Theorem 5

Tag 05SE, permanent. Cite it as [Bourbaki, Tag 05SE].

Given two exact sequences of vector spaces (over the same field K*) and linear mappings*

\begin{align*} 0 &\to E' \to E \to E'' \to 0 \\ 0 &\to F' \to F \to F'' \to 0 \end{align*}

and two vector spaces G, H over K, the corresponding sequences

\begin{align*} 0 &\to \mathrm{Hom}(E'', G) \to \mathrm{Hom}(E, G) \to \mathrm{Hom}(E', G) \to 0 \\ 0 &\to \mathrm{Hom}(H, F') \to \mathrm{Hom}(H, F) \to \mathrm{Hom}(H, F'') \to 0 \end{align*}

are exact and split.

This follows from the fact that every vector subspace is a direct factor (no. 3, Proposition 4) and from § 2, no. 1, Propositions 1 and 2.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by