{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $k$ be a [[field|04C5ZBG1WT]] and let $V$ be a finite-dimensional vector space over $k$, with [[dual space|07JBSZE0V4]] $V^*$.\nLet $R \\subseteq V \\otimes V$ be a subspace.\nLet $A = T(V)/(R)$ be the quotient of the [[tensor algebra|P4J5JZDW0B]] by the two-sided ideal generated by $R$; such an $A$ is a *quadratic algebra*.\nIdentify $V^* \\otimes V^*$ with $(V \\otimes V)^*$, and let $R^\\perp \\subseteq V^* \\otimes V^*$ be the set of forms that vanish on $R$.\nThe *quadratic dual* of $A$ is $A^! = T(V^*)/(R^\\perp)$.\n*Koszul duality* is the correspondence $A \\leftrightarrow A^!$ between [[Koszul algebras|FTKFCCZ8YA]], together with the induced relations between their module categories.\n\n## Remarks\n\nThe construction is an involution: $(A^!)^! \\cong A$, because $(R^\\perp)^\\perp = R$ in finite dimension.\nIf $A$ is Koszul, then $A^!$ is Koszul, and the algebra $\\operatorname{Ext}^\\bullet_A(k, k)$ is isomorphic to $A^!$ (some conventions give the opposite algebra).\nFor Koszul $A$ the Hilbert series satisfy $h_A(t) \\, h_{A^!}(-t) = 1$.\nExample: let $A = S(V)$ be the symmetric algebra, so $R$ is spanned by the tensors $x \\otimes y - y \\otimes x$.\nA form on $V \\otimes V$ vanishes on $R$ exactly when it is symmetric, so $R^\\perp$ is the space of symmetric tensors, spanned by the tensors $\\xi \\otimes \\xi$ and $\\xi \\otimes \\eta + \\eta \\otimes \\xi$.\nHence $A^! = \\Lambda(V^*)$ is the exterior algebra, and $h_A(t) = (1 - t)^{-n}$, $h_{A^!}(-t) = (1 - t)^n$ for $n = \\dim V$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "6AW49BZ8TM",
    "kind": "definition",
    "msc": [
      "16S37"
    ],
    "name": "Koszul duality",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Koszul_duality"
      }
    ],
    "tag": "YX0J6TVGZ8",
    "uses": [
      "04C5ZBG1WT",
      "P4J5JZDW0B",
      "07JBSZE0V4",
      "FTKFCCZ8YA"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/YX0J6TVGZ8/",
  "links": [
    {
      "field": "defines",
      "target": "6AW49BZ8TM"
    },
    {
      "field": "uses",
      "target": "04C5ZBG1WT"
    },
    {
      "field": "uses",
      "target": "P4J5JZDW0B"
    },
    {
      "field": "uses",
      "target": "07JBSZE0V4"
    },
    {
      "field": "uses",
      "target": "FTKFCCZ8YA"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/YX0J6TVGZ8.md",
  "msc": [
    "16S37"
  ],
  "name": "Koszul duality",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/YX0/YX0J6TVGZ8.md",
  "status": null,
  "tag": "YX0J6TVGZ8",
  "uri": "m2://YX0J6TVGZ8",
  "url": "https://tamnd.github.io/math2/wiki/koszul-duality-definition/",
  "writing": {
    "tag": "YX0J6TVGZ8",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}