DUALITY FOR PARTIAL GROUP ACTIONS

Year 2008, Volume: 4 Issue: 4, 53 - 62, 01.12.2008

Christian Lomp

Abstract

Given a finite group G acting as automorphisms on a ring A, the skew group ring A ∗ G is an important tool for studying the structure of G-stable ideals of A. The ring A ∗ G is G-graded, i.e. G coacts on A ∗ G. The Cohen-Montgomery duality says that the smash product A ∗G#k[G]∗ of A ∗G with the dual group ring k[G]∗ is isomorphic to the full matrix ring Mn(A) over A, where n is the order of G. In this note we show how much of the Cohen-Montgomery duality carries over to partial group actions in the sense of R.Exel. In particular we show that the smash product (A ∗α G) #k[G]∗ of the partial skew group ring A ∗α G and k[G]∗ is isomorphic to a direct product of the form K × eMn(A)e where e is a certain idempotent of Mn(A) and K is a subalgebra of (A ∗α G) #k[G]∗. Moreover A ∗α G is shown to be isomorphic to a separable subalgebra of eMn(A)e. We also look at duality for infinite partial group actions.

Keywords

Partial Group actions, Cohen-Montgomery duality