{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $A$ be a [[commutative ring|0310REJJY8]] with identity, and let $X = \\operatorname{Spec} A$ be its [[spectrum|FB45BHHA2J]] with the [[Zariski topology|8SQK7RTB63]].\nFor $f \\in A$, write $D(f) = \\set{\\mathfrak{p} \\in X : f \\notin \\mathfrak{p}}$; these sets form a basis of open sets of $X$.\nLet $A_f$ be the [[localization|4C1AE84N5N]] of $A$ at the powers of $f$.\nThe *structure sheaf* $\\mathcal{O}_X$ is the unique [[sheaf|TB6TM6R4MD]] of rings on $X$ with $\\mathcal{O}_X(D(f)) = A_f$ for every $f \\in A$.\nIts restriction maps $A_f \\to A_g$ for $D(g) \\subseteq D(f)$ are the natural maps.\nIts stalk at $\\mathfrak{p}$ is the [[localization at the prime ideal|X6JD4AG973]] $A_{\\mathfrak p}$, which is a local ring.\n\nAn *affine scheme* is a [[locally ringed space|B7FPT8JTXR]] isomorphic to $(\\operatorname{Spec} A, \\mathcal{O}_{\\operatorname{Spec} A})$ for some commutative ring $A$.\n\n## Remarks\n\nExample: $\\operatorname{Spec} \\Z$ is an affine scheme; its points are $(0)$ and $(p)$ for the primes $p$, and $\\mathcal{O}(D(n)) = \\Z[1/n]$ for $n \\ne 0$.\nExample: the affine $n$-space over a ring $R$ is $\\mathbb{A}^n_R = \\operatorname{Spec} R[x_1, \\ldots, x_n]$.\nNon-example: for a field $k$, the open subset $U = \\mathbb{A}^2_k \\setminus \\set{(x, y)}$ of the plane, with the restricted sheaf, is a locally ringed space but not an affine scheme.\nIts ring of global functions is $k[x, y]$, so if it were affine it would be $\\operatorname{Spec} k[x, y]$, which also contains the origin $(x, y)$.\nThe ring is recovered as $A \\cong \\mathcal{O}_X(X)$, the global sections.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "B3V3XJ2W8G",
    "kind": "definition",
    "msc": [
      "14A15"
    ],
    "name": "Affine scheme",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Spectrum_of_a_ring"
      }
    ],
    "tag": "CTBYV71CEJ",
    "uses": [
      "B7FPT8JTXR",
      "FB45BHHA2J",
      "0310REJJY8",
      "8SQK7RTB63",
      "4C1AE84N5N",
      "X6JD4AG973",
      "TB6TM6R4MD"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/CTBYV71CEJ/",
  "links": [
    {
      "field": "defines",
      "target": "B3V3XJ2W8G"
    },
    {
      "field": "uses",
      "target": "B7FPT8JTXR"
    },
    {
      "field": "uses",
      "target": "FB45BHHA2J"
    },
    {
      "field": "uses",
      "target": "0310REJJY8"
    },
    {
      "field": "uses",
      "target": "8SQK7RTB63"
    },
    {
      "field": "uses",
      "target": "4C1AE84N5N"
    },
    {
      "field": "uses",
      "target": "X6JD4AG973"
    },
    {
      "field": "uses",
      "target": "TB6TM6R4MD"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/CTBYV71CEJ.md",
  "msc": [
    "14A15"
  ],
  "name": "Affine scheme",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/CTB/CTBYV71CEJ.md",
  "status": null,
  "tag": "CTBYV71CEJ",
  "uri": "m2://CTBYV71CEJ",
  "url": "https://tamnd.github.io/math2/wiki/affine-scheme-definition/",
  "writing": {
    "tag": "CTBYV71CEJ",
    "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
      }
    ]
  }
}