View source for 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.



====Comments====


====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  S. MacLane,   "Homology" , Springer  (1963)  pp. 16, 260  {{ZBL|0133.26502}}</TD></TR>
</table>

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