On a conjecture of Moore

This can be regarded as a generalization of Serre’s Theorem that every torsion-free group of finite virtual cohomological dimension has finite cohomological dimension (see [2]). For suppose that G is torsion-free and that H has cohomological dimension n < co. Then the nth syzygy in any projective resolution of Z over ZG is projective as ZH-module, and Moore’s Conjecture implies at once that it is also projective as ZG-module. One of the special cases of the conjecture which we prove here also implies Serre’s Theorem. It is natural to generalize the conjecture, because, in addition to group rings, it makes perfect sense for crossed products and in fact for strongly group-graded rings. Recall that a G-graded ring is a ring A with a direct sum decomposition