# Rational cohomological dimension of a locally finite group

$$\DeclareMathOperator\cd{cd}$$Recall that the rational cohomological dimension of a group $$G$$ is the supremum of the set of integers $$k$$ such that there exists a $$\mathbb{Q}[G]$$-module $$M$$ with $$H^k(G;M) \neq 0$$. Denote this by $$\cd_{\mathbb{Q}}(G)$$.

If $$G$$ is a finite group, then it is easy to see that $$\cd_{\mathbb{Q}}(G) = 0$$.

However, it is not clear to me what the rational cohomological dimension of a locally finite group is (where "locally finite" means that all finitely generated subgroups are finite). Since homology commutes with direct limits, the rational homological dimension of these group are $$0$$.

So my question is what can be said about the rational cohomological dimension of a locally finite group $$G$$. I mostly care about countable $$G$$. In fact, the easiest example where I don't know the answer is $$G = \bigoplus_{k=1}^{\infty} \mathbb{Z}/2\mathbb{Z}.$$

## 1 Answer

Edited version. If $$G$$ is a countable infinite locally finite group, then the rational cohomological dimension is exactly $$1$$. The rational cohomological dimension for an infinite group is never $$0$$ because if $$\mathbb Q$$ is a projective $$\mathbb QG$$-module, then you need an idempotent $$e$$ with $$\mathbb QGe\cong \mathbb Q$$ and that is impossible unless $$G$$ is finite and $$e$$ is the average of all elements of $$G$$.

Here are two proofs that the projective dimesion of $$\mathbb QG$$ is at most one for a countable locally finite group. This amounts to showing the augmentation ideal is projective. Then $$\mathbb Q$$ has projective dimension at most $$1$$ since $$I\to \mathbb QG\to \mathbb Q\to 0$$ is a projective resolution where $$I$$ is the augmentation ideal.

It follows from a result in Dicks book Groups, trees and projective modules (1980) characterizing projectivity of the augmentation ideal of $$RG$$ that $$\mathbb QG$$ has a projective augmentation ideal whenever $$G$$ is countable and locally finite. See this question for more details.

The second argument is that by a result of Connell, $$\mathbb QG$$ is von Neumann regular if $$G$$ is locally finite. The augmentation ideal is countably generated as a left ideal if $$G$$ is countable and hence by a result of Kaplansky is projective (Kaplansky proved that a countably generated left ideal in a von Neumann regular ring is projective).

Thus for a countably infinite locally finite group the rational cohomological dimension is one.

It follows from Theorem 2 of Derek Holt's 1981 paper (DOI link) that if $$G$$ is a locally finite group of cardinality $$\aleph_1$$, then the rational cohomological dimension of $$G$$ is at least $$2$$.

• There's one thing I know making a big difference between countable and uncountable locally finite groups: $H^1(G,\mathbf{Z}G)$ is infinite when $G$ is infinite countable locally finite, while it's zero for $G$ locally finite uncountable. This former (easy) fact is due to Scott-Sonneborn; the latter is due du Holt.
– YCor
Sep 23 at 19:25
• @YCor, he shows H^2 is not zero when G is uncountable Sep 23 at 20:36
• Browsing subsequent papers (see other papers by Holt in 1981 and those quoting him, etc), there seems to be a few more papers on this, but I'm not sure what they exactly conjecture or say (as it doesn't only focus on rational cd, and also because some make set-theoretic assumptions). Does it conjecture that a locally finite group of cardinal $\aleph_n$ has rational cohomological dimension $n$, and infinite when $n\ge\aleph_0$? Is part of the conjecture known (e.g., is it known that the cohomological dimension is $\ge n$? is it known for abelian groups? etc.
– YCor
Sep 24 at 7:22
• @YCor, in Derek's paper or seemed to conjecture at least n. Maybe Derek will chime in. Sep 24 at 11:00