21.71 (2026)

For a ring $R$, we say that a group $G$ is of type $\mathbf{FL}(R)$ if the trivial $RG$-module $R$ admits a finite resolution by finitely generated free modules. If $R$ is a field, we define the Euler characteristic of $G$ over $R$ to be the alternating sum of $R$-ranks of homology groups $H_i(G; R)$.

Does there exist a group of type $\mathbf{FL}(\mathbb{F}_i)$ for two fields $\mathbb{F}_1$ and $\mathbb{F}_2$ such that the Euler characteristics of the group over the fields $\mathbb{F}_1$ and $\mathbb{F}_2$ differ?