Exercise 5, § 1

Commutative Algebra, I, 40

Suppose that an exact sequence $A \xrightarrow{u} A \xrightarrow{v} A'' \to 0$ and two surjective homomorphisms $B' \xrightarrow{a'} A',\ B'' \xrightarrow{a''} A''$ are given, where $A,\ A',\ A,\ B',\ B''$ are modules over the same ring. Show that, if $B''$ is a projective module, there exists a surjective homomorphism $a : B' \oplus B'' \to A$ such that the diagram

\begin{array}{ccc} B' & \xrightarrow{i} & B' \oplus B'' \xrightarrow{p} B'' \\ a' \downarrow & & a \downarrow \quad a'' \downarrow \\ A' \xrightarrow{u} & & A \xrightarrow{v} A'' \end{array}

is commutative ( $i$ and $p$ being the canonical mappings).

Solution

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