{
  "id": "1604.07743",
  "version": "v1",
  "published": "2016-04-26T16:27:30.000Z",
  "updated": "2016-04-26T16:27:30.000Z",
  "title": "Saturation and solvability in abstract elementary classes with amalgamation",
  "authors": [
    "Sebastien Vasey"
  ],
  "comment": "23 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "$\\mathbf{Theorem.}$ Let $K$ be an abstract elementary class (AEC) with amalgamation and no maximal models. Let $\\lambda > \\text{LS} (K)$. If $K$ is categorical in $\\lambda$, then the model of cardinality $\\lambda$ is Galois-saturated. This answers a question asked independently by Baldwin and Shelah. We deduce several corollaries: $K$ has a unique limit model in each cardinal below $\\lambda$, (when $\\lambda$ is big-enough) $K$ is weakly tame below $\\lambda$, and the thresholds of several existing categoricity transfers can be improved. We also prove a downward transfer of solvability (a version of superstability introduced by Shelah): $\\mathbf{Corollary.}$ Let $K$ be an AEC with amalgamation and no maximal models. Let $\\lambda > \\mu > \\text{LS} (K)$. If $K$ is solvable in $\\lambda$, then $K$ is solvable in $\\mu$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-26T16:27:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C48",
      "03C45",
      "03C52",
      "03C55"
    ],
    "keywords": [
      "abstract elementary class",
      "amalgamation",
      "solvability",
      "maximal models",
      "saturation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}