Splitting Lemma

Proposition

Let 20230815-splitting_lemma_282826e9ebf8bd11a88dcc45a6dc4bb8b788556f.svg be an additive category and

20230815-splitting_lemma_1f5a2adc3e8204446ae71135b7ba99df3585e92b.svg

be a short exact sequence.

TFAE:

there exists a split
there exists a split
such that and and that following diagram commutes

20230815-splitting_lemma_9ab3a614dc7b7e66d22d758bf156c91ca6cffbbb.svg

Proof

1) 3)

We want to define a map 20230815-splitting_lemma_7744a6a9d179d48b9fa5fb90177854b149b442dd.svg which becomes (in hindsight) an isomorphism
For this consider the endomorphism 20230815-splitting_lemma_e7345fff078458ca09834f2889f4f82a26ab4562.svg : It restricts to 20230815-splitting_lemma_0795014a406707be33cce434027d8a019ec22e65.svg – the fiber – as 20230815-splitting_lemma_e873a3dd3c0a6b614e845f2c6152d33acdd073c7.svg .
Therefore define 20230815-splitting_lemma_1abbad309e3454f28915fc8b3ac72d05ea9c7874.svg and 20230815-splitting_lemma_fe0e910bc5f515080cac4573e13a774047b4aa0d.svg .
Now it remains to calculate that

20230815-splitting_lemma_59e837bcc9cf8b42493ad4d59c0a30ad79378b7b.svg

is a commutative diagram where the middle maps are isomorphisms

2) 3)

dual to 1) 20230815-splitting_lemma_e1be93bd3820e525960794a47f6526ae6446d37a.svg 3)

3) 1), 3) 2)

follows from the retraction of the 20230815-splitting_lemma_37a20a48134e057460b4a05d96b6a0f2e22ac4b4.svg