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  "body": "\n## Summary\n\nAn exact sequence is a chain of modules and linear maps in which the image of each map equals the [[kernel|0W88CEGQP1]] of the next.\nShort exact sequences $0 \\to A \\to B \\to C \\to 0$ describe $B$ as built from a submodule $A$ and a quotient $C$.\nExact sequences are the language of homological algebra: one asks which functors, such as $\\Hom$ and the [[tensor product|F0HQXQ0HCD]], preserve them.\nThe same notion makes sense for groups and in any abelian category.\n\n## Explanations\n\n### L1 human\n\nAn exact sequence is a row of modules connected by maps, where what one map produces is exactly what the next map kills.\nSo nothing is lost and nothing extra appears between two steps.\nThe basic example is $0 \\to \\Z \\xrightarrow{2} \\Z \\to \\Z/2\\Z \\to 0$.\nHere multiplication by $2$ is injective, its image $2\\Z$ is exactly the set of integers that become $0$ modulo $2$, and reduction modulo $2$ is onto.\n",
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