{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "ES08X651A3"
    }
  ],
  "body": "\n## Definition\n\nLet $a \\in \\C^n$.\nConsider pairs $(U, A)$ where $U$ is an open [[neighbourhood|0ZCY0Z80B6]] of $a$ in $\\C^n$ and $A$ is an [[analytic subset|VMDTGTMN9P]] of $U$.\nTwo pairs $(U, A)$ and $(V, B)$ are equivalent if there is an open neighbourhood $W \\subseteq U \\cap V$ of $a$ with $A \\cap W = B \\cap W$.\nA *germ of an analytic set* at $a$ is an equivalence class for this relation; the class of $(U, A)$ is written $A_a$.\n\n## Remarks\n\nThis is the notion of [[germ|Y98MD3WVTG]] applied to analytic sets.\nInclusion, finite union and finite intersection of germs are defined through representatives on a common neighbourhood.\nTo a germ $A_a$ one attaches the ideal $\\mathcal{I}(A_a)$ of all [[holomorphic germs|2TGPZ8VW2D]] $f_a \\in \\mathcal{O}_{n,a}$ that have a representative vanishing on a representative of $A_a$; it is a radical ideal.\nConversely, an ideal $I \\subseteq \\mathcal{O}_{n,a}$ with generators $f_{1,a}, \\dots, f_{k,a}$ defines the germ $V(I)$ of the common zero set of representatives $f_1, \\dots, f_k$.\nExample: the sets $\\set{z_1 z_2 = 0}$ and $\\set{z_1 z_2 (z_1 - 1) = 0}$ in $\\C^2$ have the same germ at $0$, since they agree on the ball of radius $1$ about $0$.\nNon-example: the sets $\\set{z_1 = 0}$ and $\\set{z_1 = z_2^2}$ have different germs at $0$.\nIndeed, every neighbourhood of $0$ contains a point $(0, t)$ with $t \\ne 0$, which lies in the first set but not the second.\nThe germ of $A$ at a point $a \\notin A$ is the empty germ.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "Q9B74BYZ8C",
    "kind": "definition",
    "msc": [
      "32B10",
      "32B05"
    ],
    "name": "Germ of an analytic set",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Complex_analytic_variety"
      }
    ],
    "tag": "A2WHCVP6JN",
    "uses": [
      "VMDTGTMN9P",
      "Y98MD3WVTG",
      "0ZCY0Z80B6",
      "2TGPZ8VW2D"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/A2WHCVP6JN/",
  "links": [
    {
      "field": "defines",
      "target": "Q9B74BYZ8C"
    },
    {
      "field": "uses",
      "target": "VMDTGTMN9P"
    },
    {
      "field": "uses",
      "target": "Y98MD3WVTG"
    },
    {
      "field": "uses",
      "target": "0ZCY0Z80B6"
    },
    {
      "field": "uses",
      "target": "2TGPZ8VW2D"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/A2WHCVP6JN.md",
  "msc": [
    "32B10",
    "32B05"
  ],
  "name": "Germ of an analytic set",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/A2W/A2WHCVP6JN.md",
  "status": null,
  "tag": "A2WHCVP6JN",
  "uri": "m2://A2WHCVP6JN",
  "url": "https://tamnd.github.io/math2/wiki/germ-of-an-analytic-set-definition/",
  "writing": {
    "tag": "A2WHCVP6JN",
    "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
      }
    ]
  }
}