Split exact sequence

In mathematics, a split exact sequence is a short exact sequence in which the middle term is built out of the two outer terms in the simplest possible way.

In mathematics, a split exact sequence is a short exact sequence in which the middle term is built out of the two outer terms in the simplest possible way. ==Equivalent characterizations== A short exact sequence of abelian groups or of modules over a fixed ring, or more generally of objects in an abelian category 0 \to A \mathrel{\stackrel{a}{\to}} B \mathrel{\stackrel{b}{\to}} C \to 0 is called split exact if it is isomorphic to the exact sequence where the middle term is the direct sum of the outer ones: 0 \to A \mathrel{\stackrel{i}{\to}} A \oplus C \mathrel{\stackrel{p}{\to}} C \to 0 with i: A \to A \oplus C being the natural inclusion of A into the direct sum, and p: A \oplus C \to C denoting the natural projection of the direct sum onto the second summand. The requirement that the sequence is isomorphic means that there is an isomorphism f : B \to A \oplus C such that the composite f \circ a is the natural inclusion i: A \to A \oplus C and such that the composite p \circ f equals b. This can be summarized by a commutative diagram as: The splitting lemma provides further equivalent characterizations of split exact sequences. The sequence 0 \to A \mathrel{\stackrel{a}{\to}} B \mathrel{\stackrel{b}{\to}} C \to 0 is split exact if and only if there exists r : C \to B such that b \circ r = 1_C , which is the case if and only if there exists s : B \to A such that s \circ a = 1_A . ==Examples== A trivial example of a split short exact sequence is 0 \to M_1 \mathrel{\stackrel{q}{\to}} M_1\oplus M_2 \mathrel{\stackrel{p}{\to}} M_2 \to 0 where M_1, M_2 are R-modules, q is the canonical injection and p is the canonical projection. Any short exact sequence of vector spaces is split exact. This is a rephrasing of the fact that any set of linearly independent vectors in a vector space can be extended to a basis. The exact sequence 0 \to \mathbf{Z}\mathrel{\stackrel{2}{\to}} \mathbf{Z}\to \mathbf{Z}/ 2\mathbf{Z} \to 0 (where the first map is multiplication by 2) is not split exact. ==Related notions== Pure exact sequences can be characterized as the filtered colimits of split exact sequences. ==References== ==Sources==

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