Projective representations and relative semisimplicity

Let R be a local (commutative) ring and let p be a prime not invertible in R. Let G be a finite group of order divisible by p. It is well known that the group ring RG admits nonprojective lattices (e.g., R itself with the trivial action). For any element alpha is an element of H-2(G, R*) one can form the twisted group ring R(alpha)G. The "twisting problem" asks whether there exists a class alpha s.t. the corresponding twisted group ring admits only projective lattices. For fields of characteristic p, the answer is in E. Aljadeff and D. J. S. Robinson [J. Pure Appl. Algebra 94 (1994), 1-15]. Here we answer this question for rings of the form Z(Ps), s greater than or equal to 2. The main tools are the classification of modular representation of the Klein 4 group over Z(2) and a Chouinard-like theorem [E. Aljadeff and Y. Ginosar, J. Algebra 179 (1996), 599-606] for twisted group rings. (C) 1999 Academic Press.