Posted by Jason Polak on 21. November 2012 · Write a comment · Categories: homological-algebra · Tags: ,

Welcome ladies and gentlemen to a new feature on AZC called Wild Spectral Sequences. This will be a regular feature on spectral sequences for as long as I can find new examples of using spectral sequences. Showing a small, neat, easily digestible example of using spectral sequences will be the aim of each episode.

Now, it is widely believed that spectral sequences are scary and dangerous. Of course this is not true, and in fact spectral sequences are 100% safe for children eight and older! The prerequisites of these posts will be essentially basic knowledge of spectral sequences, such as the kind that can be found in Weibel’s book or in many other texts.

Today’s feature is the following: prove the snake lemma using spectral sequences. Recall:

Snake Lemma. Given an diagram

in an Abelian category with exact rows, there is a long exact sequence

$ 0\to \ker(f)\to\ker(g)\to\ker(h)\to\mathrm{coker}(f)\to\mathrm{coker}(g)\to\mathrm{coker}(h)\to 0$.

Read on for the solution! (But try it first: it’s fun!) More »