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 $m=\infty$, we say that a $k$-linear derivation $\delta$ of $A$ is $m$-integrable if it extends up to a Hasse--Schmidt derivation $D=(\Id,D_1=\delta,D_2,...,D_m)$ 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 $D_i=\frac{\delta^i}{i!}$; (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 $\Ider_k(A;m)$. 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 $\delta$ of $A$ is $m$-integrable if and only if the induced derivation $\delta_{\mathfrak{p}}:A_{\mathfrak{p}} \to A_{\mathfrak{p}} $ is $m$-integrable for each prime ideal $\mathfrak{p}\subset A$. In particular, for any locally finitely presented morphism of schemes $f:X \to S$ and any positive integer $m$, the $S$-derivations of $X$ which are locally $m$-integrable form a quasi-coherent submodule $\fIder_S(\OO_X;m)\subset \fDer_S(\OO_X)$ such that, for any affine open sets $U=\Spec A \subset X$ and $V=\Spec k \subset S$, with $f(U)\subset V$, we have $\Gamma(U,\fIder_S(\OO_X;m))=\Ider_k(A;m)$ and $\fIder_S(\OO_X;m)_p = \Ider_{\OO_{S,f(p)}}(\OO_{X,p};m)$ for each $p\in X$. We also give, for each positive integer $m$, an algorithm to decide whether all derivations are $m$-integrable or not.