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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]] and let $M$ and $N$ be [[$R$-modules|0NZ8JA25WK]].\nChoose a [[projective resolution|HK2AA1YSQ2]] $\\cdots \\to P_1 \\xrightarrow{d_1} P_0 \\to M \\to 0$ of $M$.\n[[Tensoring|F0HQXQ0HCD]] the complex $P_\\bullet$ with $N$ gives a chain complex\n$$\\cdots \\to P_2 \\otimes_R N \\xrightarrow{d_2 \\otimes \\id_N} P_1 \\otimes_R N \\xrightarrow{d_1 \\otimes \\id_N} P_0 \\otimes_R N \\to 0.$$\nFor $n \\ge 0$, the $n$-th *Tor module* is its homology\n$$\\operatorname{Tor}_n^R(M, N) = \\ker(d_n \\otimes \\id_N) / \\im(d_{n+1} \\otimes \\id_N),$$\nwith $d_0 \\otimes \\id_N = 0$.\n\n## Remarks\n\nUp to a canonical isomorphism, $\\operatorname{Tor}_n^R(M, N)$ does not depend on the resolution, and resolving $N$ instead of $M$ gives the same modules.\nOne has $\\operatorname{Tor}_0^R(M, N) \\cong M \\otimes_R N$ and $\\operatorname{Tor}_n^R(M, N) \\cong \\operatorname{Tor}_n^R(N, M)$.\nExample: for integers $m, n \\ge 1$, tensoring $0 \\to \\Z \\xrightarrow{\\cdot m} \\Z \\to \\Z/m\\Z \\to 0$ with $\\Z/n\\Z$ gives $\\Z/n\\Z \\xrightarrow{\\cdot m} \\Z/n\\Z$, whose kernel is cyclic of order $\\gcd(m, n)$; so $\\operatorname{Tor}_1^\\Z(\\Z/m\\Z, \\Z/n\\Z) \\cong \\Z/\\gcd(m,n)\\Z$.\nNon-example: the exact sequence $0 \\to \\Z/m\\Z \\xrightarrow{\\id} \\Z/m\\Z \\to 0$ is not a projective resolution over $\\Z$ for $m \\ge 2$, and using it would wrongly give $\\operatorname{Tor}_1 = 0$.\nA module $M$ is [[flat|73RYYEVDKA]] if and only if $\\operatorname{Tor}_1^R(M, N) = 0$ for all $R$-modules $N$.\nFor a noncommutative ring $R$, the same construction works for a right module $M$ and a left module $N$.\n",
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    "name": "Tor functor",
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        "url": "https://en.wikipedia.org/wiki/Tor_functor"
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  "name": "Tor functor",
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