{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $k$ be a separably closed field and let $X$ be an [[algebraic variety|HV6YBS0YDH]] over $k$.\nLet $\\ell$ be a prime number different from the characteristic of $k$.\nFor $i \\ge 0$, the *$\\ell$-adic cohomology groups* of $X$ are\n$$H^i(X, \\Z_\\ell) = \\varprojlim_n H^i_{\\text{ét}}(X, \\Z/\\ell^n\\Z), \\qquad H^i(X, \\Q_\\ell) = H^i(X, \\Z_\\ell) \\otimes_{\\Z_\\ell} \\Q_\\ell.$$\nHere $H^i_{\\text{ét}}$ is [[étale cohomology|7YX76G0838]].\nThe [[inverse limit|DZB378CNCK]] is taken along the maps induced by the reductions $\\Z/\\ell^{n+1}\\Z \\to \\Z/\\ell^n\\Z$.\n$\\Z_\\ell$ is the ring of [[$\\ell$-adic integers|GBN9F8FAQ3]], and $\\Q_\\ell$ is the field of [[$\\ell$-adic numbers|CHT5JYM6QW]].\n\n## Remarks\n\nExample: let $k$ be algebraically closed and $X = \\mathbb{P}^1_k$.\nThen $H^0_{\\text{ét}}(X, \\Z/\\ell^n\\Z) = \\Z/\\ell^n\\Z$, $H^1_{\\text{ét}} = 0$ and $H^2_{\\text{ét}} \\cong \\Z/\\ell^n\\Z$, compatibly in $n$.\nSo $H^i(X, \\Q_\\ell)$ is $\\Q_\\ell$ for $i = 0, 2$ and $0$ otherwise, as for the sphere $\\mathbb{P}^1(\\C)$.\nNon-example: the condition $\\ell \\ne \\operatorname{char} k$ is needed.\nLet $k$ have characteristic $p \u003e 0$ and let $E$ be an elliptic curve over $k$.\nThen $H^1(E, \\Q_p)$ has dimension $1$ if $E$ is ordinary and $0$ if $E$ is supersingular, not the expected dimension $2$.\nFor $X$ proper over $k$, the groups $H^i(X, \\Q_\\ell)$ are finite-dimensional and vanish for $i \u003e 2 \\dim X$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "E1DVHN06CR",
    "kind": "definition",
    "msc": [
      "14F20",
      "14G15"
    ],
    "name": "ℓ-adic cohomology",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/%C3%89tale_cohomology"
      }
    ],
    "tag": "TGKPC4GV7R",
    "uses": [
      "7YX76G0838",
      "HV6YBS0YDH",
      "GBN9F8FAQ3",
      "CHT5JYM6QW",
      "DZB378CNCK"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/TGKPC4GV7R/",
  "links": [
    {
      "field": "defines",
      "target": "E1DVHN06CR"
    },
    {
      "field": "uses",
      "target": "7YX76G0838"
    },
    {
      "field": "uses",
      "target": "HV6YBS0YDH"
    },
    {
      "field": "uses",
      "target": "GBN9F8FAQ3"
    },
    {
      "field": "uses",
      "target": "CHT5JYM6QW"
    },
    {
      "field": "uses",
      "target": "DZB378CNCK"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/TGKPC4GV7R.md",
  "msc": [
    "14F20",
    "14G15"
  ],
  "name": "ℓ-adic cohomology",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/TGK/TGKPC4GV7R.md",
  "status": null,
  "tag": "TGKPC4GV7R",
  "uri": "m2://TGKPC4GV7R",
  "url": "https://tamnd.github.io/math2/wiki/adic-cohomology-definition/",
  "writing": {
    "tag": "TGKPC4GV7R",
    "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
      }
    ]
  }
}