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  "body": "\n## Statement\n\nLet $p$ be a prime and let $X$ be a space whose integral [[singular homology|M0Q5CFJ7ZH]] groups $H_n(X; \\Z)$ are finitely generated for every $n$.\nThen there is a spectral sequence $(E^r_n, d^r)_{r \\ge 1}$ with $d^r \\colon E^r_n \\to E^r_{n-1}$ such that:\n\n1. $E^1_n = H_n(X; \\F_p)$, the [[homology with coefficients|XT7VWNTKGB]] in $\\F_p$, and $d^1$ is the [[Bockstein homomorphism|H7PCM776GR]] of $0 \\to \\Z/p \\to \\Z/p^2 \\to \\Z/p \\to 0$;\n2. $E^\\infty_n = E^r_n$ for $r$ large, and $E^\\infty_n \\cong \\big(H_n(X; \\Z) / \\mathrm{torsion}\\big) \\otimes \\F_p$;\n3. each direct summand $\\Z/p^r$ of $H_{n-1}(X; \\Z)$ gives classes in $E^r_n$ and $E^r_{n-1}$ joined by a nonzero $d^r$.\nAll nonzero $d^r$ arise in this way.\n\n## Remarks\n\nThe spectral sequence comes from an [[exact couple|VXDPP4EJPG]] with $D = H_*(X; \\Z)$ and $E = H_*(X; \\F_p)$.\nThe map $D \\to D$ is multiplication by $p$, and the other two maps come from the long exact sequence of $0 \\to \\Z \\xrightarrow{p} \\Z \\to \\Z/p \\to 0$.\n\nExample: for $X = \\R P^2$ and $p = 2$, one has $H_1(X; \\Z) = \\Z/2$ and $H_2(X; \\Z) = 0$.\nHere $E^1_1 = E^1_2 = \\F_2$ and $d^1 \\colon E^1_2 \\to E^1_1$ is an isomorphism, so $E^2_n = 0$ for $n = 1, 2$, which matches $E^\\infty$ since the free part of $H_n(X; \\Z)$ vanishes for $n = 1, 2$.\n\nThe finite generation hypothesis ensures that the spectral sequence stops at a finite page in each degree.\nFor a summand such as $\\Z[1/p]$ or $\\Z/p^\\infty$ the description in terms of $E^\\infty$ changes.\n\n## Explanations\n\n### L1 human\n\nMod $p$ homology is easy to compute but forgets how large the torsion of integral homology is.\nThe Bockstein spectral sequence recovers this: a summand $\\Z/p^r$ shows up as a pair of mod $p$ classes that survive exactly $r$ pages before cancelling.\nWhat survives to the end is the free part of integral homology.\n\n### L2 human\n\nThe spectral sequence is used to detect higher torsion, for instance in the homology of Lie groups, loop spaces and Eilenberg–MacLane spaces.\nIn cohomology it is multiplicative, and the higher Bocksteins interact with Steenrod operations, which gives restrictions on how torsion can appear.\nIt is one of the simplest examples of a spectral sequence built from an exact couple.\n",
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