06822nam 2201705 450 991082203230332120210506032322.01-4008-8122-610.1515/9781400881222(CKB)3710000000537982(EBL)4198288(OCoLC)933388580(MiAaPQ)EBC4198288(StDuBDS)EDZ0001756478(DE-B1597)467880(OCoLC)979911327(DE-B1597)9781400881222(Au-PeEL)EBL4198288(CaPaEBR)ebr11135512(CaONFJC)MIL882376(EXLCZ)99371000000053798220160115h20162016 uy 0engurnnu---|u||urdacontentrdamediardacarrierNon-archimedean tame topology and stably dominated types /Ehud Hrushovski, François LoeserPrinceton, New Jersey ;Oxford, [England] :Princeton University Press,2016.©20161 online resource (227 p.)Annals of Mathematics Studies ;Number 192Description based upon print version of record.0-691-16169-0 0-691-16168-2 Includes bibliographical references and index.Front matter --Contents --1. Introduction --2. Preliminaries --3. The space v̂ of stably dominated types --4. Definable compactness --5. A closer look at the stable completion --6. Γ-internal spaces --7. Curves --8. Strongly stably dominated points --9. Specializations and ACV2F --10. Continuity of homotopies --11. The main theorem --12. The smooth case --13. An equivalence of categories --14. Applications to the topology of Berkovich spaces --Bibliography --Index --List of notationsOver 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.Annals of mathematics studies ;Number 192.Tame algebrasAbhyankar property.Berkovich space.Galois orbit.Riemann-Roch.Zariski dense open set.Zariski open subset.Zariski topology.algebraic geometry.algebraic variety.algebraically closed valued field.analytic geometry.birational invariant.canonical extension.connectedness.continuity criteria.continuous definable map.continuous map.curve fibration.definable compactness.definable function.definable homotopy type.definable set.definable space.definable subset.definable topological space.definable topology.definable type.definably compact set.deformation retraction.finite simplicial complex.finite-dimensional vector space.forward-branching point.fundamental space.g-continuity.g-continuous.g-open set.germ.good metric.homotopy equivalence.homotopy.imaginary base set.ind-definable set.ind-definable subset.inflation homotopy.inflation.inverse limit.iso-definability.iso-definable set.iso-definable subset.iterated place.linear topology.main theorem.model theory.morphism.natural functor.non-archimedean geometry.non-archimedean tame topology.o-minimal formulation.o-minimality.orthogonality.path.pro-definable bijection.pro-definable map.pro-definable set.pro-definable subset.pseudo-Galois covering.real numbers.relatively compact set.residue field extension.retraction.schematic distance.semi-lattice.sequence.smooth case.smoothness.stability theory.stable completion.stable domination.stably dominated point.stably dominated type.stably dominated.strong stability.substructure.topological embedding.topological space.topological structure.topology.transcendence degree.v-continuity.valued field.Γ-internal set.Γ-internal space.Γ-internal subset.Tame algebras.512.4SI 830rvkHrushovski Ehud725941Loeser FrançoisMiAaPQMiAaPQMiAaPQBOOK9910822032303321Non-archimedean tame topology and stably dominated types4046851UNINA