Math Reviews: 51:10442
Suppose that G is a finitely generated nilpotent group, and that M is a finitely generated module over Z[G]. The main theorem of the paper states that if the zero dimensional group homology of G with coefficients in M vanishes, then all of the higher homology groups of G with coefficients in M also vanish. In the special case in which G is an infinite cyclic group, this is closely related to the statement that if f is a surjective endomorphism of a finitely generated module over a noetherian ring, then f is actually an automorphism (think of the increasing sequence of kernels of powers of f). The proof relies heavily on the fact that Z[G] is noetherian.
The results in this paper were used in Exotic convergence of the Eilenberg-Moore spectral sequence and other places.
Related and in some cases stronger vanishing theorems were obtained about the same time by Brown and Dror (52:10809 Brown, Kenneth S.; Dror, Emmanuel The Artin-Rees property and homology. Israel J. Math. 22 (1975), no. 2, 93--109) and by Robinson (53:8280 Robinson, Derek J. S. The vanishing of certain homology and cohomology groups. J. Pure Appl. Algebra 7 (1976), no. 2, 145--167).