{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]] and let $f \\in R$.\nThe *basic open set* (or *principal open set*) of $f$ is the subset\n$$D(f) = \\set{\\mathfrak p \\in \\operatorname{Spec} R : f \\notin \\mathfrak p}$$\nof the [[spectrum|FB45BHHA2J]] of $R$.\nIt is the complement of the closed set $V(\\set{f})$ of the [[Zariski topology|8SQK7RTB63]], hence open.\n\n## Remarks\n\nThe basic open sets have these properties, for $f, g \\in R$:\n\n1. $D(f) \\cap D(g) = D(fg)$, since a [[prime ideal|00MHHZ9SNE]] contains $fg$ exactly when it contains $f$ or $g$;\n2. each open set $X \\setminus V(E)$ equals $\\bigcup_{f \\in E} D(f)$, so the sets $D(f)$ form a [[basis|0NA0732YAG]] of the topology;\n3. $D(f) = \\emptyset$ exactly when $f$ is [[nilpotent|0ZB2JG1M9C]], and $D(f) = \\operatorname{Spec} R$ exactly when $f$ is a [[unit|0XVEZDE9J8]];\n4. the map $\\operatorname{Spec} R_f \\to \\operatorname{Spec} R$ induced by $R \\to R_f$ is a homeomorphism onto $D(f)$; see [[prime ideals of a localization|W8QNZZ4QJS]].\n\nExample: in $\\operatorname{Spec} \\Z$, the set $D(6)$ consists of $(0)$ and all $p\\Z$ with $p \\ne 2, 3$.\n\nNon-example: not every open set is basic.\nLet $k$ be a field and $R = k[x, y]$.\nThe set $U = \\operatorname{Spec} R \\setminus \\set{(x, y)}$ is open, because $\\set{(x,y)} = V(\\set{x, y})$.\nIf $U = D(f)$, then $f \\ne 0$ and $f$ is not a unit, since $U$ is neither empty nor everything.\nThen an irreducible factor $g$ of $f$ generates a prime ideal $(g)$ that contains $f$, and $(g) \\ne (x, y)$ because $(x,y)$ is not principal.\nSo $(g) \\notin D(f)$ but $(g) \\in U$, a contradiction.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "N6JX9BCKQP",
    "kind": "definition",
    "msc": [
      "14A05",
      "13A15"
    ],
    "name": "Basic open set",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Spectrum_of_a_ring"
      }
    ],
    "tag": "PS8DVS9WTA",
    "uses": [
      "FB45BHHA2J",
      "8SQK7RTB63",
      "00MHHZ9SNE",
      "0NA0732YAG"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/PS8DVS9WTA/",
  "links": [
    {
      "field": "defines",
      "target": "N6JX9BCKQP"
    },
    {
      "field": "uses",
      "target": "FB45BHHA2J"
    },
    {
      "field": "uses",
      "target": "8SQK7RTB63"
    },
    {
      "field": "uses",
      "target": "00MHHZ9SNE"
    },
    {
      "field": "uses",
      "target": "0NA0732YAG"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/PS8DVS9WTA.md",
  "msc": [
    "14A05",
    "13A15"
  ],
  "name": "Basic open set",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/PS8/PS8DVS9WTA.md",
  "status": null,
  "tag": "PS8DVS9WTA",
  "uri": "m2://PS8DVS9WTA",
  "url": "https://math2.tamnd.com/wiki/basic-open-set-definition/",
  "writing": {
    "tag": "PS8DVS9WTA",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}