{
  "id": "math/0308036",
  "version": "v1",
  "published": "2003-08-05T13:24:31.000Z",
  "updated": "2003-08-05T13:24:31.000Z",
  "title": "Representability of Hom implies flatness",
  "authors": [
    "Nitin Nitsure"
  ],
  "comment": "9 pages, LaTeX",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $X$ be a projective scheme over a noetherian base scheme $S$, and let $F$ be a coherent sheaf on $X$. For any coherent sheaf $E$ on $X$, consider the set-valued contravariant functor $Hom_{E,F}$ on $S$-schemes, defined by $Hom_{E,F}(T) = Hom(E_T,F_T)$ where $E_T$ and $F_T$ are the pull-backs of $E$ and $F$ to $X_T = X\\times_S T$. A basic result of Grothendieck ([EGA] III 7.7.8, 7.7.9) says that if $F$ is flat over $S$ then $Hom_{E,F}$ is representable for all $E$. We prove the converse of the above, in fact, we show that if $L$ is a relatively ample line bundle on $X$ over $S$ such that the functor $Hom_{L^{-n},F}$ is representable for infinitely many positive integers $n$, then $F$ is flat over $S$. As a corollary, taking $X=S$, it follows that if $F$ is a coherent sheaf on $S$ then the functor $T\\mapsto H^0(T, F_T)$ on the category of $S$-schemes is representable if and only if $F$ is locally free on $S$. This answers a question posed by Angelo Vistoli. The techniques we use involve the proof of flattening stratification, together with the methods used in proving the author's earlier result (see arXiv.org/abs/math.AG/0204047) that the automorphism group functor of a coherent sheaf on $S$ is representable if and only if the sheaf is locally free.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-08-05T13:24:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14A15",
      "14F05",
      "14L15"
    ],
    "keywords": [
      "hom implies flatness",
      "coherent sheaf",
      "representability",
      "noetherian base scheme",
      "authors earlier result"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......8036N"
    }
  }
}