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 SequenceThis entry contributed by Todd Rowland
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Cite this as:
Rowland, Todd. "Exact Sequence." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ExactSequence.html