446 research outputs found
Imaginaries in separably closed valued fields
We show that separably closed valued fields of finite imperfection degree
(either with lambda-functions or commuting Hasse derivations) eliminate
imaginaries in the geometric language. We then use this classification of
interpretable sets to study stably dominated types in those structures. We show
that separably closed valued fields of finite imperfection degree are
metastable and that the space of stably dominated types is strict
pro-definable
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
Special transformations in algebraically closed valued fields
We present two of the three major steps in the construction of motivic
integration, that is, a homomorphism between Grothendieck semigroups that are
associated with a first-order theory of algebraically closed valued fields, in
the fundamental work of Hrushovski and Kazhdan. We limit our attention to a
simple major subclass of V-minimal theories of the form ACVF_S(0, 0), that is,
the theory of algebraically closed valued fields of pure characteristic
expanded by a (VF, Gamma)-generated substructure S in the language L_RV. The
main advantage of this subclass is the presence of syntax. It enables us to
simplify the arguments with many different technical details while following
the major steps of the Hrushovski-Kazhdan theory.Comment: This is the published version of a part of the notes on the
Hrushovski-Kazhdan integration theory. To appear in the Annals of Pure and
Applied Logi
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
- …