# Thread: snake lemma, exact at Coker f

1. ## 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.

2. Originally Posted by Erdos32212
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)