Horrocks' theorem for noncommutative rings (Q922626)

The author proves an analog of \textit{G. Horrocks'} theorem [Proc. Lond. Math. Soc., III. Ser. 14, 714-718 (1964; Zbl 0132.281] for skew polynomial rings. Let B be a left noetherian semilocal ring with the Jacobson radical J, \(\alpha\) and d be an automorphism and \(\alpha\)- derivation of B such that d(J)\(\subset J\). Let, further, \(A=B[T;\alpha,d]\) be a skew polynomial ring over A in the indeterminate T, P a finitely generated projective A-module and S be the set of all elements \(f\in A\) such that A/Af is a finitely generated left B-module and f is no zero divisor in A. It is proved, in particular, that if \(S^{-1}P\) and P/JP are free modules of rank t over \(S^{-1}A\) and A/JA respectively then \(P=P_ 0\oplus A^{t-1}\) where \(S^{-1}P_ 0\simeq S^{-1}A\) and \(P_ 0/JP_ 0\simeq A/JA\).