Splitting lemma

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

Given a short exact sequence with maps q and r:

one writes the additional arrows t and u for maps that may not exist:

Then the following are equivalent:

The short exact sequence is called split if any of the above statements hold.

It allows one to refine the first isomorphism theorem:

  • the first isomorphism theorem states that in the above short exact sequence, C \cong B/q(A)
  • if the sequence splits, then B \cong q(A) \oplus u(C) \cong A \oplus C, and the first isomorphism theorem is just the projection onto C.

It is a categorical generalization of the rank–nullity theorem (in the form V \approx \ker T \oplus \operatorname{im}\,T) in linear algebra.

Contents

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

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

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 r = im q. The subsequence BC → 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 tC is an isomorphism; and ker t is isomorphic to C.

Finally, im q is isomorphic to A due to the exactness of 0 → AB; 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.

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