Extension of derivations to forms

Abstract

The problem of extending derivations of a field $F$ to an $F-$algebra $B$ is widely studied in commutative algebra and non-commutative ring theory. For example, every derivation of $F$ extends to $B$ if $B$ is a separable algebraic extension or a central simple algebra over $F.$ We unify and generalize these results by showing that a derivation $d$ of $F$ with the field of constants $C$ extends to a finite dimensional algebra $B$ if $B$ is a form of some $C-$algebra having a smooth automorphism scheme $\rm G$. Furthermore, we show that the set of derivations of $B$ that extend the derivation $d$ of $F$ is in bijection with the set of derivations $δ$ such that $(Y,δ)$ is a differential $\rm G_F-$torsor where $Y$ is the $\rm G_F-$torsor corresponding to $B$.