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  "body": "\n## Summary\n\nThe Tor functors $\\operatorname{Tor}_n^R(M, N)$ are the left derived functors of the [[tensor product|F0HQXQ0HCD]] $M \\otimes_R N$.\nThey measure how far tensoring is from being exact.\nThe vanishing of $\\operatorname{Tor}_1^R(M, -)$ characterizes the [[flat modules|73RYYEVDKA]], and Tor groups over $\\Z$ appear in the universal coefficient and Künneth theorems of topology.\n\n## Explanations\n\n### L1 human\n\nTensoring with a module can turn an injective map into one that is not injective.\nThe Tor groups record what is lost.\nTo compute them, replace $M$ by a [[projective resolution|HK2AA1YSQ2]], tensor with $N$, and take homology.\nFor example, $\\operatorname{Tor}_1^\\Z(\\Z/2\\Z, \\Z/2\\Z) \\cong \\Z/2\\Z$, because multiplication by $2$ is injective on $\\Z$ but becomes the zero map on $\\Z/2\\Z$.\nThe name comes from torsion: $\\operatorname{Tor}_1^\\Z(\\Z/n\\Z, A)$ is the set of elements of the abelian group $A$ killed by $n$.\n",
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