Exact sequence in an abelian category

A sequence is exact at an object when the image of the incoming morphism equals the kernel of the outgoing morphism. It is exact when this holds everywhere.

A short exact sequence in an abelian category has one nonzero morphism entering the middle object and one leaving it: $0\to A\stackrel{i}{\to }B\stackrel{p}{\to }C\to 0$. Equivalently, $i$ is the kernel in a category of the epimorphism $p$. It splits if $p$ has a section; in that case the additive category splitting argument identifies $B$ with the biproduct $A\oplus C$. Categories of modules give familiar examples.

Two projective presentations of the same object in an abelian category, with kernels $K,K^{\prime }$ and middle objects $P,P^{\prime }$, satisfy $K\oplus P^{\prime }\cong K^{\prime }\oplus P$. Form their pullback in a category. Its two induced short exact sequences in an abelian category split because each quotient is a projective object in a category. The two resulting biproduct descriptions of the pullback give the isomorphism.

Pulling back the final arrow of a short exact sequence in an abelian category preserves its kernel object and yields another short exact sequence in an abelian category. The new kernel inclusion is specified by its old kernel component and zero component into the new quotient. The pullback stability of epimorphisms in an abelian category ensures the new final arrow remains epic.

In a commutative diagram of two exact five-term sequences, if the first and fourth vertical maps are epimorphisms and the second and fifth are monomorphisms, then an isomorphism in the middle follows; in particular, four surrounding isomorphisms force the fifth.

For commuting five-term exact sequences in an abelian category, the first vertical map being epic and the second invertible give an invertible induced map on the images entering the middle objects, because these images are the cokernels of the first horizontal maps. The fourth vertical map being invertible and the fifth monic similarly give an invertible map on the images leaving the middle objects, because these are the kernels of the last horizontal maps. The image factorization in an abelian category packages each middle object between these two images in a short exact sequence. The short five lemma then makes the middle vertical map invertible. The proof is categorical and does not require an embedding into a category of modules.

A morphism between two short exact sequences in an abelian category induces an exact sequence