Proposition 15

Given an exact sequence of $A$-modules

the following propositions are equivalent: (a) The submodule $f(E)$ of $F$ is a direct factor. (b) There exists a linear retraction $r : F \to E$ associated with $f$ (Set Theory, II, § 3, no. 8, Definition 11). (c) There exists a linear section $s : G \to F$ associated with $g$ (Set Theory, II, § 3, no. 8, Definition 11). When this is so, $f + s : E \oplus G \to F$ is an isomorphism.

If there exists a projector $e$ in $\mathrm{End}(F)$ such that $e(F) = f(E)$, the homomorphism $f^{-1} \circ e : F \to E$ is a linear retraction associated with $f$; conversely, if there exists such a retraction $r$, it is immediate that $f \circ r$ is a projector in $F$ whose image is $f(E)$, hence (a) and (b) are equivalent (Proposition 14). If $f(E)$ admits a supplement $E'$ in $F$ and $j : E' \to F$ is the canonical injection, $g \circ j$ is an isomorphism of $E'$ onto $G$ and the inverse isomorphism, considered as a mapping of $G$ into $F$, is a linear section associated with $g$. Conversely, if such a section $s$ exists, $s \circ g$ is a projector in $F$ whose kernel is $f(E)$, hence (a) and (c) are equivalent (Proposition 14). Moreover $s$ is a bijection of $G$ onto $s(G)$ and as $s(G)$ is supplementary to $f(E)$, $f + s$ is an isomorphism.

Note that being given $r$ (resp. $s$) is equivalent to being given a supplement of $f(E)$ in $F$, namely the kernel of $r$ (resp. image of $s$).

When the exact sequence (33) satisfies the conditions of Proposition 15, it is said to split or $(F, f, g)$ is said to be a trivial extension of $G$ by $E$ (I, § 6, no. 1).