Hasse--Schmidt derivations versus classical derivations

n-平凡扩张与多哈塞-施密特导出

n-trivial extensions and multi Hasse-Schmidt derivations

本文提出了一种哈塞-施密特导出的推广，该推广与安德森-比尼-法伊德-沙伊亚提出的n-平凡扩张概念等价，如同导出与平凡扩张等价。文章提供了这种推广的许多例子，并证明了其一些基本性质。

Higher derivations of modules and the Hasse-Schmidt module

In this paper we revisit Ribenboim's notion of higher derivations of modules and relate it to the recent work of De Fernex and Docampo on the sheaf of differentials of the arc space. In particular, we derive their formula for the Kähler differentials of the Hasse-Schmidt algebra as a consequence of the fact that the Hasse-Schmidt algebra functors commute.

On Hasse--Schmidt derivations: the action of substitution maps

We study the action of substitution maps between power series rings as an additional algebraic structure on the groups of Hasse--Schmidt derivations. This structure appears as a counterpart of the module structure on classical derivations.

Generalised Hasse-Schmidt varieties and their jet spaces

This work provides a unified formalism for studying difference and (Hasse-) differential algebraic geometry, by introducing a theory of "iterative Hasse rings and schemes". As an application, Hasse jet spaces are constructed generally, allowing the development of the theory for arbitrary systems of algebraic partial difference/differential equations, where constructions by earlier authors applied only to the finite dimensional case. In particular, it is shown that under appropriate separability assumptions a Hasse variety is determined by its jet spaces at a point.

Rings of differential operators as enveloping algebras of Hasse--Schmidt derivations

Let $k$ be a commutative ring and $A$ a commutative $k$-algebra. In this paper we introduce the notion of enveloping algebra of Hasse--Schmidt derivations of $A$ over $k$ and we prove that, under suitable smoothness hypotheses, the canonical map from the above enveloping algebra to the ring of differential operators $\mathcal{D}_{A/k}$ is an isomorphism. This result generalizes the characteristic 0 case in which the ring $\mathcal{D}_{A/k}$ appears as the enveloping algebra of the Lie-Rinehart algebra of the usual $k$-derivations of $A$ provided that $A$ is smooth over $k$.

On the bracket of integrable derivations

We prove that any multi-variate Hasse-Schmidt derivation can be decomposed in terms of substitution maps and uni-variate Hasse-Schmidt derivations. As a consequence we prove that the bracket of two $m$-integrable derivations is also $m$-integrable, for $m$ a positive integer or infinity.

Prolongations in differential algebra

We develop the theory of higher prolongations of algebraic varieties over fields in arbitrary characteristic with commuting Hasse-Schmidt derivations. Prolongations were introduced by Buium in the context of fields of characteristic 0 with a single derivation. Inspired by work of Vojta, we give a new construction of higher prolongations in a more general context. Generalizing a result of Buium in characteristic 0, we prove that these prolongations are represented by a certain functor, which shows that they can be viewed as `twisted jet spaces.' We give a new proof of a theorem of Moosa, Pillay…

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 $m=\infty$, we say that a $k$-linear derivation $δ$ 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 $D_i=\frac{δ^i}{i!}$; (2) $k$ is arbitrary and $A$ is a smooth $k$-algebra. The set of $m$-integrable derivations of $A$ over $k$ is an …