Homologically trivial actions on chain complexes and finiteness properties of infinite groups (Q1328775)

The author uses the technique for grafting together finite chain complexes with the homology of a sphere introduced by \textit{R. G. Swan} [Ann. Math. (2) 72, 267--291 (1960; Zbl 0096.01701)] to prove the following result: Let \(G\) be a group and \(R\) a commutative Noetherian ring. Then \(R\) as trivial \(G\)-module has an \(RG\)-projective resolution by finitely generated modules \((: G\) is of type \(FP (\infty, R))\) if and only if there is an \(R\)-augmented chain complex of finitely generated projective \(RG\)-modules such that the homology groups are finitely generated over \(R\) and trivial over \(G\).