 
Summary: PERIODIC FLAT MODULES, AND FLAT MODULES FOR FINITE GROUPS
D. J. BENSON AND K. R. GOODEARL
Abstract. If R is a ring of coefficients and G a finite group, then a flat RGmodule which is
projective as an Rmodule is necessarily projective as an RGmodule. More generally, if H is a
subgroup of finite index in an arbitrary group \Gamma, then a flat R\Gammamodule which is projective as an
RHmodule is necessarily projective as an R\Gammamodule. This follows from a generalization of the
first theorem to modules over strongly Ggraded rings. These results are proved using the following
theorem about flat modules over an arbitrary ring S: If a flat Smodule 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.
1. Introduction
In the representation theory of finite groups, a great deal of attention has been given to the
problem of determining whether a module over a group ring is projective. For example, a well
known theorem of Chouinard [13] states that a module is projective if and only if its restriction
to each elementary abelian subgroup is projective. A theorem of Dade [16] states that over an
algebraically closed field of characteristic p, a finitely generated module for an elementary abelian
pgroup is projective if and only if its restriction to each cyclic shifted subgroup is projective,
where a cyclic shifted subgroup is a certain kind of cyclic subgroup of the group algebra. For an
infinitely generated module, the statement is no longer valid, but in [8] it is proved that an infinitely
