Short Exact Sequence
A short exact sequence of groups A, B, and C is given by two maps alpha:A->B and beta:B->C and is written
| 0->A->B->C->0. |
(1)
|
Because it is an exact sequence, alpha is injective, and beta is surjective. Moreover, the group kernel of beta is the image of alpha. Hence, the group A can be considered as a (normal) subgroup of B, and C is isomorphic to B/A.
A short exact sequence is said to split if there is a map gamma:C->B such that beta degreesgamma is the identity on C. This only happens when B is the direct product of A and C.
The notion of a short exact sequence also makes sense for modules and sheaves. Given a module M over a unit ring R, all short exact sequences
| 0-->A-->B-->M-->0 |
(2)
|
are split iff M is projective, and all short exact sequences
| 0-->M-->B-->C-->0 |
(3)
|
A short exact sequence of vector spaces is always split.
See also
Exact Sequence, Group Extension, Long Exact Sequence, Module, Principal Bundle, Split Exact SequenceThis entry contributed by Todd Rowland
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Rowland, Todd. "Short Exact Sequence." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ShortExactSequence.html