{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $\\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]].\nLet $0 \\to A' \\xrightarrow{i} A \\xrightarrow{\\pi} A'' \\to 0$ be a [[short exact sequence|KJKNJMGKZZ]] in $\\mathcal{A}$.\nLet $P'_\\bullet \\to A'$ and $P''_\\bullet \\to A''$ be [[projective resolutions|HK2AA1YSQ2]].\nThen there is a projective resolution $P_\\bullet \\to A$ with $P_n = P'_n \\oplus P''_n$ for every $n \\ge 0$ with the following property.\nThe inclusions $P'_n \\to P_n$ and the projections $P_n \\to P''_n$ form a short exact sequence of [[chain complexes|WQ82C0M3QB]]\n$$0 \\to P'_\\bullet \\to P_\\bullet \\to P''_\\bullet \\to 0$$\nthat lifts the sequence $0 \\to A' \\to A \\to A'' \\to 0$.\n\n## Remarks\n\nThe proof builds $P_\\bullet$ one degree at a time.\nSince $P''_0$ is [[projective|YXMXAGVC1Q]], the map $P''_0 \\to A''$ lifts through $\\pi$ to a map $P''_0 \\to A$.\nTogether with $P'_0 \\to A' \\to A$ this gives $P'_0 \\oplus P''_0 \\to A$, which is an epimorphism by the snake lemma.\nThe kernels again form a short exact sequence, and the same step is repeated with them.\nThe differential of $P_\\bullet$ is in general not the direct sum of the differentials: it has an extra component $P''_n \\to P'_{n-1}$.\nFor example, over $\\Z$ take $0 \\to \\Z \\xrightarrow{\\cdot 2} \\Z \\to \\Z/2\\Z \\to 0$ with $P'_\\bullet = \\Z$ and $P''_\\bullet = (\\Z \\xrightarrow{\\cdot 2} \\Z)$; the lemma gives a resolution of $\\Z$ by $\\Z^2$ in degree $0$ and $\\Z$ in degree $1$.\nThe dual statement holds for injective resolutions.\n\n## Explanations\n\n### L1 human\n\nIf the two outer terms of a short exact sequence have projective resolutions, the middle term gets one too.\nThe new resolution is built from the two old ones, placed side by side.\nThe picture of the two resolutions on the left and right, with the new one in the middle, looks like a horseshoe.\n\n### L2 human\n\nThe lemma is the key step to show that a short exact sequence gives a long exact sequence of left [[derived functors|NPMC3K6J2F]].\nApplying an additive functor to the split sequences $0 \\to P'_n \\to P_n \\to P''_n \\to 0$ keeps them exact, and the long exact sequence in homology does the rest.\nThe same construction, applied degree by degree, is used to build [[Cartan–Eilenberg resolutions|SX8V8GR5ZG]].\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "HK2AA1YSQ2"
    ],
    "kind": "statement",
    "msc": [
      "18G10",
      "18G05"
    ],
    "name": "Horseshoe lemma",
    "role": "lemma",
    "same_as": {
      "wikidata": "Q5906080",
      "wikipedia": "Horseshoe lemma"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Horseshoe_lemma"
      }
    ],
    "tag": "VNMBSYX9PY",
    "uses": [
      "Y9CZ3MTNF1",
      "KJKNJMGKZZ",
      "HK2AA1YSQ2",
      "YXMXAGVC1Q",
      "WQ82C0M3QB"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/VNMBSYX9PY/",
  "links": [
    {
      "field": "about",
      "target": "HK2AA1YSQ2"
    },
    {
      "field": "uses",
      "target": "Y9CZ3MTNF1"
    },
    {
      "field": "uses",
      "target": "KJKNJMGKZZ"
    },
    {
      "field": "uses",
      "target": "HK2AA1YSQ2"
    },
    {
      "field": "uses",
      "target": "YXMXAGVC1Q"
    },
    {
      "field": "uses",
      "target": "WQ82C0M3QB"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/VNMBSYX9PY.md",
  "msc": [
    "18G10",
    "18G05"
  ],
  "name": "Horseshoe lemma",
  "prov": null,
  "role": "lemma",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/VNM/VNMBSYX9PY.md",
  "status": null,
  "tag": "VNMBSYX9PY",
  "uri": "m2://VNMBSYX9PY",
  "url": "https://tamnd.github.io/math2/wiki/horseshoe-lemma/",
  "writing": {
    "tag": "VNMBSYX9PY",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}