{
  "id": "math/0511310",
  "version": "v2",
  "published": "2005-11-11T19:35:01.000Z",
  "updated": "2006-11-20T14:01:37.000Z",
  "title": "Stable domination and independence in algebraically closed valued fields",
  "authors": [
    "Deirdre Haskell",
    "Ehud Hrushovski",
    "Dugald Macpherson"
  ],
  "categories": [
    "math.LO",
    "math.AG"
  ],
  "abstract": "We seek to create tools for a model-theoretic analysis of types in algebraically closed valued fields (ACVF). We give evidence to show that a notion of 'domination by stable part' plays a key role. In Part A, we develop a general theory of stably dominated types, showing they enjoy an excellent independence theory, as well as a theory of definable types and germs of definable functions. In Part B, we show that the general theory applies to ACVF. Over a sufficiently rich base, we show that every type is stably dominated over its image in the value group. For invariant types over any base, stable domination coincides with a natural notion of `orthogonality to the value group'. We also investigate other notions of independence, and show that they all agree, and are well-behaved, for stably dominated types. One of these is used to show that every type extends to an invariant type; definable types are dense. Much of this work requires the use of imaginary elements. We also show existence of prime models over reasonable bases, possibly including imaginaries.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-11-20T14:01:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "12J10",
      "03C45",
      "03C60"
    ],
    "keywords": [
      "algebraically closed valued fields",
      "stable domination",
      "invariant type",
      "stably dominated types",
      "value group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....11310H"
    }
  }
}