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

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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 $2$.

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  • $\begingroup$ 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. $\endgroup$
    – YCor
    Sep 23 at 19:25
  • $\begingroup$ @YCor, he shows H^2 is not zero when G is uncountable $\endgroup$ Sep 23 at 20:36
  • $\begingroup$ 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. $\endgroup$
    – YCor
    Sep 24 at 7:22
  • $\begingroup$ @YCor, in Derek's paper or seemed to conjecture at least n. Maybe Derek will chime in. $\endgroup$ Sep 24 at 11:00

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