Last time in Wild Spectral Squences 2, we saw how to prove the five lemma using a spectral sequence. Today, we’ll see a very simple application of spectral sequences to the concept of cohomological dimension in group cohomology.

We will define the well-known concept of cohomological dimension of a group $ G$, and then show how the dimension of $ G$ relates to the dimension of $ G/N$ and $ N$ for a normal subgroup $ N$. We do this with a spectral sequence. Although this application will appear to be very simple, it might be a good exercise for those just learning about spectral sequences.

### The Category

For the sake of conreteness, let us work in the category of $ G$-modules where $ G$ is a profinite group. The $ G$-modules are the $ \mathbb{Z}G$-modules $ A$ with a continuous action of $ G$ where $ A$ is given the discrete topology, but we could be working with any group $ G$ with suitable, minor modifications, or Lie algebras, etc.

Like in all homology theories, there is the notion of cohomological dimension: a profinite group $ G$ has cohomological dimension $ n\in \mathbb{N}$ if for every $ r > n$ and every torsion $ G$-module $ A$, the group $ H^r(G,A)$ is trivial. If no such $ n$ exists, we say that $ G$ has cohomological dimension $ \infty$. The usual arithmetic rules in working with $ \infty$ apply.

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