{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "F0M6BY5BH7"
    },
    {
      "field": "uses",
      "target": "VFPP3RZGK3"
    }
  ],
  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]].\nConsider a finite or infinite sequence of [[$R$-modules|0NZ8JA25WK]] and $R$-[[linear maps|0NRYHJPQG2]]\n$$\\cdots \\to M_{i-1} \\xrightarrow{f_{i-1}} M_i \\xrightarrow{f_i} M_{i+1} \\to \\cdots.$$\nThe sequence is *exact at* $M_i$ if $\\im f_{i-1} = \\ker f_i$, where $\\im$ is the [[image|0S21GNW9ZB]] and $\\ker$ the [[kernel|0W88CEGQP1]].\nThe sequence is *exact* if it is exact at every module that has both an incoming and an outgoing map.\nA *short exact sequence* is an exact sequence of the form $0 \\to A \\xrightarrow{f} B \\xrightarrow{g} C \\to 0$.\n\n## Remarks\n\nA sequence $0 \\to A \\xrightarrow{f} B \\xrightarrow{g} C \\to 0$ is exact exactly when $f$ is injective, $g$ is surjective, and $\\im f = \\ker g$; then $C \\cong B / f(A)$.\nExactness of $0 \\to A \\xrightarrow{f} B$ means that $f$ is injective, and exactness of $B \\xrightarrow{g} C \\to 0$ means that $g$ is surjective.\nExactness at $M_i$ implies $f_i \\circ f_{i-1} = 0$, but not conversely.\n\nExample: $0 \\to \\Z \\xrightarrow{2} \\Z \\xrightarrow{\\pi} \\Z/2\\Z \\to 0$, with $\\pi$ the reduction, is a short exact sequence of $\\Z$-modules.\nMultiplication by $2$ is injective, $\\pi$ is onto, and $\\im(2) = 2\\Z = \\ker \\pi$.\n\nNon-example: the sequence $\\Z \\xrightarrow{2} \\Z \\xrightarrow{\\rho} \\Z/4\\Z$, with $\\rho$ the reduction modulo $4$, is not exact at the middle term.\nThe image of the first map is $2\\Z$, but $\\ker \\rho = 4\\Z \\ne 2\\Z$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "JB7ZBFBA9C",
    "kind": "definition",
    "msc": [
      "18G35",
      "13D02",
      "18E10"
    ],
    "name": "Exact sequence",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Exact_sequence"
      }
    ],
    "tag": "CBDM280TDF",
    "uses": [
      "0310REJJY8",
      "0NZ8JA25WK",
      "0NRYHJPQG2",
      "0W88CEGQP1",
      "0S21GNW9ZB"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/CBDM280TDF/",
  "links": [
    {
      "field": "defines",
      "target": "JB7ZBFBA9C"
    },
    {
      "field": "uses",
      "target": "0310REJJY8"
    },
    {
      "field": "uses",
      "target": "0NZ8JA25WK"
    },
    {
      "field": "uses",
      "target": "0NRYHJPQG2"
    },
    {
      "field": "uses",
      "target": "0W88CEGQP1"
    },
    {
      "field": "uses",
      "target": "0S21GNW9ZB"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/CBDM280TDF.md",
  "msc": [
    "18G35",
    "13D02",
    "18E10"
  ],
  "name": "Exact sequence",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/CBD/CBDM280TDF.md",
  "status": null,
  "tag": "CBDM280TDF",
  "uri": "m2://CBDM280TDF",
  "url": "https://math2.tamnd.com/wiki/exact-sequence-definition/",
  "writing": {
    "tag": "CBDM280TDF",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}