In mathematics, and more specifically in homological algebra, the splitting lemma states that the following statements regarding the below short exact sequence in any abelian category are equivalent: Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ... In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have nice properties. ... In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ...

1. there exists a map t: B → A such that tq is the identity on A,
2. there exists a map u: C → B such that ru is the identity on C,
3. B is isomorphic to the direct sum of A and C, with q being the natural injection of A and r being the natural projection onto C.

The short exact sequence is called split if any the above statements hold. An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...

Proof

First, to show that (3) implies both (1) and (2), we assume (3) and take as t the natural projection of the direct sum onto A, and take as u the natural injection of C into the direct sum.

To prove that (1) implies (3), first note that any member of B is in the set (ker t + im q). This follows since for all b in B, b = (b - qt(b)) + qt(b); qt(b) is obviously in im q, and (b - qt(b)) is in ker t, since In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. ... It has been suggested that this article or section be merged into image (disambiguation). ...

t(b - qt(b)) = t(b) - tqt(b) = t(b) - (tq)t(b) = t(b) - t(b) = 0.

Next, the intersection of im q and ker t is 0, since if exists a in A such that q(a) = b, and t(b) = 0, then 0 = tq(a) = a; and therefore, b = 0.

(By exactness, im q = ker r, so im q is a normal subgroup of B in the case of groups.) In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are...

This proves that B is the direct sum of im q and ker t. So, for all b in B, b can be uniquely identified by some a in A, k in ker t, such that b = q(a) + k.

By exactness, ker rq = A, and so ker r = im q. The subsequence B → C → 0 implies that r is onto; therefore for any c in C there exists some b = q(a) + k such that c = r(b) = r(q(a) + k) = r(k). Therefore, for any c in C, exists k in ker t such that c = r(k), and r(ker t) = C.

If r(k) = 0, then k is in im q; since the intersection of im q and ker t = 0, then k = 0. Therefore the restriction of the morphism r : ker t → C is an isomorphism; and ker t is isomorphic to C.

Finally, im q is isomorphic to A due to the exactness of 0 → A → B; so B is isomorphic to the direct sum of A and C, which proves (3).

To show that (2) implies (3), we follow a similar argument. Any member of B is in the set ker r + im u; since for all b in B, b = (b - ur(b)) + ur(b), which is in ker r + im u. The intersection of ker r and im u is 0, since if r(b) = 0 and u(c) = b, then 0 = ru(c) = c.

By exactness, im q = ker r, and since q is an injection, im q is isomorphic to A, so A is isomorphic to ker r. Since ru is a bijection, u is an injection, and thus im u is isomorphic to C. So B is again the direct sum of A and C.

Non-abelian groups

A different version of the splitting lemma applies to non-abelian groups.