Splitting lemma

In mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements are equivalent for a short exact sequence

<math>0 \longrightarrow A \mathrel{\overset{q}{\longrightarrow}} B \mathrel{\overset{r}{\longrightarrow}} C \longrightarrow 0.</math>

If any of these statements holds, the sequence is called a split exact sequence, and the sequence is said to split.

In the above short exact sequence, where the sequence splits, it allows one to refine the first isomorphism theorem, which states that:

Template:Math (i.e., Template:Math isomorphic to the coimage of Template:Math or cokernel of Template:Math)

to:

Template:Math

where the first isomorphism theorem is then just the projection onto Template:Math.

It is a categorical generalization of the rank–nullity theorem (in the form Template:Math in linear algebra.

Proof for the category of abelian groups

Template:Math and Template:Math

First, to show that 3. implies both 1. and 2., we assume 3. and take as Template:Math the natural projection of the direct sum onto Template:Math, and take as Template:Math the natural injection of Template:Math into the direct sum.

Template:Math

To prove that 1. implies 3., first note that any member of B is in the set (Template:Math). This follows since for all Template:Math in Template:Math, Template:Math; Template:Math is in Template:Math, and Template:Math is in Template:Math, since

Template:Math

Next, the intersection of Template:Math and Template:Math is 0, since if there exists Template:Math in Template:Math such that Template:Math, and Template:Math, then Template:Math; and therefore, Template:Math.

This proves that Template:Math is the direct sum of Template:Math and Template:Math. So, for all Template:Math in Template:Math, Template:Math can be uniquely identified by some Template:Math in Template:Math, Template:Math in Template:Math, such that Template:Math.

By exactness Template:Math. The subsequence Template:Math implies that Template:Math is onto; therefore for any Template:Math in Template:Math there exists some Template:Math such that Template:Math. Therefore, for any c in C, exists k in ker t such that c = r(k), and r(ker t) = C.

If Template:Math, then Template:Math is in Template:Math; since the intersection of Template:Math and Template:Math, then Template:Math. Therefore, the restriction Template:Math is an isomorphism; and Template:Math is isomorphic to Template:Math.

Finally, Template:Math is isomorphic to Template:Math due to the exactness of Template:Math; so B is isomorphic to the direct sum of Template:Math and Template:Math, which proves (3).

Template:Math

To show that 2. implies 3., we follow a similar argument. Any member of Template:Math is in the set Template:Math; since for all Template:Math in Template:Math, Template:Math, which is in Template:Math. The intersection of Template:Math and Template:Math is Template:Math, since if Template:Math and Template:Math, then Template:Math.

By exactness, Template:Math, and since Template:Math is an injection, Template:Math is isomorphic to Template:Math, so Template:Math is isomorphic to Template:Math. Since Template:Math is a bijection, Template:Math is an injection, and thus Template:Math is isomorphic to Template:Math. So Template:Math is again the direct sum of Template:Math and Template:Math.

An alternative "abstract nonsense" proof of the splitting lemma may be formulated entirely in category theoretic terms.

Non-abelian groups

In the form stated here, the splitting lemma does not hold in the full category of groups, which is not an abelian category.

Partially true

It is partially true: if a short exact sequence of groups is left split or a direct sum (1. or 3.), then all of the conditions hold. For a direct sum this is clear, as one can inject from or project to the summands. For a left split sequence, the map Template:Math gives an isomorphism, so Template:Math is a direct sum (3.), and thus inverting the isomorphism and composing with the natural injection Template:Math gives an injection Template:Math splitting Template:Math (2.).

However, if a short exact sequence of groups is right split (2.), then it need not be left split or a direct sum (neither 1. nor 3. follows): the problem is that the image of the right splitting need not be normal. What is true in this case is that Template:Math is a semidirect product, though not in general a direct product.

Counterexample

To form a counterexample, take the smallest non-abelian group Template:Math, the symmetric group on three letters. Let Template:Math denote the alternating subgroup, and let Template:Math}. Let Template:Math and Template:Math denote the inclusion map and the sign map respectively, so that

<math>0 \longrightarrow A \mathrel{\stackrel{q}{\longrightarrow}} B \mathrel{\stackrel{r}{\longrightarrow}} C \longrightarrow 0 </math>

is a short exact sequence. 3. fails, because Template:Math is not abelian, but 2. holds: we may define Template:Math by mapping the generator to any two-cycle. Note for completeness that 1. fails: any map Template:Math must map every two-cycle to the identity because the map has to be a group homomorphism, while the order of a two-cycle is 2 which can not be divided by the order of the elements in A other than the identity element, which is 3 as Template:Math is the alternating subgroup of Template:Math, or namely the cyclic group of order 3. But every permutation is a product of two-cycles, so Template:Math is the trivial map, whence Template:Math is the trivial map, not the identity.

References

• Saunders Mac Lane: Homology. Reprint of the 1975 edition, Springer Classics in Mathematics, Template:ISBN, p. 16
• Allen Hatcher: Algebraic Topology. 2002, Cambridge University Press, Template:ISBN, p. 147