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 α H2(G, R*) one can form the twisted group ring RαG. The “twisting problem” asks whether there exists a class α 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. Algebra94 (1994), 1–15]. Here we answer this question for rings of the form ps, s ≥ 2. The main tools are the classification of modular representation of the Klein 4 group over 2 and a Chouinard-like theorem [E. Aljadeff and Y. Ginosar, J. Algebra179 (1996), 599–606] for twisted group rings.