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Split sequence

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split exact sequence, split short exact sequence

An exact sequence

(*)

in an Abelian category which is isomorphic to the direct sum sequence,

by an isomorphism which induces the identity on and on . Sufficient conditions for an exact sequence (*) to be split are the existence of a right inverse for , or of a left inverse for . The class of split exact sequences is the zero of the group (see Baer multiplication). In a category of vector spaces (that is, of modules over a fixed field) every exact sequence splits.

For relative homological algebra, the typical situation is to consider exact sequences in one category which split in another.


Comments

References

[a1] S. MacLane, "Homology" , Springer (1963) pp. 16, 260
How to Cite This Entry:
Split sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Split_sequence&oldid=41781
This article was adapted from an original article by V.E. Govorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article