{
  "id": "1603.05863",
  "version": "v1",
  "published": "2016-03-18T13:01:13.000Z",
  "updated": "2016-03-18T13:01:13.000Z",
  "title": "Duality and contravariant functors in the representation theory of artin algebras",
  "authors": [
    "Samuel Dean"
  ],
  "categories": [
    "math.RT",
    "math.CT",
    "math.LO"
  ],
  "abstract": "The model theory of modules leads to a way of obtaining definable categories of modules over a ring $R$ as the kernels of certain functors $(R\\textbf{-Mod})^{\\text{op}}\\to\\textbf{Ab}$ rather than of functors $R\\textbf{-Mod}\\to\\textbf{Ab}$ which are given by a pp pair. This paper will give various algebraic characterisations of these functors in the case that $R$ is an artin algebra. Suppose that $R$ is an artin algebra. An additive functor $G:(R\\textbf{-Mod})^{\\text{op}}\\to\\textbf{Ab}$ preserves inverses limits and $G|_{(R\\textbf{-mod})^{\\text{op}}}:(R\\textbf{-mod})^{\\text{op}}\\to\\textbf{Ab}$ is finitely presented if and only if there is a sequence of natural transformations $(-,A)\\to(-,B)\\to G\\to 0$ for some $A,B\\in R\\textbf{-mod}$ which is exact when evaluated at any left $R$-module. Any additive functor $(R\\textbf{-Mod})^{\\text{op}}\\to\\textbf{Ab}$ with one of these equivalent properties has a definable kernel, and every definable subcategory of $R\\textbf{-Mod}$ can be obtained as the kernel of a family of such functors. In the final section a generalised setting is introduced, so that our results apply to more categories than those of the form $R\\textbf{-Mod}$ for an artin algebra $R$. That is, our results are extended to those locally finitely presented $K$-linear categories whose finitely presented objects form a dualising $K$-variety, where $K$ is a commutative artinian ring.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-03-18T13:01:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "18A25"
    ],
    "keywords": [
      "artin algebra",
      "contravariant functors",
      "representation theory",
      "additive functor",
      "categories"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160305863D"
    }
  }
}