Duality for Partial Group Actions

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] ∗. MoreoverA∗αG is shown to be isomorphic to a separable subalgebra of eMn(A)e. We also look at duality for infinite partial group actions and for partial Hopf actions.