If R is a ring of coefficients
and G a finite group, then a flat RG-module which is projective as an R-module is
necessarily projective as an RG-module. More generally, if H is a subgroup of finite
index in an arbitrary group Γ, then a flat RΓ-module which is projective as an
RH-module is necessarily projective as an RΓ-module. This follows from a
generalization of the first theorem to modules over strongly G-graded rings. These
results are proved using the following theorem about flat modules over an arbitrary
ring S: If a flat S-module M sits in a short exact sequence 0 → M → P → M → 0
with P projective, then M is projective. Some other properties of flat and projective
modules over group rings of finite groups, involving reduction modulo primes, are
also proved.
