{
  "id": "0809.1305",
  "version": "v3",
  "published": "2008-09-08T10:42:53.000Z",
  "updated": "2009-09-05T08:16:05.000Z",
  "title": "Semistar-Krull and Valuative Dimension of Integral Domains",
  "authors": [
    "Parviz Sahandi"
  ],
  "comment": "Final version: Remark 2.2 change to Ptoposition 2.2 and added Example 4.4",
  "categories": [
    "math.AC"
  ],
  "abstract": "Given a stable semistar operation of finite type $\\star$ on an integral domain $D$, we show that it is possible to define in a canonical way a stable semistar operation of finite type $\\star[X]$ on the polynomial ring $D[X]$, such that, if $n:=\\star$-$\\dim(D)$, then $n+1\\leq \\star[X]\\text{-}\\dim(D[X])\\leq 2n+1$. We also establish that if $D$ is a $\\star$-Noetherian domain or is a Pr\\\"{u}fer $\\star$-multiplication domain, then $\\star[X]\\text{-}\\dim(D[X])=\\star\\text{-}\\dim(D)+1$. Moreover we define the semistar valuative dimension of the domain $D$, denoted by $\\star$-$\\dim_v(D)$, to be the maximal rank of the $\\star$-valuation overrings of $D$. We show that $\\star$-$\\dim_v(D)=n$ if and only if $\\star[X_1,...,X_n]$-$\\dim_v(D[X_1,...,X_n])=2n$, and that if $\\star$-$\\dim_v(D)<\\infty$ then $\\star[X]$-$\\dim_v(D[X])=\\star$-$\\dim_v(D)+1$. In general $\\star$-$\\dim(D)\\leq\\star$-$\\dim_v(D)$ and equality holds if $D$ is a $\\star$-Noetherian domain or is a Pr\\\"{u}fer $\\star$-multiplication domain. We define the $\\star$-Jaffard domains as domains $D$ such that $\\star$-$\\dim(D)<\\infty$ and $\\star$-$\\dim(D)=\\star$-$\\dim_v(D)$. As an application, $\\star$-quasi-Pr\\\"{u}fer domains are characterized as domains $D$ such that each $(\\star,\\star')$-linked overring $T$ of $D$, is a $\\star'$-Jaffard domain, where $\\star'$ is a stable semistar operation of finite type on $T$. As a consequence of this result we obtain that a Krull domain $D$, must be a $w_D$-Jaffard domain.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2009-09-05T08:16:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13G05",
      "13A15",
      "13C15",
      "13M10"
    ],
    "keywords": [
      "integral domain",
      "stable semistar operation",
      "jaffard domain",
      "finite type",
      "semistar-krull"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0809.1305S"
    }
  }
}