  We show that a virtually RFRS group $G$ of type $\mathrm{FP}_n(\mathbb{Q})$
virtually algebraically fibres with kernel of type $\mathrm{FP}_n(\mathbb{Q})$
if and only if the first $n$ $\ell^2$-Betti numbers of $G$ vanish, that is,
$b_p^{(2)}(G) = 0$ for $0 \leqslant p \leqslant n$. We also offer a variant of
this result over other fields, in particular in positive characteristic.
  As an application of the main result, we show that virtually amenable RFRS
groups of type $\mathrm{FP}(\mathbb{Q})$ are polycyclic-by-finite. It then
follows that if $G$ is a virtually RFRS group of type $\mathrm{FP}(\mathbb{Q})$
such that $\mathbb{Z}G$ is Noetherian, then $G$ is polycyclic-by-finite. This
answers a longstanding conjecture of Baer for virtually RFRS groups of type
$\mathrm{FP}(\mathbb{Q})$.
