I am reading a paper where they write the following:
Since $S_{0}$ is a group of type $FP_{n}(\mathbb{Q})$ and there is a series $$S_{0}\triangleleft S_{1}\triangleleft \cdots \triangleleft S_{l}=T$$ where each $S_{i+1}/S_{i}$ is finite or infinite cyclic, then by the obvious induction, $T$ is of type $FP_{n}(\mathbb{Q})$.
I understand that if $S_{i+1}/S_{i}$ is finite and $S_{i}$ of type $FP_{n}(\mathbb{Q})$, then so is $S_{i+1}$, because that property is inherited in finite extensions. Nevertheless, I really can't prove the same when $S_{i+1}/S_{i}$ is infinite cyclic.
Why is this true? Maybe it does not always hold, but in my case yes: I am working in direct products of limit groups, so $S_{0}$ is a full subdirect product of limit groups, $S_{0}\leq \Gamma_{1}\times \cdots \times \Gamma_{n}$.
If $0\mapsto A\mapsto B\mapsto C \mapsto $ is a short exact sequence and $A$, $C$ are of type $FP_{n}$, then so is $B$ (https://arxiv.org/pdf/1611.03759.pdf Prop 2.2).