{
  "id": "2008.01757",
  "version": "v1",
  "published": "2020-08-04T18:20:53.000Z",
  "updated": "2020-08-04T18:20:53.000Z",
  "title": "Functorial properties of pro-$p$-Iwahori cohomology",
  "authors": [
    "Karol Koziol"
  ],
  "comment": "32 pages. Comments welcome!",
  "categories": [
    "math.RT",
    "math.NT"
  ],
  "abstract": "Suppose $F$ is a finite extension of $\\mathbb{Q}_p$, $G$ is the group of $F$-points of a connected reductive $F$-group, and $I_1$ is a pro-$p$-Iwahori subgroup of $G$. We construct two spectral sequences relating derived functors on mod-$p$ representations of $G$ to the analogous functors on Hecke modules coming from pro-$p$-Iwahori cohomology. More specifically: (1) using results of Ollivier--Vign\\'eras, we provide a link between the right adjoint of parabolic induction on pro-$p$-Iwahori cohomology and Emerton's functors of derived ordinary parts; and (2) we establish a \"Poincar\\'e duality spectral sequence\" relating duality on pro-$p$-Iwahori cohomology to Kohlhaase's functors of higher smooth duals. As applications, we calculate various examples of the Hecke modules $\\textrm{H}^i(I_1,\\pi)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-04T18:20:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "iwahori cohomology",
      "functorial properties",
      "poincare duality spectral sequence",
      "hecke modules",
      "spectral sequences relating derived functors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}