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 $D=(\Id,D_1=δ,D_2,...,D_m)$ 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 $Γ(U,\fIder_S(\OO_X;m))=\Ider_k(A;m)$ 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.

Final version; in the previous version, the last example 3.5 was incomplete and a reference was missing

References in corpus (2)

• Jets via Hasse-Schmidt Derivations
• Hasse--Schmidt derivations, divided powers and differential smoothness

Cited by in corpus (8)

• On Hasse--Schmidt derivations: the action of substitution maps
• Integrable derivations in the sense of Hasse-Schmidt for some binomial plane curve
• Pairs of Lie-type and large orbits of group actions on filtered modules. (A characteristic-free approach to finite determinacy.)
• Leaps of modules of integrable derivations in the sense of Hasse-Schmidt
• Rings of differential operators as enveloping algebras of Hasse--Schmidt derivations
• Hasse--Schmidt modules versus integrable connections
• On the bracket of integrable derivations
• On the behavior of modules of $m$-integrable derivations in the sense of Hasse-Schmidt under base change