Reduction module p of cyclic group actions of order p(r)

An action of a local multiplicative group scheme mu(pr),(k) on an affine sc heme Spec (A) over bar defined over a field k of characteristic p is descri bed in terms of a locally finite higher derivation of multiplicative type o n a k-algebra A. We consider a lifting of such an action of mu(p)(r),(k) to a field K of characteristic zero, which is then an action of a cyclic grou p of order p(r) provided K contains a primitive p(r)th root of unity. We de velop a general theory of the lifting by considering an action of a finite group scheme mu(p)(r),(e) over a p(r)-good discrete valuation ring O. The p resent article is a sequel to the second author's previous paper (Miyanishi and Nomura, J. Pure Appl. Algebra 71 (1991) 249-264), where the case r=1 i s treated. (C) 1999 Published by Elsevier Science B.V. All rights reserved.