# On a commutative diagram

Let a commutative diagram:

\begin{array}{ccccccccc} 0 & \longrightarrow & A & \overset{f}{\longrightarrow} & B & \overset{g}{\longrightarrow} & C & \longrightarrow & 0\\ & & \alpha\downarrow & & \beta\downarrow & & \gamma\downarrow\\ 0 & \longrightarrow & A' & \overset{f'}{\longrightarrow} & B' & \overset{g'}{\longrightarrow} & C' & \longrightarrow & 0 \end{array}

The two lines are exact sequences and $\beta$ is $R$-isomorphism

Prove that $\alpha$ is monomorphism, $\gamma$ is epimorphism. Furthermore $\alpha$ is isomorphism iff $\gamma$ is isomorphism

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Not again 5 lemma question. The proof is there math.msu.edu/~rotthaus/teaching/910/homework/solhw5.pdf. Not hard to find in google. –  user52045 Nov 26 '13 at 9:12
what have you tried?? what do you know about this particular commutative diagram –  Praphulla Koushik Nov 26 '13 at 9:12
You don't want the 5 lemma, you want the snake lemma: en.wikipedia.org/wiki/Snake_lemma –  Kevin Nov 26 '13 at 9:22
Can you explain more precise? –  user109584 Nov 26 '13 at 15:00
I have made it precise, resolving this question –  Alexander Grothendieck Dec 7 '13 at 13:02