Rational cohomological dimension of a locally finite group

Question

$\DeclareMathOperator\cd{cd}$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 $$G = \bigoplus_{k=1}^{\infty} \mathbb{Z}/2\mathbb{Z}.$$

Answer 1

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 $2$.