{
  "id": "1902.09311",
  "version": "v1",
  "published": "2019-02-18T07:24:53.000Z",
  "updated": "2019-02-18T07:24:53.000Z",
  "title": "On the Surjectivity of Certain Maps III: For Symplectic Groups Over Rings and Generalized Projective Spaces",
  "authors": [
    "C. P. Anil Kumar"
  ],
  "comment": "36 pages",
  "categories": [
    "math.NT",
    "math.AC",
    "math.AG"
  ],
  "abstract": "We prove three main theorems on the surjectivity of certain maps for symplectic groups over commutative rings with unity in two different contexts. In the first context, we prove in Theorem $\\Lambda$, the surjectivity of the reduction map of strong approximation type for a ring quotiented by an ideal which satisfies unital set condition in the case of symplectic groups. In the second context of the surjectivity of the map from $(2k\\times 2k)$-order symplectic group over a ring to the product of generalized projective spaces of $2k$-mutually co-maximal ideals associating the $2k$-rows or $2k$-columns, we prove the remaining two main Theorems $[\\Omega,\\Sigma]$, under certain conditions, either on the ring or on the generalized projective spaces. Finally in the second context, we give counter examples where, the surjectivity fails for $(p,q)$-indefinite orthogonal groups over integers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-18T07:24:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13A15",
      "51N30",
      "11D79",
      "11B25",
      "16U60"
    ],
    "keywords": [
      "generalized projective spaces",
      "surjectivity",
      "main theorems",
      "second context",
      "satisfies unital set condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}