D. J. Benson and K. R. Goodearl

Periodic flat modules, and flat modules for finite groups

\begin{abstract}
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 $\Gamma$, then a flat
$R\Gamma$-module which is projective as an $RH$-module is necessarily
projective as an
$R\Gamma$-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 $0 \ra M \ra P \ra M \ra 0$ 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.
\end{abstract}