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



Rational cohomological dimension of a locally finite group
**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](https://mathoverflow.net/questions/297043/using-dunwoodys-results-on-cohomological-dimension-to-learn-about-a-von-neumann?rq=1) 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)](https://doi.org/10.1112/blms/13.6.557) that if $G$ is a locally finite group of cardinality $\aleph\_1$, then the rational cohomological dimension of $G$ is at least $2$.


