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

Let $k$ be a commutative ring and $A$ a commutative $k$-algebra. Given a positive integer $m$, or $m=\mathrm{\infty }$, we say that a $k$-linear derivation $\delta$ of $A$ is $m$-integrable if it extends up to a Hasse–Schmidt derivation $D=(\text{Id},D_1=\delta ,D_2,\mathrm{.}\mathrm{.}\mathrm{.},D_m)$ 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 ${\text{Ider}}_k(A;m)$. 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 ${\text{fIder}}_S({\text{OO}}_X;m)\subset {\text{fDer}}_S({\text{OO}}_X)$ such that, for any affine open sets $U=\text{Spec}A\subset X$ and $V=\text{Spec}k\subset S$, with $f(U)\subset V$, we have $\mathrm{\Gamma }(U,{\text{fIder}}_S({\text{OO}}_X;m))={\text{Ider}}_k(A;m)$ and ${\text{fIder}}_S({\text{OO}}_X;m{)}_p={\text{Ider}}_{{\text{OO}}_{S,f(p)}}({\text{OO}}_{X,p};m)$ 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.