{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "4DD3GM2K4T"
    },
    {
      "field": "uses",
      "target": "5ZJDCNSGHJ"
    },
    {
      "field": "uses",
      "target": "6T74XFKWRA"
    },
    {
      "field": "uses",
      "target": "D34QHZVPES"
    },
    {
      "field": "uses",
      "target": "DD4D0EKX74"
    },
    {
      "field": "uses",
      "target": "ETSXVDDCYK"
    },
    {
      "field": "uses",
      "target": "F0B6BRJA5Z"
    },
    {
      "field": "uses",
      "target": "VA3DY2YX83"
    },
    {
      "field": "defines",
      "target": "FYQX9CJAXG"
    }
  ],
  "body": "\n## Summary\n\nThe Ext functors $\\operatorname{Ext}^n_R(M, N)$ are the right derived functors of $\\Hom_R(-, N)$.\nThey measure how far $\\Hom$ is from being exact.\nThe group $\\operatorname{Ext}^1_R(M, N)$ classifies the extensions of $M$ by $N$, and the vanishing of Ext detects [[projective modules|BFFA03GE3V]], injective modules and the [[projective dimension|MR9QCH1PNA]].\n\n## Explanations\n\n### L1 human\n\nApplying $\\Hom(-, N)$ to a short exact sequence loses exactness on the right.\nThe Ext groups record exactly what is lost, in a long sequence of correction terms.\nTo compute them, replace $M$ by a [[projective resolution|HK2AA1YSQ2]], apply $\\Hom(-, N)$, and take cohomology.\nFor example, $\\operatorname{Ext}^1_\\Z(\\Z/n\\Z, \\Z) \\cong \\Z/n\\Z$, which reflects the extension $0 \\to \\Z \\xrightarrow{n} \\Z \\to \\Z/n\\Z \\to 0$ that does not split.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "13D07",
      "18G15"
    ],
    "name": "Ext functor",
    "same_as": {
      "wikidata": "Q1384902",
      "wikipedia": "Ext functor"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Ext_functor"
      }
    ],
    "tag": "4P7GBWJ4RR"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/4P7GBWJ4RR/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/4P7GBWJ4RR.md",
  "msc": [
    "13D07",
    "18G15"
  ],
  "name": "Ext functor",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/4P7/4P7GBWJ4RR.md",
  "status": null,
  "tag": "4P7GBWJ4RR",
  "uri": "m2://4P7GBWJ4RR",
  "url": "https://math2.tamnd.com/wiki/ext-functor/",
  "writing": {
    "tag": "4P7GBWJ4RR",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}