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  "body": "\n## Statement\n\nLet $R$ be a [[principal ideal domain|0CMD0BCH8T]], let $C_\\bullet$ be a [[chain complex|WQ82C0M3QB]] of free $R$-modules, and let $M$ be an $R$-module.\nThen for every $n$ there are natural short exact sequences\n$$0 \\to H_n(C) \\otimes_R M \\to H_n(C \\otimes_R M) \\to \\operatorname{Tor}_1^R(H_{n-1}(C), M) \\to 0,$$\n$$0 \\to \\operatorname{Ext}^1_R(H_{n-1}(C), M) \\to H^n(\\Hom_R(C, M)) \\to \\Hom_R(H_n(C), M) \\to 0,$$\nand both split, though not naturally.\nIn particular, for a [[topological space|0CQHFPGJ45]] $X$ and an abelian group $G$, applied to the singular chain complex of $X$:\n$$H_n(X; G) \\cong (H_n(X) \\otimes G) \\oplus \\operatorname{Tor}_1^\\Z(H_{n-1}(X), G), \\qquad H^n(X; G) \\cong \\Hom(H_n(X), G) \\oplus \\operatorname{Ext}^1_\\Z(H_{n-1}(X), G).$$\n\n## Remarks\n\nHere $H_n(X)$ is [[singular homology|M0Q5CFJ7ZH]] with integer coefficients and $H^n(X; G)$ is [[singular cohomology|6N7BC70CX0]]; [[Tor|DDQRERAQCV]] and [[Ext|4P7GBWJ4RR]] are the derived functors of $\\otimes$ and $\\Hom$.\nFor the proof, the cycles $Z_n$ and boundaries $B_n$ of $C$ are free, since submodules of free modules over a principal ideal domain are free.\nSo $0 \\to Z_\\bullet \\to C_\\bullet \\to B_{\\bullet-1} \\to 0$ splits in each degree.\nApply $- \\otimes M$ or $\\Hom(-, M)$ and take the long exact sequence; its connecting maps are induced by the inclusions $B_n \\to Z_n$.\nTheir kernels and cokernels are the Tor and Ext terms.\nExample: for the real projective plane $P$, $H_0(P) = \\Z$, $H_1(P) = \\Z/2\\Z$, $H_2(P) = 0$.\nSo $H^1(P; \\Z) \\cong \\Hom(\\Z/2\\Z, \\Z) \\oplus \\operatorname{Ext}^1(\\Z, \\Z) = 0$ and $H^2(P; \\Z) \\cong \\Hom(0, \\Z) \\oplus \\operatorname{Ext}^1(\\Z/2\\Z, \\Z) \\cong \\Z/2\\Z$.\nThis shows that cohomology is not simply the dual of homology: the torsion of $H_1$ moves up to $H^2$.\n\n## Explanations\n\n### L1 human\n\nHomology with integer coefficients determines homology and cohomology with any other coefficients.\nThe answer is almost the naive one, the tensor product or the dual, plus a correction that comes from torsion.\nFor the projective plane, the integer homology in degree one is $\\Z/2\\Z$, and this torsion reappears in integer cohomology one degree higher.\n\n### L2 human\n\nThe homology part of the theorem is a special case of the [[Künneth theorem|77K0BZQ4MK]], with the second complex a module in degree $0$.\nIt is used constantly, for example to compute cohomology from homology.\nIt also shows that a map inducing isomorphisms on integral homology induces isomorphisms on homology and cohomology with all coefficients.\nOver a field $k$, it gives $H^n(X; k) \\cong \\Hom_k(H_n(X; k), k)$.\n",
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