 
Summary: PERIODIC FLAT MODULES, AND FLAT MODULES FOR FINITE GROUPS
D. J. BENSON AND K. R. GOODEARL
Abstract. The main theorem says that if R is a ring of coefficients and G a finite group, then a
flat RGmodule M which is projective as an Rmodule is necessarily projective as an RGmodule.
This is proved using the following theorem about flat modules over an arbitrary ring R. If a flat
Rmodule M sits in a short exact sequence 0 ! M ! P ! M ! 0 with P projective, then M is
projective.
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 the group ring is projective. For example, a well
known theorem of Chouinard [9] states that a module is projective if and only if its restriction
to each elementary abelian subgroup is projective. A theorem of Dade [10] 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infinitely generated module, the statement is no longer valid, but in [6] it is proved that an infinitely
generated module is projective if and only if its restriction to each cyclic shifted subgroup defined
over each extension field is projective. These theorems have formed the basis for the development
of the theory of varieties for modules [2, 5, 6, 7].
The main theorem of this paper is the following characterization of projectivity for flat modules
