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=δii!; (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 m-integrable form a quasi-coherent submodule IderS(OX;m)⊂DerS(OX) such that, for any affine open sets U=SpecA⊂X and V=Speck⊂S, with f(U)⊂V, we have Γ(U,IderS(OX;m))=Iderk(A;m) and IderS(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.