770 research outputs found
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
Valued fields, Metastable groups
We introduce a class of theories called metastable, including the theory of
algebraically closed valued fields (ACVF) as a motivating example. The key
local notion is that of definable types dominated by their stable part. A
theory is metastable (over a sort ) if every type over a sufficiently
rich base structure can be viewed as part of a -parametrized family of
stably dominated types. We initiate a study of definable groups in metastable
theories of finite rank. Groups with a stably dominated generic type are shown
to have a canonical stable quotient. Abelian groups are shown to be
decomposable into a part coming from , and a definable direct limit
system of groups with stably dominated generic. In the case of ACVF, among
definable subgroups of affine algebraic groups, we characterize the groups with
stably dominated generics in terms of group schemes over the valuation ring.
Finally, we classify all fields definable in ACVF.Comment: 48 pages. Minor corrections and improvements following a referee
repor
Beautiful pairs
We introduce an abstract framework to study certain classes of stably
embedded pairs of models of a complete -theory , called
beautiful pairs, which comprises Poizat's belles paires of stable structures
and van den Dries-Lewenberg's tame pairs of o-minimal structures. Using an
amalgamation construction, we relate several properties of beautiful pairs with
classical Fra\"{i}ss\'{e} properties.
After characterizing beautiful pairs of various theories of ordered abelian
groups and valued fields, including the theories of algebraically, -adically
and real closed valued fields, we show an Ax-Kochen-Ershov type result for
beautiful pairs of henselian valued fields. As an application, we derive strict
pro-definability of particular classes of definable types. When is one of
the theories of valued fields mentioned above, the corresponding classes of
types are related to classical geometric spaces such as Berkovich and Huber's
analytifications. In particular, we recover a result of Hrushovski-Loeser on
the strict pro-definability of stably dominated types in algebraically closed
valued fields.Comment: 40 page
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
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
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