3
$\begingroup$

Is it true that if I have exact sequences of abelian groups

$0 \rightarrow A\rightarrow B \rightarrow C\rightarrow 0$

and

$ A_1\rightarrow B_1 \rightarrow C_1 $ (exact only in the middle)

and morphisms

$a:A\rightarrow A_1$, $b:B\rightarrow B_1$ and $c:C\rightarrow C_1$ such that $a$ and $c$ are the zero morphism, then also $b$ is the zero morphism?

Thx

  • 1
    Morphisms $a$, $b$ and $c$ in your question are arbitrary. Maybe you also imply that the appropriate diagram is commutative?2012-12-11

1 Answers 1

3

No this is not true. Consider the diagram below: enter image description here