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Exact Sequence


An exact sequence is a sequence of maps

 alpha_i:A_i->A_(i+1)
(1)

between a sequence of spaces A_i, which satisfies

 Im(alpha_i)=Ker(alpha_(i+1)),
(2)

where Im denotes the image and Ker the group kernel. That is, for a in A_i, alpha_i(a)=0 iff a=alpha_(i-1)(b) for some b in A_(i-1). It follows that alpha_(i+1) degreesalpha_i=0. 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

 ...->A_(i-1)->A_i->A_(i+1)->....
(3)

An exact sequence may be of either finite or infinite length. The special case of length five,

 0->A->B->C->0,
(4)

beginning and ending with zero, meaning the zero module {0}, is called a short exact sequence. An infinite exact sequence is called a long exact sequence. For example, the sequence where A_i=Z/4Z and alpha_i is given by multiplying by 2,

 ...-->^(×2)Z/4Z-->^(×2)Z/4Z-->^(×2)...,
(5)

is a long exact sequence because at each stage the kernel and image are equal to the subgroup {0,2}.

Special information is conveyed when one of the spaces A_i is the zero module. For instance, the sequence

 0->A->B
(6)

is exact iff the map A->B is injective. Similarly,

 A->B->0
(7)

is exact iff the map A->B is surjective.


See also

Chain Complex, Homology, Long Exact Sequence, Short Exact Sequence, Spectral Sequence

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Exact Sequence." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ExactSequence.html

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