Totally projective unit groups in modular abelian group algebras

Let p be a prime, R a commutative ring of characteristic p, G a p-primary abelian group, and V(RG) the group of normalized units in the group algebra RG. It is largely determined when V(RG) is totally projective. This generalizes the well-known special case when R is a perfect field. Moreover, the current proof has the advantage of greater clarity.