On the modules of m-integrable derivations in non-zero characteristic

Abstract

Let k be a commutative ring and A a commutative k-algebra. Given a positive integer m, or m = ∞, we say that a k-linear derivation δ of A is m-integrable if it extends up to a Hasse–Schmidt derivation D = (Id, D1 = δ, D2, . . . , Dm) of A over k of length m. This condition is automatically satisfied for any m under one of the following orthogonal hypotheses: (1) k contains the rational numbers and A is arbitrary, since we can take Di = δ i i! ; (2) k is arbitrary and A is a smooth k-algebra. The set of m-integrable derivations of A over k is an A-module which will be denoted by Iderk(A; m). In this paper we prove that, if A is a finitely presented k-algebra and m is a positive integer, then a k-linear derivation δ of A is m-integrable if and only if the induced derivation δp : Ap → Ap is m-integrable for each prime ideal p ⊂ A. In particular, for any locally finitely presented morphism of schemes f : X → S and any positive integer m, the S-derivations of X which are locally mintegrable form a quasi-coherent submodule Ider S(OX; m) ⊂ Der S(OX) such that, for any affine open sets U = Spec A ⊂ X and V = Spec k ⊂ S, with f(U) ⊂ V , we have Γ(U,Ider S(OX; m)) = Iderk(A; m) and Ider S(OX; m)p = IderOS,f(p) (OX,p; m) for each p ∈ X. We also give, for each positive integer m, an algorithm to decide whether all derivations are m-integrable or not.