counterexample to "symmetric" nine lemma

Question

Consider a commutative diagram of $R$-modules

$$\require{AMScd}\begin{CD} @. 0 @. 0 @. 0\\ @. @VVV @VVV @VVV\\ 0@>>>A_1@>a>>A_2@>a'>>A_3@>>>0\\ @. @VVfV@VVgV @VVhV\\ 0@>>>B_1@>b>>B_2@>b'>>B_3@>>>0\\ @. @VVf'V @VVg'V @VVh'V\\ 0@>>>C_1@>c>>C_2@>c'>>C_3@>>>0\\ @. @VVV @VVV @VVV\\ @. 0 @. 0 @. 0\\ \end{CD}$$

such that the three rows and the first and third columns are short exact sequences.

I am looking for a counterexample to the claim that the middle column is exact.

I know it to be true that if $\text{Im } g \subset \ker g',$ then the middle column is in fact exact.

I was able to show via diagram chase that if $C_1$ is $0$, then the middle column is exact.

AMScd is not so good. It can't draw diagonal arrows. The best modern solutions are tikz-cd and _the psmatrix environment from pst-node. — Bernard, Jan 06 '15 at 17:11
I am always confused when s.e.s. are written $A\to B\to C$ insetad of $0\to A\to B\to C\to 0$ ... — Hagen von Eitzen, Jan 11 '15 at 13:18

score 13 · Accepted Answer · answered Jan 09 '15 at 10:36

13

$$\require{AMScd}\begin{CD} 0@>>>R@>1>>R\\ @VVV@VV\begin{pmatrix}1\\0\end{pmatrix}V @VV1V\\ R@>\begin{pmatrix}1\\1\end{pmatrix}>>R^2@>\begin{pmatrix}1&-1\end{pmatrix}>>R\\ @VV1V @VV\begin{pmatrix}1&0\end{pmatrix}V @VVV\\ R@>1>>R@>>>0\\ \end{CD}$$

answered Jan 09 '15 at 10:36

Jeremy Rickard

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counterexample to "symmetric" nine lemma

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