Critical and injective modules over skew polynomial rings

Abstract

Let R be a commutative local k-algebra of Krull dimension one, where k is a field. Let α be a k-algebra automorphism of R, and define S to be the skew polynomial algebra R[θ;α]. We offer, under some additional assumptions on R, a criterion for S to have injective hulls of all simple S-modules locally Artinian - that is, for S to satisfy property (⋄). It is easy and well known that if α is of finite order, then S has this property, but in order to get the criterion when α has infinite order we found it necessary to classify all cyclic (Krull) critical S-modules in this case, a result which may be of independent interest. With the help of the above we show that Sˆ=k[[X]][θ,α] satisfies (⋄) for all k-algebra automorphisms α of k[[X]].