{
  "id": "1601.07146",
  "version": "v1",
  "published": "2016-01-26T19:55:16.000Z",
  "updated": "2016-01-26T19:55:16.000Z",
  "title": "Bases of T-equivariant cohomology of Bott-Samelson varieties",
  "authors": [
    "Vladimir Shchigolev"
  ],
  "categories": [
    "math.RT",
    "math.AG"
  ],
  "abstract": "We construct combinatorial bases of the $T$-equivariant ($T$ is the maximal torus) cohomology $H^\\bullet_T(\\Sigma,k)$ of the Bott-Samelson variety $\\Sigma$ under some mild restrictions on the field of coefficients $k$. This bases allow us to prove the surjectivity of the restrictions $H^\\bullet_T(\\Sigma,k)\\to H^\\bullet_T(\\pi^{-1}(x),k)$ and $H^\\bullet_T(\\Sigma,k)\\to H^\\bullet_T(\\Sigma\\setminus\\pi^{-1}(x),k)$, where $\\pi:\\Sigma\\to G/B$ is the canonical resolution. In fact, we also construct bases of the targets of these restrictions by picking up certain subsets of certain bases of $H^\\bullet_T(\\Sigma,k)$ and restricting them to $\\pi^{-1}(x)$ or $\\Sigma\\setminus\\pi^{-1}(x)$ respectively. As an application, we calculate the cohomology of the costalk-to-stalk embedding for the direct image $\\pi_*{\\underline k}_\\Sigma$. This algorithm avoids division by 2, which allows us to reestablish 2-torsion for parity sheaves in Braden's example.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-26T19:55:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bott-samelson variety",
      "t-equivariant cohomology",
      "construct combinatorial bases",
      "algorithm avoids division",
      "mild restrictions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}