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  1. What is an intuitive, straightforward explanation of the cohomological dimension of a group ?

  2. How does one compute the cohomological dimension of a group ?

  3. Is there a good reference that explains this concept and provides examples ?

I am particularly interested in the cohomological dimension of the braid group $\mathcal{B}_n$ and some of its subgroups.

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The first three questions can be answered in two words: read Brown. – t.b. Jun 29 '11 at 8:19

Recall that group cohomology of $G$ are just cohomology of topological space $BG$ (in general, with coefficients in some local system). In particular, if there is a model of $BG$ that is $n$-dimensional CW-complex, then cohomological dimension of $G$ is $\le n$.

Now, there is a very nice model for $BB_n$: the space of $n$ (indistinguishable distinct) points of $\mathbb C$. Vieta map identify this space with $\mathbb C^n\setminus\Delta$ (where $\Delta$ is some kind of discriminant subset). In particular, cohomological dimension of $B_n$ is $\le 2n-1$ (oh... I think, the answer is, in fact, more like $n$). Canonical reference here is Arnold's “On some topological invariants of algebraic functions”, AFAIR.

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One reason to care about the cohomological dimension of a group comes from etale cohomology, because etale cohomology over a field is the same thing as Galois cohomology. (Namely, if $k$ is a field, etale sheaves on $\operatorname{Spec} k$ are the same thing as continuous discrete $G$-sets for $G$ the Galois group of the separable closure, and cohomology of an abelian sheaf is (profinite) group cohomology.) As a result, if one wants to prove vanishing theorems in etale cohomology (the most important of which states that for a variety of dimension $n$ over an algebraically closed field, the cohomology groups of any torsion sheaf vanish in degrees $>2n$) the basic case one is often reduced to is that of a field. It thus becomes necessary to find bounds for the cohomological dimension of Galois groups.

(Henceforth, I am using the definition of cohomological dimension for torsion modules.)

To actually compute this, one can use the following fact: $G$ has cohomological dimension $\le n$ if and only if, for each $p$, there is a $p$-Sylow subgroup $G_p \subset G$ such that $H^{n+1}(G_p, \mathbb{Z}/p) = 0$. The justification is that any finitely generated $p$-torsion $G_p$-module has a finite filtration with quotients isomorphic to $\mathbb{Z}/p$, and after that one can use restriction and inflation to get the result for $G$. In practice one way to show the vanishing of these groups is to use certain exact sequences, for instance

$$0 \to \mathbb{Z}/p \to k^{sep* } \to k^{sep *} \to 0$$

where the last map is raising to the $p$th power. (When $p$ is the characteristic, this should be replaced by $a \mapsto a^p -a $ and one uses the additive group.) Since there are many theorems on the cohomology of $k^{sep*}$ (keywords: Brauer group, Tate's theorem, Hilbert's theorem 90) and that of $k_{sep}$ (this is actually trivial by the normal basis theorem), one can often use them to get results about cohomological dimension.

A very fun reference for bounding cohomological dimension (but with no mention of etale cohomology) is Serre's book "Galois cohomology."

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I imagine that this characterization is only supposed to apply to finite groups $G$? – Tom Church Mar 30 '13 at 22:33
@Tom : It can be generalized to profinite groups (if I remember correctly Serre talks about the notion of a Sylow subgroup of a profinite group in "Galois cohomology"). – Akhil Mathew Mar 31 '13 at 23:07

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