Split sequence

split exact sequence, split short exact sequence

An exact sequence \begin{equation} 0 \rightarrow A \stackrel{f}{\rightarrow} B \stackrel{g}{\rightarrow} C \rightarrow 0 \label{eq:1} \end{equation} in an Abelian category which is isomorphic to the direct sum sequence, $$ 0 \rightarrow A \rightarrow A \oplus C \rightarrow C \rightarrow 0 $$ by an isomorphism $B \rightarrow A \oplus C$ which induces the identity on $A$ and on $C$. Sufficient conditions for an exact sequence \eqref{eq:1} to be split are the existence of a right inverse $f'$ for $f$, or of a left inverse $g'$ for $g$. The class of split exact sequences is the zero of the group $\mathrm{Ext}_R^1(A,C)$ (see Baer multiplication). In a category of vector spaces (that is, of modules over a fixed field) every exact sequence splits.

For relative homological algebra, the typical situation is to consider exact sequences in one category which split in another.

References

[a1]	S. MacLane, "Homology" , Springer (1963) pp. 16, 260 Zbl 0133.26502