{
  "id": "1707.06536",
  "version": "v1",
  "published": "2017-07-19T16:56:26.000Z",
  "updated": "2017-07-19T16:56:26.000Z",
  "title": "Some results on the annihilators and attached primes of local cohomology modules",
  "authors": [
    "Ali Atazadeh",
    "Monireh Sedghi",
    "Reza Naghipour"
  ],
  "comment": "13 pages, to appear in Archiv der Mathematik. arXiv admin note: substantial text overlap with arXiv:1312.1253",
  "categories": [
    "math.AC"
  ],
  "abstract": "Let $(R, \\frak m)$ be a local ring and $M$ a finitely generated $R$-module. It is shown that if $M$ is relative Cohen-Macaulay with respect to an ideal $\\frak a$ of $R$, then $\\text{Ann}_R(H_{\\mathfrak{a}}^{\\text{cd}(\\mathfrak{a}, M)}(M))=\\text{Ann}_RM/L=\\text{Ann}_RM$ and $\\text{Ass}_R(R/\\text{Ann}_RM)\\subseteq \\{\\mathfrak{p} \\in \\text{Ass}_R M|\\,{\\rm cd}(\\mathfrak{a}, R/\\mathfrak{p})=\\text{cd}(\\mathfrak{a}, M)\\},$ where $L$ is the largest submodule of $M$ such that ${\\rm cd}(\\mathfrak{a}, L)< {\\rm cd}(\\mathfrak{a}, M)$. We also show that if $H^{\\dim M}_{\\mathfrak{a}}(M)=0$, then $\\text{Att}_R(H^{\\dim M-1}_{\\mathfrak{a}}(M))= \\{\\mathfrak{p} \\in \\text{Supp} (M)|\\,{\\rm cd}(\\mathfrak{a}, R/\\mathfrak{p})=\\dim M-1\\},$ and so the attached primes of $H^{\\dim M-1}_{\\mathfrak{a}}(M)$ depends only on $\\text{Supp} (M)$. Finally, we prove that if $M$ is an arbitrary module (not necessarily finitely generated) over a Noetherian ring $R$ with ${\\rm cd}(\\mathfrak{a}, M)={\\rm cd}(\\mathfrak{a}, R/\\text{Ann}_RM)$, then $\\text{Att}_R(H^{{\\rm cd}(\\mathfrak{a}, M)}_{\\mathfrak{a}}(M))\\subseteq\\{\\mathfrak{p} \\in V(\\text{Ann}_RM)|\\,{\\rm cd}(\\mathfrak{a}, R/\\mathfrak{p})={\\rm cd}(\\mathfrak{a}, M)\\}.$ As a consequence of this it is shown that if $\\dim M=\\dim R$, then $\\text{Att}_R(H^{\\dim M}_{\\mathfrak{a}}(M))\\subseteq\\{\\mathfrak{p} \\in \\text{Ass}_R M|\\,{\\rm cd}(\\mathfrak{a}, R/\\mathfrak{p})=\\dim M\\}.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-19T16:56:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13D45",
      "14B15",
      "13E05"
    ],
    "keywords": [
      "local cohomology modules",
      "attached primes",
      "annihilators",
      "largest submodule",
      "arbitrary module"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}