Exercise 2, § 1

Commutative Algebra, I, 39

Consider a commutative diagram of commutative groups

\begin{array}{ccccc} A & \xrightarrow{u} & B & \xrightarrow{v} & C \\ a \downarrow & & b \downarrow & & c \downarrow \\ A' & \xrightarrow{u'} & B' & \xrightarrow{v'} & C' \\ a' \downarrow & & b' \downarrow & & \\ A'' & \xrightarrow{u''} & B'' \end{array}

Suppose that: (1) $(u, v)$ and $(b, b')$ are exact sequences; (2) $v' \circ u' = 0$ and $a' \circ a = 0$ ; (3) $c$ and $u'$ are injective and $a'$ is surjective. Show that under these conditions $u''$ is injective.

Solution

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