# Wild Spectral Sequences Ep. 3: Cohomological Dimension

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.

(If we drop "torsion", we get the notion of strict cohomolgical dimension, so the terminology here is unfortunate, but it has stuck through history.)

We denote the cohomological dimension of $G$ by $\mathrm{cd}(G)$.

Let $N$ be a normal subgroup of $G$. Of course, we would like to compare the three numbers $\mathrm{cd}(G)$, $\mathrm{cd}(G/N)$ and $\mathrm{cd}(N)$. In general, they will not be equal, but at the least we would like $\mathrm{cd}(G) \leq \mathrm{cd}(G/N) + \mathrm{cd}(N)$. Under certain conditions we will get equality, but the purpose of today is just to prove the inequality, and this is something we can do with a spectral sequence.

### The Spectral Sequence

We use the Hochschild-Serre spectral sequence for the group $G$, the normal subgroup $N\leq G$, and $G$-module $A$:

$E_2^{p,q} = H^p(G/N,H^q(N,A)) \Rightarrow H^{p+q}(G,A)$.

This is a special case of the Grothendieck spectral sequence for the composition of functors $-^G = ((-)^N)^{G/N}$. Now let $m = \mathrm{cd}(G/N)$ and $n = \mathrm{cd}(N)$. Notice that if $p > m$ then $E_2^{p,q} = 0$ and if $q > n$ then $E_2^{p,q} = 0$ as well! However, even if $p > m$, this does not imply that $H^p(G,A) = 0$, since there for a fixed $r$ there are many ways to write $r = p + q$.

We recall that the $H^{r}(G,A)$ piece will have a canonical filtration, whose subquotients are isomoprhic to the $E_2^{p,q}$ terms for all $p + q = r$. Hence in order to conclude $H^r(G,A) = 0$, we need $E_2$ to vanish for all possible $p$ and $q$ such that $r = p + q$.

In other words, for such an $r$ and every $p$ and $q$ such that $r = p + q$, either $p > m$ or $q > n$. If $r > m + n$, then certainly at least one of $p > m$ or $q > n$ must be true for every $r = p + q$. If $r \leq m + n$ then we cannot guarantee this without further information.

Thus the best we can do without any further information is that $r > m + n$. In other words, we known that $H^r(G,A) = 0$ for every torsion $A$ as long as $r > m + n$, which by definition shows that

$\mathrm{cd}(G)\leq \mathrm{cd}(N) + \mathrm{cd}(G/N)$.

Exercise. Provide a proof of this without using spectral sequences.