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Let $A\xrightarrow{f} B\xrightarrow{g} C$ be an exact sequence of finitely generated abelian groups with $B\cong A\oplus C$.

Question: Is the sequence short exact?

If $B$ is finite, the answer is clear positive.

Is there a counterexample to show that the finiteness assumption on $B$ is vital? Thanks in advance!

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  • $\begingroup$ Add one more assumption: $A,C$ are nontrivial. $\endgroup$
    – LipCaty
    Mar 8, 2022 at 11:36
  • $\begingroup$ This might be of help. $\endgroup$
    – mrtaurho
    Mar 8, 2022 at 23:04

1 Answer 1

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Specialize the claim to $A=0$ and $B=C$. This means that all injective homomorphisms are surjective, which needs not be true (see multiplication by a constant on $\Bbb Z$).

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  • $\begingroup$ Thanks, you are right. I want to add some ''nice'' conditions to make the original sequence short exact, so I think I forgot the assumption that $A,C$ are nontrivial. $\endgroup$
    – LipCaty
    Mar 8, 2022 at 11:31

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