An exact sequence is a sequence of maps
(1)
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between a sequence of spaces , which satisfies
(2)
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where denotes the image and the group kernel. That is, for , iff for some . It follows that . The notion of exact sequence makes sense when the spaces are groups, modules, chain complexes, or sheaves. The notation for the maps may be suppressed and the sequence written on a single line as
(3)
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An exact sequence may be of either finite or infinite length. The special case of length five,
(4)
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beginning and ending with zero, meaning the zero module , is called a short exact sequence. An infinite exact sequence is called a long exact sequence. For example, the sequence where and is given by multiplying by 2,
(5)
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is a long exact sequence because at each stage the kernel and image are equal to the subgroup .
Special information is conveyed when one of the spaces is the zero module. For instance, the sequence
(6)
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is exact iff the map is injective. Similarly,
(7)
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is exact iff the map is surjective.