Reduction modulo p of cyclic group actions of order pr

AbstractAn action of a local multiplicative group scheme μpr,k on an affine scheme Spec Ā defined over a field k of characteristic p is described in terms of a locally finite higher derivation of multiplicative type on a k-algebra Ā. We consider a lifting of such an action of μpr,k to a field K of characteristic zero, which is then an action of a cyclic group of order pr provided K contains a primitive prth root of unity. We develop a general theory of the lifting by considering an action of a finite group scheme μpr,O over a pr-good discrete valuation ring O. The present 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 is treated