A few weeks back in the Berkeley model theory seminar, Will Johnson gave a talk on elimination of imaginaries in algebraically closed valued fields. In case you missed the talk and aren’t familiar with the subject, I will make a few comments (and give a few links) before linking to Will’s article.

You probably know about *elimination of imaginaries *if you are a model theorist. If not, let me give a very short explanation. Suppose that you have a structure with at least two -definable elements (I am writing to mean definable without parameters). If you are a non-model theorist, you are probably already interested in structures which have at least two -definable elements, since fields in the language of rings are an example. Now suppose that you have a -definable equivalence relation on . Then has *elimination of imaginaries* if there is an -definable function for some such that . Again, since you are reading this paragraph, you are likely a non-model theorist, so it might not be clear why this is useful. The most obvious answer is that this allows one to view naturally occurring quotients which your structure interprets as definable sets (in the above formulation, this is the image of the function ).

Let be an algebraically closed field with a nontrivial valuation. Let denote the valuation ring and the maximal ideal. denotes the residue field. Here is the basic problem: in ACVF there are certain naturally interpretable sets which can not be viewed as definable sets (if you are a model theorist or you just read the previous paragraph, then you know what I mean). The first example of this situation in ACVF consists of the which are isomorphic to . Such objects are called lattices. denotes the set of lattices in . This set can be identified with . One also has to add sorts which consists of .

The main theorem which Will talked about is due to Haskell, Hrushovski and McPhereson who write that they first got the idea for their approach from Holly. The theorem says that ACVF with additional sorts and eliminates imaginaries. The theorem has a certain tightness as well; ACVF does not eliminate imaginaries in any finite set of sorts. Will has produced a shortened and perhaps more conceptual version of the proof of the theorem.