{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $M$ be a [[smooth manifold|F45VQ47GPV]], and let $E_0, E_1, \\ldots, E_N$ be smooth [[vector bundles|P4DZM975BV]] over $M$.\nFor $0 \\le k \u003c N$, let $P_k \\colon \\Gamma(E_k) \\to \\Gamma(E_{k+1})$ be a [[differential operator|CGBX1N9V8H]] of order $m_k$, with [[principal symbol|S8GC34D5ZG]] $\\sigma(P_k) = \\sigma_{m_k}(P_k)$.\nThe sequence\n$$0 \\to \\Gamma(E_0) \\xrightarrow{P_0} \\Gamma(E_1) \\xrightarrow{P_1} \\cdots \\xrightarrow{P_{N-1}} \\Gamma(E_N) \\to 0$$\nis an *elliptic complex* if both of the following hold.\n\n1. $P_{k+1} \\circ P_k = 0$ for $0 \\le k \u003c N - 1$.\n2. For every $x \\in M$ and every nonzero $\\xi \\in T^*_x M$, the sequence of linear maps\n$$0 \\to E_{0,x} \\xrightarrow{\\sigma(P_0)(x, \\xi)} E_{1,x} \\to \\cdots \\xrightarrow{\\sigma(P_{N-1})(x, \\xi)} E_{N,x} \\to 0$$\nis exact.\n\n## Remarks\n\nFor $N = 1$ the complex is a single operator $P_0$, and it is elliptic exactly when $P_0$ is an [[elliptic operator|MT9P2X85V8]].\nWhen $M$ is compact without boundary, the cohomology groups $\\ker P_k / \\operatorname{im} P_{k-1}$ are finite-dimensional, and the index of the complex is $\\sum_k (-1)^k \\dim(\\ker P_k / \\operatorname{im} P_{k-1})$.\nSuppose that all $P_k$ have the same order and that $M$ and the bundles carry metrics, and write $P = \\sum_k P_k$ and $P^* = \\sum_k P_k^*$ for the formal adjoint.\nThen $P + P^* \\colon \\Gamma(\\bigoplus_k E_{2k}) \\to \\Gamma(\\bigoplus_k E_{2k+1})$ is elliptic and has the same index as the complex.\n\nExample: the de Rham complex $0 \\to \\Omega^0(M) \\xrightarrow{d} \\Omega^1(M) \\xrightarrow{d} \\cdots \\xrightarrow{d} \\Omega^n(M) \\to 0$ of a manifold of dimension $n$, with the [[exterior derivative|0ZHSTAG8ZY]].\nIt satisfies $d \\circ d = 0$, and the symbol of $d$ at $\\xi$ is $\\omega \\mapsto \\xi \\wedge \\omega$.\nFor $\\xi \\ne 0$, complete $\\xi$ to a basis $\\xi = e_1, e_2, \\ldots, e_n$ of $T^*_x M$.\nA form $\\omega$ with $\\xi \\wedge \\omega = 0$ is then a sum of terms containing $e_1$.\nSo it equals $\\xi \\wedge \\eta$ for some $\\eta$, which proves exactness.\nIts cohomology is the [[de Rham cohomology|GX74C07T0G]], and its index is the [[Euler characteristic|RNDDWDAFJ5]] of $M$.\n\nNon-example: on a manifold of dimension $n \\ge 2$, the two-term complex $0 \\to \\Omega^0(M) \\xrightarrow{d} \\Omega^1(M) \\to 0$ satisfies clause 1 but not clause 2.\nThe symbol sequence $0 \\to \\R \\to T^*_x M \\to 0$ is not exact at $T^*_x M$, because the image of $c \\mapsto c \\, \\xi$ is a line in a space of dimension $n$.\n\n## Explanations\n\n### L1 human\n\nAn elliptic complex is a chain of differential operators that compose to zero and that, at top order in each nonzero direction, fit together exactly.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "M8GMH8GQR6",
    "kind": "definition",
    "msc": [
      "58J10",
      "58J20"
    ],
    "name": "Elliptic complex",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Elliptic_complex"
      }
    ],
    "tag": "8722T9KCP5",
    "uses": [
      "CGBX1N9V8H",
      "S8GC34D5ZG",
      "MT9P2X85V8",
      "P4DZM975BV",
      "F45VQ47GPV",
      "0ZHSTAG8ZY",
      "GX74C07T0G",
      "RNDDWDAFJ5"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/8722T9KCP5/",
  "links": [
    {
      "field": "defines",
      "target": "M8GMH8GQR6"
    },
    {
      "field": "uses",
      "target": "CGBX1N9V8H"
    },
    {
      "field": "uses",
      "target": "S8GC34D5ZG"
    },
    {
      "field": "uses",
      "target": "MT9P2X85V8"
    },
    {
      "field": "uses",
      "target": "P4DZM975BV"
    },
    {
      "field": "uses",
      "target": "F45VQ47GPV"
    },
    {
      "field": "uses",
      "target": "0ZHSTAG8ZY"
    },
    {
      "field": "uses",
      "target": "GX74C07T0G"
    },
    {
      "field": "uses",
      "target": "RNDDWDAFJ5"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/8722T9KCP5.md",
  "msc": [
    "58J10",
    "58J20"
  ],
  "name": "Elliptic complex",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/872/8722T9KCP5.md",
  "status": null,
  "tag": "8722T9KCP5",
  "uri": "m2://8722T9KCP5",
  "url": "https://tamnd.github.io/math2/wiki/elliptic-complex-definition/",
  "writing": {
    "tag": "8722T9KCP5",
    "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
      }
    ]
  }
}