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I encountered this question when I am working on Proposition 2.9 from Atiyah-Macdonald.

Let $M_1 \xrightarrow{f} M_2 \xrightarrow{g} M_3$ be a sequence of $R$-module.

Let $D$ be any $R$-module, and $Hom(M_3,D)\xrightarrow{\tilde{g}}Hom(M_2,D) \xrightarrow{\tilde{f}} Hom(M_1,D)$ be the induced sequence.

Hopefully, if my calculations are correct, I know that:

  1. If the induced sequence is exact for ANY $R$-module $D$, then the original sequence is exact.

  2. Furthermore, if the original sequence is exact and $g$ is surjective, then the induced sequence is exact for any $R$-module $D$.

So, I would like to know: Is there a counter-example where $g$ is not surjective, the original sequence is exact, but the induced sequence is not exact for some $D$?

Thanks.

GradStudent001
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1 Answers1

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The sequence is exact for every $D$ iff $M_1 \to M_2 \to M_3$ is exact and $\mathrm{im}(g)$ is a direct summand of $M_3$. See here.

Community
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Martin Brandenburg
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  • Hi Martin, thanks for the help!

    I am actually hoping to find an exact sequence, such that the induced sequence is not exact for some D.

    Thanks for the hint! I will continue to work on it.

     – GradStudent001 Oct 23 '13 at 14:41
  • Well, this is also answered using the criterion. If $\mathrm{im}(g)$ is not a direct summand, then you can find some $D$ such that the sequence is not exact. – Martin Brandenburg Oct 23 '13 at 22:51