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Split Exact Sequence


A short exact sequence of groups

 0-->A-->B-->C-->0
(1)

is called split if it essentially presents B as the direct sum of the groups A and C.

SplitExactSequence

More precisely, one can construct a commutative diagram as diagrammed above, where i is the injection of the first summand A and p is the projection onto the second summand C, and the vertical maps are isomorphisms.

SplitExactSequence2

Not all short exact sequences of groups are split. For example the short exact sequence diagrammed above cannot be split, since Z_4 and Z_2 direct sum Z_2 are non isomorphic finite groups. Note that this is also a short exact sequence of Z-modules: this shows that being split is a distinguished property of short exact sequences also in the category of modules. In fact, it is related to particular classes of modules.

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

Commutative Diagram, Direct Sum, Short Exact Sequence

This entry contributed by Margherita Barile

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References

Hilton, P. J. and Stammbach, U. A Course in Homological Algebra, 2nd ed. New York: Springer-Verlag, pp. 24-25, 1997.Mac Lane, S. Homology. Berlin: Springer-Verlag, pp. 16-17, 1967.Munkres, J. R. "The Zig-Zag Lemma." §24 in Elements of Algebraic Topology. New York: Perseus Books Pub.,pp. 131-132, 1993.Passman, D. S. A Course in Ring Theory. Pacific Grove, CA: Wadsworth & Brooks/Cole, pp. 14-16, 1991.Reid, M. Undergraduate Commutative Algebra. Cambridge, England: Cambridge University Press, pp. 45-46, 1995.Rowen, L. H. Ring Theory, Vol. 1. San Diego, CA: Academic Press, pp. 68-71, 1988.Sharp, R. Y. Steps in Commutative Algebra, 2nd ed. Cambridge, England: Cambridge University Press, p. 117, 2000.

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Split Exact Sequence

Cite this as:

Barile, Margherita. "Split Exact Sequence." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/SplitExactSequence.html

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