We establish a sufficient condition for some modules M over the group algebra ℤ[G ] to be of homological type FP2, where G is a finitely generated split extension of abelian groups. This generalizes a result of Bieri and Strebel [R. Bieri and R. Strebel. Valuations and finitely presented metabelian groups. Proc. London Math. Soc. (3) 41 (1980), 439–464] when M is the trivial module ℤ and it establishes a special case of [D. H. Kochloukova. A new characterisation of m -tame groups over finitely generated abelian groups. J. London Math. Soc. (2) 60 (1999), 802–816, Conjecture K. S. Brown. Cohomology of groups (Springer-Verlag, 1982)].
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