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Math Help - snake lemma, exact at Coker f

  1. #1
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    snake lemma, exact at Coker f

    Prove that the long exact sequence in the statement of the snake lemma is exact at the term \text{coker}\hspace{.03in}f. The long exact sequence is: \text{ker}\hspace{.03in}f \xrightarrow{\alpha_0} \text{ker}\hspace{.03in} g \xrightarrow{\beta_0} \text{ker}\hspace{.03in} h \xrightarrow{\delta} \text{coker}\hspace{.03in} f \xrightarrow{\overline{\alpha'}} \text{coker} \hspace{.03in}g \xrightarrow{\overline{\beta'}} \text{coker}\hspace{.03in} h. The diagram is below.

    Last edited by Erdos32212; November 20th 2008 at 08:53 PM.
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  2. #2
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    Quote Originally Posted by Erdos32212 View Post
    Prove that the long exact sequence in the statement of the snake lemma is exact at the term \text{coker}\hspace{.03in}f. The long exact sequence is: \text{ker}\hspace{.03in}f \xrightarrow{\alpha_0} \text{ker}\hspace{.03in} g \xrightarrow{\beta_0} \text{ker}\hspace{.03in} h \xrightarrow{\delta} \text{coker}\hspace{.03in} f \xrightarrow{\overline{\alpha'}} \text{coker} \hspace{.03in}g \xrightarrow{\overline{\beta'}} \text{coker}\hspace{.03in} h. The diagram is below.

    what i don't like about homological algebra is "diagram chasing" ... it's really annoying! anyway, since the proof is quite long and it can be found almost anywhere, i won't bring it in here.

    there's a very detailed and extremely easy-to-understand proof of the lemma in here. (page 3, from line 6 to the end of the page is the solution to your problem)
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