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Register a new userImaginaries and definable types in algebraically closed valued fields
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Authors
Ehud Hrushovski
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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.
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We answer two open questions about the model theory of valued differential fields introduced by Scanlon. We show that they eliminate imaginaries in the geometric language introduced by Haskell, Hrushovski and Macpherson and that they have the invariant extension property. These two result follow from an abstract criterion for the density of definable types in enrichments of algebraically closed valued fields. Finally, we show that this theory is metastable.
The following strong form of density of definable types is introduced for theories T admitting a fibered dimension function d: given a model M of T and a definable subset X of M^n, there is a definable type p in X, definable over a code for X and of the same d-dimension as X. Both o-minimal theories and the theory of closed ordered differential fields (CODF) are shown to have this property. As an application, we derive a new proof of elimination of imaginaries for CODF.
In this paper we study elimination of imaginaries in some classes of henselian valued fields of equicharacteristic zero and residue field algebraically closed. The results are sensitive to the complexity of the value group. We focus first in the case where the ordered abelian group has finite spines, and then prove a better result for the dp-minimal case. An ordered abelian with finite spines weakly eliminates imaginaries once we add sorts for the quotient groups $Gamma/ Delta$ for each definable convex subgroup $Delta$, and sorts for the quotient groups $Gamma/ Delta+ lGamma$ where $Delta$ is a definable convex subgroup and $l in mathbb{N}_{geq 2}$. We refer to these sorts as the quotient sorts. We prove the following two theorems: Theorem: Let $K$ be a valued field of equicharacteristic zero, residue field algebraically closed and value group with finite spines. Then $K$ admits weak elimination of imaginaries once we add codes for all the definable $mathcal{O}$-submodules of $K^{n}$ for each $n in mathbb{N}$, and the quotient sorts for the value group. Theorem: Let $K$ be a henselian valued field of equicharacteristic zero, residue field algebraically closed and dp-minimal value group. Then $K$ eliminates imaginaries once we add codes for all the definable $mathcal{O}$-submodules of $K^{n}$ for each $n in mathbb{N}$, the quotient sorts for the value group and constants to distinguish representatives of the cosets of $Delta+lGamma$ in $Gamma$, where $Delta$ is a convex definable subgroup and $l in mathbb{N}_{geq 2}$.
For certain theories of existentially closed topological differential fields, we show that there is a strong relationship between $mathcal Lcup{D}$-definable sets and their $mathcal L$-reducts, where $mathcal L$ is a relational expansion of the field language and $D$ a symbol for a derivation. This enables us to associate with an $mathcal Lcup{D}$-definable group in models of such theories, a local $mathcal L$-definable group. As a byproduct, we show that in closed ordered differential fields, one has the descending chain condition on centralisers.
We continue the study of a class of topological $mathcal{L}$-fields endowed with a generic derivation $delta$, focussing on describing definable groups. We show that one can associate to an $mathcal{L}_{delta}$ definable group a type $mathcal{L}$-definable topological group. We use the group configuration tool in o-minimal structures as developed by K. Peterzil.
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