{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\mathcal{A}$ and $\\mathcal{B}$ be [[abelian categories|Y9CZ3MTNF1]], and assume that $\\mathcal{A}$ has enough injectives.\nLet $F \\colon \\mathcal{A} \\to \\mathcal{B}$ be a left [[exact|S1G61TDW0K]] [[additive functor|KAN07YC9T2]].\nAn object $X$ of $\\mathcal{A}$ is *$F$-acyclic* if $R^n F(X) = 0$ for every $n \\ge 1$, where $R^n F$ are the right [[derived functors|NPMC3K6J2F]] of $F$.\nDually, for a right exact $F$ on a category with enough projectives, $X$ is $F$-acyclic if $L_n F(X) = 0$ for every $n \\ge 1$.\n\n## Remarks\n\nExample: every injective object $I$ is $F$-acyclic, because $0 \\to I \\to I \\to 0$ is an injective resolution, and applying $F$ gives a complex with no cohomology in degrees $n \\ge 1$.\nNon-example: for $F = \\Hom_\\Z(\\Z/2\\Z, -)$ on abelian groups, the group $\\Z$ is not $F$-acyclic, since $R^1 F(\\Z) = \\operatorname{Ext}^1_\\Z(\\Z/2\\Z, \\Z) \\cong \\Z/2\\Z \\ne 0$.\nFor $F = - \\otimes_R N$, every [[flat module|73RYYEVDKA]] is $F$-acyclic, since $\\operatorname{Tor}_n^R(M, N) = 0$ for flat $M$ and $n \\ge 1$.\nIf $X \\to J^\\bullet$ is a resolution by $F$-acyclic objects, then $R^n F(X) \\cong H^n(F(J^\\bullet))$ for every $n$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "F2CMPBHQ02",
    "kind": "definition",
    "msc": [
      "18G10"
    ],
    "name": "Acyclic object",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Acyclic_object"
      }
    ],
    "tag": "HEVF97NZ14",
    "uses": [
      "Y9CZ3MTNF1",
      "KAN07YC9T2",
      "S1G61TDW0K",
      "NPMC3K6J2F"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/HEVF97NZ14/",
  "links": [
    {
      "field": "defines",
      "target": "F2CMPBHQ02"
    },
    {
      "field": "uses",
      "target": "Y9CZ3MTNF1"
    },
    {
      "field": "uses",
      "target": "KAN07YC9T2"
    },
    {
      "field": "uses",
      "target": "S1G61TDW0K"
    },
    {
      "field": "uses",
      "target": "NPMC3K6J2F"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/HEVF97NZ14.md",
  "msc": [
    "18G10"
  ],
  "name": "Acyclic object",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/HEV/HEVF97NZ14.md",
  "status": null,
  "tag": "HEVF97NZ14",
  "uri": "m2://HEVF97NZ14",
  "url": "https://tamnd.github.io/math2/wiki/acyclic-object-definition/",
  "writing": {
    "tag": "HEVF97NZ14",
    "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
      }
    ]
  }
}