{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $f \\colon X \\to Y$ be a continuous map of topological spaces and let $\\mathcal{F}$ be a sheaf of abelian groups on $X$.\nFor $q \\ge 0$, let $R^q f_* \\mathcal{F}$ be the $q$-th higher direct image, that is, the $q$-th right derived functor of the direct image $f_*$.\nIt is the sheaf on $Y$ associated with the presheaf $V \\mapsto H^q(f^{-1}(V); \\mathcal{F})$.\nThen there is a first-quadrant spectral sequence with\n$$E_2^{p,q} = H^p(Y; R^q f_* \\mathcal{F}) \\Longrightarrow H^{p+q}(X; \\mathcal{F}),$$\nwhere $H^*$ denotes [[sheaf cohomology|KB76923A1M]], and the differentials are $d_r \\colon E_r^{p,q} \\to E_r^{p+r, q-r+1}$.\n\n## Remarks\n\nSince $\\Gamma(Y, f_* \\mathcal{F}) = \\Gamma(X, \\mathcal{F})$ and $f_*$ sends injective sheaves to flasque sheaves, which are acyclic for $\\Gamma(Y, -)$, this is the Grothendieck spectral sequence of the composite functor $\\Gamma(Y, -) \\circ f_*$.\n\nExample: if $R^q f_* \\mathcal{F} = 0$ for all $q \u003e 0$, the spectral sequence collapses and $H^p(X; \\mathcal{F}) \\cong H^p(Y; f_* \\mathcal{F})$ for all $p$.\nThis applies when $f$ is the inclusion of a closed subspace, because then $f_*$ is exact.\n\nLet $f$ be a fibre bundle with fibre $F$ over a nice base, and let $\\mathcal{F}$ be a constant sheaf.\nThen $R^q f_* \\mathcal{F}$ is locally constant with stalks $H^q(F)$, and one recovers a form of the Serre spectral sequence.\n\n## Explanations\n\n### L1 human\n\nThe Leray spectral sequence computes the cohomology of a space $X$ by looking at it through a map $f$ to another space $Y$.\nOver each small open set of $Y$, one first computes the cohomology of the part of $X$ above it.\nThese local answers form sheaves on $Y$, and the cohomology of $Y$ with values in them leads to the cohomology of $X$.\n\n### L2 human\n\nLeray introduced sheaves and spectral sequences together for this purpose.\nIn algebraic geometry the same spectral sequence, for a morphism of schemes and étale or coherent cohomology, is a basic tool.\nIt is the prototype of the Grothendieck spectral sequence for a composite of derived functors.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "KB76923A1M"
    ],
    "kind": "statement",
    "msc": [
      "55T05",
      "18G40",
      "55N30"
    ],
    "name": "Leray spectral sequence",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q6528749",
      "wikipedia": "Leray spectral sequence"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Leray_spectral_sequence"
      }
    ],
    "tag": "8BMQ90EJPR",
    "uses": [
      "KB76923A1M"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/8BMQ90EJPR/",
  "links": [
    {
      "field": "about",
      "target": "KB76923A1M"
    },
    {
      "field": "uses",
      "target": "KB76923A1M"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/8BMQ90EJPR.md",
  "msc": [
    "55T05",
    "18G40",
    "55N30"
  ],
  "name": "Leray spectral sequence",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/8BM/8BMQ90EJPR.md",
  "status": null,
  "tag": "8BMQ90EJPR",
  "uri": "m2://8BMQ90EJPR",
  "url": "https://math2.tamnd.com/wiki/leray-spectral-sequence/",
  "writing": {
    "tag": "8BMQ90EJPR",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}