On F-integrable actions of the restricted Lie algebra of a formal group F in characteristic p > 0

Abstract

Let k be an integral domain, let F = (F1X, Y),…, Fn(X, Y)), X = (X1,…, Xn), Y = (Y1,…, Yn), be an n-dimensional formal group over k, and let L(F) be the Lie algebra of all F-invariant k-derivations of the ring of formal power series k[X] (cf. § 2). If A is a (commutative) k-algebra and Derk (A) denotes the Lie algebra of all k-derivations d: A → A, then by an action of L(F) on A we mean a morphism of Lie algebras φ: L(F) → Derk (A) such that φ(dp) = φ(d)p, provided char (k) = p > 0. An action of the formal group F on A is a morphism of k-algebras D: A-→A[X] such that D(a)≡a mod (X) for a ∊ A, and FA ∘ D = DY ∘ D, where FA: A[X] → A[X, Y], DF: A[X] → A[X, Y] are morphisms of k-algebras given by FA(g(X)) = g(F), DY(Σa aaXa) = Σa D(aa)Y; for a motivation of this notion, see [15]. Let D: A → A[X] be such an action.