Modules of Type FP2 over the Integral Group Algebra of a Metabelian Group

We demonstrate a sufficient condition for some modules $M$ over the group algebra $\BZ[G]$ to be of homological type $FP_2$, where $G$ is a finitely generated split extension of abelian groups. This generalises a result of Bieri-Strebel when $M$ is the trivial module $\BZ$ \cite{B-S1} and is a special case of a conjecture suggested in \cite[Conj.~7]{K4}.