Cohomological dimension

Cohomological dimension arises in many things I read, but my familiarity with it is superficial. What's a good source for understanding cohomological dimension on its various examples, fascets, theorems, and applications, basically providing the maximal intuition-building narative?

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Cohomological dimension of what? There is a notion of cohomological dimension for each cohomology theory, and there are far too many cohomology theories to give a sensible answer. – Angelo May 31 '10 at 4:44

4 Answers

I'm not sure if I know a good source, but I can make a remark.

In the category of topological spaces up to homotopy, there is a highly nontrivial geometric invariant that people have taken as interesting: What is the lowest dimension of a topological representative of a given homotopy type? For instance if you have a group $G$, what is the lowest possible dimension of a classifying space $B_G$? The cohomological dimension is a lower bound for this geometric invariant which is more algebraic and sometimes more tractable.

For example, if you have a group $G$, it may not be easy to tell whether it is the fundamental group of a compact, hyperbolic $n$-manifold. (Or a Euclidean manifold or a non-positively curved manifold.) If you can compute its group cohomology, then you'll learn something about it, because there is such an $n$-manifold, then the cohomological dimension of $G$ is exactly $n$.

As another example, if $G$ is a non-trivial finite group, then it is not hard to compute that its cohomology is periodic and thus known that its cohomological dimension is infinite. Thus, any finite-dimensional CW complex with a non-trivial but finite fundamental group has to have higher homotopy groups.

Maybe a good reference is Brown's GTM book on cohomology of groups.

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Very few finite groups have periodic cohomology. The conclusion that all non-trivial finite $G$ have cohomology in arbitrarily high dimension is of course still true. – Torsten Ekedahl May 30 '10 at 19:20
Hmmm....I don't seem to know as much about this as I thought. – Greg Kuperberg May 30 '10 at 19:53
Then you will no doubt enjoy learning about groups with periodic cohomology. They are completely classified and Swan proved that a finite group has periodic cohomology iff it admits a free action on a CW-complex homotopic to a sphere. – Torsten Ekedahl May 30 '10 at 20:28
A simple explanation for all nontrivial finite groups having infinite homological dimension is that the homological dimension of any Frobenius algebra is either zero or infinity. – Leonid Positselski May 30 '10 at 22:17
Indeed,there is a wonderful proof that is short enough to fit into one of these comments... Consider a finite projective resolution of a module. Shifted it is also a finite resolution of the left hand side which is injective. As all but the last modules in that resolution are injective the last on must also be. Hence any module with a finite projective resolution must be injective and hence projective. – Torsten Ekedahl May 31 '10 at 6:18

Cohomological dimension arises in different contexts, algebraic topology, algebraic geometry, group theory... So I think that it would be difficult to give a single reference for this broad topic.

It was already mentioned that Brown "cohomology of groups" has a chapter on it, and I think this is a good reference for cohomological dimension of groups. However cohomological dimension also arises in the context of topology, and when the space is well-behaved, say a manifold, it just coincides with the usual topological dimension. So it is perhaps better to start building intuition in that context.

Topological dimension is defined with covers, so Cech cohomology, which is also defined using covers, is perhaps the best cohomology to start with in order to understand the relation between the two notions. Cech cohomology is used for practical computations in algebraic geometry, so it will be useful if you are interested in that subject. There is a chapter on Cech cohomology in Hartshorne, but there may be better references more focused on dimension in the algebraic setting. If you are interested in the differential viewpoint, then Bott-Tu "Differential forms in algebraic topology" is a good reference.

Bredon, "sheaf theory", has a chapter on dimension from the topological viewpoint. Here is what I learnt from it. You may expect that if X is a well-behaved subset of Y, the cohomological dimension of X (that is, the greatest integer, or $\infty$) is less than or equal to the dimension of Y. This is true for some cohomologies, like the sheaf cohomology with constant coefficients (X,Y compact), but it is false for others. In particular, singular cohomology behaves a bit erratically with respect to cohomological dimension. Barratt and Milnor built in 62 an example of a compact manifold in R^3 (infinitely spheres of radius 1/n tangent to xy plane at the origin) with infinite singular cohomological dimension. This is definitely surprising !

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In the context of rings and modules, cohomological dimension is a measure of the complexity you need in presenting modules. Using that idea as a guide, you'll see sense in lots of results concerning dimension.

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This question is overly broad (should we introduce a tag "tell me about X" for these occasions?), but in the case of cohomological dimension 1, where the groups in question are close to free groups, here is a good reference:

MR0584790 Dicks, Warren. Groups, trees and projective modules. Lecture Notes in Mathematics, 790.

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I assume you've already seen en.wikipedia.org/wiki/Cohomological_dimension – Victor Protsak May 31 '10 at 4:42