NIP Henselian valued fields.(English)Zbl 1444.03130

Let $$(K, v)$$ denote a valued field, with underlying field $$K$$ and valuation $$v$$, let $$Kv$$ denote the residue feild, and $$K^\times$$ the multiplicative group of $$K$$, and $$\mathcal U$$ some sufficiently saturated elementary extension of $$K$$. Recall that a structure is called NIP if its theory does not have the independence property.
From the Introduction : “…we consider the question of when NIP transfers from the residue field to the valued field. Our approach generalizes well known results of Delon and Gurevich-Schmitt for Henselian fields of equicharacteristic $$0$$, and of Bélair for certain perfect fields of positive characteristic, in a uniform way.”

The two main results are the following.
Theorem 3.3. A complete theory of separably algebraically maximal Kaplansky fields of finite degree of imperfection is NIP if the corresponding theories of residue fields and value groups are both NIP.
Note that in this theorem, additional NIP structure on the value group and the residue field are allowed, something which was not addressed explicitely in the original results.
Theorem 4.6. Let $$(K, v)$$ be a Henselian valued field of residue characteristic $$\operatorname{char}(Kv)=p$$. In case $$(K, v)$$ has mixed characteristic, assume that $$K^\times / (K^\times)^p$$ is finite. If the characteristic of $$K$$ is positive, assume that $$K$$ has finite degree of imperfection. Then
$$(K, v)$$ is NIP $$\iff$$ $$Kv$$ is NIP and $$(K, v)$$ is roughly separably tame.
For Theorem 3.3, a somewhat more general result is proved. We briefly mention it here. Let $$T$$ be a complete theory of valued fields with possible additional structures. Consider the following two properties (SE) and (Im).
(SE) The residue field and the value group are stably embedded.
(Im) If $$K$$ is a model of $$T$$ and $$a\in \mathcal U$$ is a singleton such that $$K(a)/K$$ is an immediate extension, then the type of $$a$$ over K is implied by instances of NIP formulas.
Theorem 2.3. Under assumptions (SE) and (Im), the theory $$T$$ is NIP if and only if the theories of the residue field and value group are.

MSC:

 03C60 Model-theoretic algebra 03C45 Classification theory, stability, and related concepts in model theory 12J10 Valued fields 12L12 Model theory of fields
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References:

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