal subsets -- 6.3 Guessing definable maps by regular algebraic maps -- 6.4 Relatively Г-internal subsets -- 7 Curves -- 7.1 Definability of Ĉ for a curve C -- 7.2 Definable types on curves -- 7.3 Lifting paths -- 7.4 Branching points -- 7.5 Construction of a deformation retraction -- 8 Strongly stably dominated points -- 8.1 Strongly stably dominated points -- 8.2 A Bertini theorem -- 8.3 Г-internal sets and strongly stably dominated points -- 8.4 Topological properties of V^# -- 9 Specializations and ACV^2F -- 9.1 g-topology and specialization -- 9.2 v-topology and specialization -- 9.3 ACV^2F -- 9.4 The map R^20 21 : V̂20 → V̂21 -- 9.5 Relative versions 3. The space v̂ of stably dominated types 7. Curves 5. A closer look at the stable completion 11. The main theorem 12. The smooth case 4. Definable compactness Index 10. Continuity of homotopies - 14. Applications to the topology of Berkovich spaces 13. An equivalence of categories / Contents List of notations 1. Introduction Frontmatter -- 2. Preliminaries 9. Specializations and ACV2F 6. Γ-internal spaces 8. Strongly stably dominated points Bibliography |
| Title | Non-Archimedean Tame Topology and Stably Dominated Types (AM-192) |
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