Galois actions on rings and finite Galois coverings.

Abstract

In this paper, the notion of a Galois extension for rings isdeveloped which when suitably specialized gives a purely ringtheoretic formulation of finite Galois coverings in the sense ofBongartz, Gabriel and Riedtmann . Let Γ=Λ\sp G bethe fixed ring under the action of a finite group G ofautomorphisms of Λ such that Λ is a finitelygenerated Γ module. The extension Λ ofΛ\sp G is called pre Galois with group G if Λis a projective generator of the skew group ring \Lambda[G],equivalently, the fixed point functor from\roman{Mod}\,\Lambda[G] to \roman{Mod}\,Λ\sp G is anequivalence of categories. Moreover, Λ over Γ iscalled Galois if additionally Γ/{ \roman{Ann}}\sb ΓSis a semisimple Artin ring for each simple Λ module S.The notions pre Galois and Galois are discussed with respect toΛ\sp {\roman{op}}, arbitrary respectively normalsubgroups of G and G stable ideals of \Lambda[G]. IfΛ is an R algebra for a commutative ring R such thatΛ\sb {\germ m}/\roman{Rad}\,Λ\sb {\germ m} isfinite dimensional over R/\germ m for each maximal ideal\germ m of R and G is as above, then Λ is Galoisover Γ if and only if the induced action of G on theisomorphism classes of simple Λ modules is free. Theconnection to the classical covering theory is established, andequivalent conditions are given for the multiplication map\Lambda[G]\otimes\sb Λ\Lambda[G]\to \Lambda[G] being asplit epimorphism