Editing Splitting lemma (section)

{{Short description|About direct sums and exact sequences}}
{{Distinguish|text=the [[splitting lemma (functions)|splitting lemma]] in [[singularity theory]]}}

In [[mathematics]], and more specifically in [[homological algebra]], the '''splitting lemma''' states that in any [[abelian category]], the following statements are [[logical equivalence|equivalent]] for a [[short exact sequence]] 
: <math>0 \longrightarrow A \mathrel{\overset{q}{\longrightarrow}} B \mathrel{\overset{r}{\longrightarrow}} C \longrightarrow 0.</math>
{{ordered list|{{glossary}}{{term|Left split}}{{defn|There exists a [[morphism]] {{math|''t'': ''B'' → ''A''}} such that {{math|''tq''}} is the [[identity function|identity]] {{math|id{{sub|''A''}}}} on {{math|''A''}},}}{{glossary end}}|{{glossary}}{{term|Right split}}{{defn|There exists a morphism {{math|''u'': ''C'' → ''B''}} such that {{math|''ru''}} is the identity {{math|id{{sub|''C''}}}} on {{math|''C''}},}}{{glossary end}}|{{glossary}}{{term|Direct sum}}{{defn|There is an [[isomorphism (category theory)|isomorphism]] {{mvar|h}} from {{math|''B''}} to the [[biproduct|direct sum]] of {{math|''A''}} and {{math|''C''}}, such that {{math|''hq''}} is the natural injection of {{math|''A''}} into the direct sum, and <math>rh^{-1}</math> is the natural projection of the direct sum [[surjective|onto]] {{math|''C''}}.}}{{glossary end}}
}}

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:
: {{math|''C'' ≅ ''B''/ker ''r'' ≅ ''B''/''q''(''A'')}} (i.e., {{math|''C''}} isomorphic to the [[coimage]] of {{math|''r''}} or [[cokernel]] of {{math|''q''}})
to:
: {{math|''B'' {{=}} ''q''(''A'') ⊕ ''u''(''C'') ≅ ''A'' ⊕ ''C''}}
where the first isomorphism theorem is then just the projection onto {{math|''C''}}.

It is a [[category theory|categorical]] generalization of the [[rank–nullity theorem]] (in the form {{math|V ≅ ker&thinsp;''T'' ⊕ im&thinsp;''T'')}} in [[linear algebra]].