Duality for partial group actions

• Type: Article in International Scientific Journal
• Date: 2008

Authors

Christian Lomp (Author) FCUP

Journal

Title: International Electronic Journal of Algebra

Vol. 4 No. 1

Pages: 53-62

Publisher: Hacettepe University

Scientific classification

FOS: Natural sciences > Mathematics

CORDIS: Physical sciences > Mathematics > Algebra

Other information

Resumo (PT): 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 M_n(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 alpha in the sense of R.Exel. In particular we show that the smash product (A *_alpha G)#k[G]^* of the partial skew group ring A*_alpha G and k[G]^* is isomorphic to a direct product of the form K x eM_n(A)e where e is a certain idempotent of M_n(A) and K is a subalgebra of (A *_alpha G)#k[G]^*. Moreover A*_alpha G is shown to be isomorphic to a separable subalgebra of eM_n(A)e. We also look at duality for infinite partial group actions.

Abstract (EN): 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 M_n(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 alpha in the sense of R.Exel. In particular we show that the smash product (A *_alpha G)#k[G]^* of the partial skew group ring A*_alpha G and k[G]^* is isomorphic to a direct product of the form K x eM_n(A)e where e is a certain idempotent of M_n(A) and K is a subalgebra of (A *_alpha G)#k[G]^*. Moreover A*_alpha G is shown to be isomorphic to a separable subalgebra of eM_n(A)e. We also look at duality for infinite partial group actions.

Language: English

Type (Professor's evaluation): Scientific