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  "body": "\n## Statement\n\nLet $G$ be a group, let $N$ be a [[normal subgroup|0CB0DX5BVV]] of $G$, and let $M$ be a $G$-module.\nFor each $q \\ge 0$, the cohomology group $H^q(N, M)$ is a module over the [[quotient group|0KYX2N69J0]] $G/N$, through the action of $G$ by conjugation.\nThen there is a first-quadrant cohomological spectral sequence\n$$E_2^{p,q} = H^p\\big(G/N, H^q(N, M)\\big) \\Longrightarrow H^{p+q}(G, M).$$\n\n## Remarks\n\nThe spectral sequence is the Grothendieck spectral sequence of the composite of the functors $M \\mapsto M^N$ and $A \\mapsto A^{G/N}$, whose composite is $M \\mapsto M^G$.\nIt applies because $M \\mapsto M^N$ sends injective $G$-modules to $G/N$-modules that are acyclic for $A \\mapsto A^{G/N}$.\nThere is a homological version $E^2_{p,q} = H_p(G/N, H_q(N, M)) \\Rightarrow H_{p+q}(G, M)$ for [[group homology|CFAG3VRXEA]].\n\nThe terms of low degree give the inflation-restriction exact sequence\n$$0 \\to H^1(G/N, M^N) \\to H^1(G, M) \\to H^1(N, M)^{G/N} \\to H^2(G/N, M^N) \\to H^2(G, M).$$\n\nExample: suppose that $H^q(N, M) = 0$ for all $q \\ge 1$, as happens when $N$ is finite and multiplication by $\\abs{N}$ is bijective on $M$.\nThen the $E_2$ page is concentrated in the row $q = 0$, the sequence collapses, and $H^p(G, M) \\cong H^p(G/N, M^N)$ for all $p$.\n\n## Explanations\n\n### L1 human\n\nWhen a group is built from a normal subgroup and a quotient, its cohomology is assembled from the cohomology of the two pieces.\nThe spectral sequence is the bookkeeping device for this assembly.\nIt does not always give the answer at once, but it bounds it and often determines it.\n\n### L2 human\n\nThe five-term exact sequence of the homological version already gives [[Hopf's formula|S31M1ZYJRK]] for $H_2$.\nFor a [[polycyclic group|9RBA56NXX3]], one can apply the spectral sequence step by step along a normal series with cyclic factors.\nIts topological counterpart is the Serre spectral sequence of the fibration $BN \\to BG \\to B(G/N)$ of [[classifying spaces|WA796ZCNRX]].\n",
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  "name": "Lyndon–Hochschild–Serre spectral sequence",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/VC3/VC3QNG6TKR.md",
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