A note on twisted group rings and semilinearization

TL;DR

This paper constructs a right adjoint to the functor linking rings with group actions to their twisted group rings, interpreting it as semilinearization, and offers a new proof of a classical module action result.

Contribution

It introduces a right adjoint to the twisted group ring functor and interprets it as semilinearization, providing new insights and proofs in module theory.

Findings

Construction of a right adjoint to the twisted group ring functor

Interpretation of the right adjoint as semilinearization

New proof that modules over twisted group rings have semilinear actions

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.