Let $G$ be a finite group and let $R$ be a commutative ring.


I'd like to ask, if there is a theorem of the following kind:



> 
> The augmentation ideal $I\_G$ is projective as RG-module, if and only if ... ?
> 
> 
> 


This should happen only in rare cases, but I was wondering, if there exists an if-and-only-if criterion.


Thank you very much for the help.



When is the augmentation ideal projective as RG-module?
Okay, this happens precisely in the obvious case, namely if all primes dividing the order $|G|$ are invertible in $R$.


To see this, note that $\operatorname{Ext}^\*\_{R[G]}(R,R)$ is group cohomology of $G$ with coefficients in $R$. If $I\_G$ were projective, $I\_G\to R[G]$ would be a projective resolution of $R$ and thus the cohomology would be $1$-dimensional. However, the cohomology of $G$ with coefficients in $\mathbb{Z}$ has $p$-torsion in arbitrarily high degrees for all primes dividing $|G|$ (this is a well-known fact about cohomology of finite groups). So by the universal coefficient theorem, the cohomology with $R$-coefficients is unbounded, except if $p$ is invertible in $R$ for each $p$ dividing $|G|$.


