121 research outputs found
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
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
Generalizations of Kochen and Specker's Theorem and the Effectiveness of Gleason's Theorem
Kochen and Specker's theorem can be seen as a consequence of Gleason's
theorem and logical compactness. Similar compactness arguments lead to stronger
results about finite sets of rays in Hilbert space, which we also prove by a
direct construction. Finally, we demonstrate that Gleason's theorem itself has
a constructive proof, based on a generic, finite, effectively generated set of
rays, on which every quantum state can be approximated.Comment: 14 pages, 6 figures, read at the Robert Clifton memorial conferenc
An invariant for difference field extensions
In this paper we introduce a new invariant (the distant degree) for
difference field extensions of finite transcendence degree, and we explore some
of its properties. We also discuss a generalisation of this invariant and of
the limit degree to groups with an automorphism.Comment: After posting the previous version, we discovered the work of Willis
on totally disconnected locally compact groups. Over a large area of overlap,
our "distant degree" invariant of an automorphism agrees with Willis' {\em
scale}. In the new version we describe the relations between the two
frameworks. 14pp. July 2011: Small change
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
On subgroups of semi-abelian varieties defined by difference equations
Consider the algebraic dynamics on a torus T=G_m^n given by a matrix M in
GL_n(Z). Assume that the characteristic polynomial of M is prime to all
polynomials X^m-1. We show that any finite equivariant map from another
algebraic dynamics onto (T,M) arises from a finite isogeny T \to T. A similar
and more general statement is shown for Abelian and semi-abelian varieties.
In model-theoretic terms, our result says: Working in an existentially closed
difference field, we consider a definable subgroup B of a semi-abelian variety
A; assume B does not have a subgroup isogenous to A'(F) for some twisted fixed
field F, and some semi-Abelian variety A'. Then B with the induced structure is
stable and stably embedded. This implies in particular that for any n>0, any
definable subset of B^n is a Boolean combination of cosets of definable
subgroups of B^n.
This result was already known in characteristic 0 where indeed it holds for
all commutative algebraic groups ([CH]). In positive characteristic, the
restriction to semi-abelian varieties is necessary.Comment: Revised version, to appea
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