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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)

are split iff M is injective.

A short exact sequence of vector spaces is always split.


See also

Exact Sequence, Group Extension, Long Exact Sequence, Module, Principal Bundle, Split Exact Sequence

This entry contributed by Todd Rowland

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Cite this as:

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

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