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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Hlavní autoři:	Hrushovski, Ehud, Loeser, François
Médium:	E-kniha Kniha
Jazyk:	angličtina
Vydáno:	United States Princeton University Press 2016
Vydání:	1
Edice:	Annals of Mathematics Studies
Témata:	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 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)
ISBN:	9781400881222, 1400881226, 9780691161686, 0691161682, 9780691161693, 0691161690
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Popis
Shrnutí:	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.
Bibliografie:	Includes bibliographical references (p. [207]-210) and index
ISBN:	9781400881222 1400881226 9780691161686 0691161682 9780691161693 0691161690
DOI:	10.1515/9781400881222