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  "body": "\n## Statement\n\nLet $R$ be a [[principal ideal domain|0CMD0BCH8T]].\n\n1. Let $C_\\bullet$ and $D_\\bullet$ be [[chain complexes|WQ82C0M3QB]] of $R$-modules such that every $C_n$ is a free $R$-module. Then for every $n$ there is a natural short exact sequence\n$$0 \\to \\bigoplus_{i+j=n} H_i(C) \\otimes_R H_j(D) \\to H_n(C \\otimes_R D) \\to \\bigoplus_{i+j=n-1} \\operatorname{Tor}_1^R(H_i(C), H_j(D)) \\to 0,$$\nand it splits, though not naturally.\n2. Let $X$ and $Y$ be [[topological spaces|0CQHFPGJ45]]. Then for every $n$ there is a natural short exact sequence, which splits but not naturally,\n$$0 \\to \\bigoplus_{i+j=n} H_i(X; R) \\otimes_R H_j(Y; R) \\to H_n(X \\times Y; R) \\to \\bigoplus_{i+j=n-1} \\operatorname{Tor}_1^R(H_i(X; R), H_j(Y; R)) \\to 0.$$\n\nHere $H$ is the [[homology|2J4JJJYP22]] of a complex, and $H_n(-; R)$ is [[singular homology|M0Q5CFJ7ZH]] with coefficients in $R$.\nThe symbol $\\otimes_R$ is the [[tensor product|F0HQXQ0HCD]] and $\\operatorname{Tor}_1^R$ is the [[Tor functor|DDQRERAQCV]].\n\n## Remarks\n\nFor the algebraic part, note that the cycles $Z_\\bullet$ and boundaries $B_\\bullet$ of $C$ are free, since $R$ is a principal ideal domain.\nSo $0 \\to Z_\\bullet \\to C_\\bullet \\to B_{\\bullet - 1} \\to 0$ is a split sequence of free complexes.\nThe long exact sequence after tensoring with $D$ gives the result.\nThe topological part follows from the algebraic one and the Eilenberg–Zilber theorem.\nThat theorem gives a chain homotopy equivalence between the singular chains of $X \\times Y$ and the tensor product of the singular chains of $X$ and of $Y$.\nIf $R$ is a field, all Tor terms vanish and $H_n(X \\times Y; R) \\cong \\bigoplus_{i+j=n} H_i(X; R) \\otimes_R H_j(Y; R)$.\nOver $\\Z$ the Tor term matters.\nFor the real projective plane $P$, one has $H_0(P) = \\Z$, $H_1(P) = \\Z/2\\Z$ and $H_2(P) = 0$.\nSo the tensor part in degree $3$ is zero, and the formula gives $H_3(P \\times P; \\Z) \\cong \\operatorname{Tor}_1^\\Z(\\Z/2\\Z, \\Z/2\\Z) \\cong \\Z/2\\Z$.\n\n## Explanations\n\n### L1 human\n\nThe Künneth theorem computes the holes of a product space from the holes of its factors.\nWith coefficients in a field, the answer is the simplest one: the homology of the product is the tensor product of the homologies.\nFor the torus, the product of two circles, this gives one class in degree $0$, two in degree $1$ and one in degree $2$.\nWith integer coefficients there is a correction term built from torsion.\n\n### L2 human\n\nThe theorem is a direct application of the derived functor Tor: the failure of $H(C \\otimes D) \\cong H(C) \\otimes H(D)$ is measured by $\\operatorname{Tor}_1$.\nTaking $D$ to be a module $M$ in degree $0$ gives the homology form of the [[universal coefficient theorem|GTA8ZFGZ3N]].\nThere are also cohomology versions, with a cross product into $H^n(X \\times Y; R)$, and spectral sequence versions over rings that are not principal ideal domains.\n",
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