Periodic flat modules, and flat modules for finite groups

If R is a ring of coefficients and G a finite group, then a at RG-module wh ich 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, then a at R module which is projective as an RH-module is necessarily proje ctive as an R Gamma -module. This follows from a generalization of the rst theorem to modules over strongly G-graded rings. These results are proved u sing the following theorem about at modules over an arbitrary ring S: If a at 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 at and projec tive modules over group rings of finite groups, involving reduction modulo primes, are also proved.