137 research outputs found
Stable domination and independence in algebraically closed valued fields
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
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
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
Let be a first-order theory. A correspondence is established between
internal covers of models of and definable groupoids within . 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
Let 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 of the Berkovich analytification of ,
and deduce several new results on Berkovich spaces from it. In particular we
show that retracts to a finite simplicial complex and is locally
contractible, without any smoothness assumption on . When varies in an
algebraic family, we show that the homotopy type of takes only a
finite number of values. The space is obtained by defining a
topology on the pro-definable set of stably dominated types on . The key
result is the construction of a pro-definable strong retraction of
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
- …