137 research outputs found

    Stable domination and independence in algebraically closed valued fields

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    We seek to create tools for a model-theoretic analysis of types in algebraically closed valued fields (ACVF). We give evidence to show that a notion of 'domination by stable part' plays a key role. In Part A, we develop a general theory of stably dominated types, showing they enjoy an excellent independence theory, as well as a theory of definable types and germs of definable functions. In Part B, we show that the general theory applies to ACVF. Over a sufficiently rich base, we show that every type is stably dominated over its image in the value group. For invariant types over any base, stable domination coincides with a natural notion of `orthogonality to the value group'. We also investigate other notions of independence, and show that they all agree, and are well-behaved, for stably dominated types. One of these is used to show that every type extends to an invariant type; definable types are dense. Much of this work requires the use of imaginary elements. We also show existence of prime models over reasonable bases, possibly including imaginaries

    On finite imaginaries

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    We study finite imaginaries in certain valued fields, and prove a conjecture of Cluckers and Denef.Comment: 15p

    Imaginaries and definable types in algebraically closed valued fields

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    The text is based on notes from a class entitled {\em Model Theory of Berkovich Spaces}, given at the Hebrew University in the fall term of 2009, and retains the flavor of class notes. It includes an exposition of material from \cite{hhmcrelle}, \cite{hhm} and \cite{HL}, regarding definable types in the model completion of the theory of valued fields, and the classification of imaginary sorts. The latter is given a new proof, based on definable types rather than invariant types, and on the notion of {\em generic reparametrization}. I also try to bring out the relation to the geometry of \cite{HL} - stably dominated definable types as the model theoretic incarnation of a Berkovich point

    Groupoids, imaginaries and internal covers

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    Let TT be a first-order theory. A correspondence is established between internal covers of models of TT and definable groupoids within TT. We also consider amalgamations of independent diagrams of algebraically closed substructures, and find strong relation between: covers, uniqueness for 3-amalgamation, existence of 4-amalgamation, imaginaries of T^\si, and definable groupoids. As a corollary, we describe the imaginary elements of families of finite-dimensional vector spaces over pseudo-finite fields.Comment: Local improvements; thanks to referee of Turkish Mathematical Journal. First appeared in the proceedings of the Paris VII seminar: structures alg\'ebriques ordonn\'ee (2004/5

    Non-archimedean tame topology and stably dominated types

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    Let VV be a quasi-projective algebraic variety over a non-archimedean valued field. We introduce topological methods into the model theory of valued fields, define an analogue V^\hat {V} of the Berkovich analytification VanV^{an} of VV, and deduce several new results on Berkovich spaces from it. In particular we show that VanV^{an} retracts to a finite simplicial complex and is locally contractible, without any smoothness assumption on VV. When VV varies in an algebraic family, we show that the homotopy type of VanV^{an} takes only a finite number of values. The space V^\hat {V} is obtained by defining a topology on the pro-definable set of stably dominated types on VV. The key result is the construction of a pro-definable strong retraction of V^\hat {V} to an o-minimal subspace, the skeleton, definably homeomorphic to a space definable over the value group with its piecewise linear structure.Comment: Final versio
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