Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.
  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.

share|improve this question
6  
The first three questions can be answered in two words: read Brown. –  t.b. Jun 29 '11 at 8:19
    

2 Answers 2

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.

share|improve this answer

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."

share|improve this answer
    
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.