A note on twisted group rings and semilinearization

Abstract

In this short note, we construct a right adjoint to the functor which associates to a ring $R$ equipped with a group action its twisted group ring. This right adjoint admits an interpretation as semilinearization, in that it sends an $R$-module to the group of semilinear $R$-module automorphisms of the module. As an immediate corollary, we provide a novel proof of the classical observation that modules over a twisted group ring are modules over the base ring together with a semilinear action.

This note, to appear in Communications in Algebra, was initially was an appendix to "Homotopy Mackey functors of equivariant algebraic K-theory," but was more elaborate than what was strictly needed in that paper. Since this may be an observation of independent interest in algebra, it has been written up separately