Topic Archive: valuations
(Joint work with Jochen Koenigsmann.) There has been a lot of recent progress in the area of definable henselian valuations. Here, a valuation is called definable if its valuation ring is a first-order definable subset of the field in the language of rings. Applications of results concerning definable henselian valuations typically include showing decidability of the theory of a field or facts about its absolute Galois group.
We study the question of which henselian fields admit definable henselian valuations with and without parameters. In equicharacteristic 0, we give a complete characterization of henselian fields admitting parameter-definable (non-trivial) henselian valuations. We also give a partial characterization result for the parameter-free case.
Zariski introduced the space of all valuations on a given field K and named it after his mentor the `Riemann manifold’. This terminology is justified because of the following two facts he proved about it: (1) one can define a (quasi-)compact topology on this space (and we honor him embracingly by calling it the Zariski-Riemann space), and (2) if K is the function field of a curve, then this space is isomorphic to a non-singular curve with the same function field. Hence (2) gives resolution of singularities in dimension one, and he then used fact (1) to show the same in dimension two. Abhyankar then followed suit by proving that in dimension two, any point (=valuation) in this space is attainable by blowing ups, but his student Shannon showed that this failed in dimension three and higher. This latter negative result somehow ended the Zariski project of using the space of all valuations to prove resolution of singularities in higher dimensions.
However, there is a resurgence of this space and its use in resolution of singularities during the last decades, and often, these results are combined with model-theoretic techniques (Kuhlmann, Knaf, Pop, Cutkosky, Cossart, Piltant, Teissier, Scanlon,…). However, the model-theoretic setting always departs from Robinson’s point of view of a valued field: a field together with a valuation. However, the Zariski-Riemann space talks about not just one valuation, but all, so that we need a new framework. I will present some preliminary remarks of how this could be done using either a simple-minded one-sorted language or a more sophisticated two-sorted language. As a simple application of the one-sorted case, I will reprove fact (1) by simply relating it to the compactness of the Stone space of types.