Extension of derivations to forms
Manujith K. Michel, Chitrarekha Sahu
TL;DR
The paper addresses extending a derivation $d$ of a field $F$ to a finite-dimensional $F$-algebra that is a form of a $C$-algebra with a smooth automorphism scheme. It unifies classical results by employing the automorphism scheme ${\rm Aut}_A$ and Galois cohomology $H^1({\rm Gal}(\bar{F}/F), {\rm G}(\bar{F}))$ to classify forms $A_f$ and to relate derivation extension to differential ${\rm G}_F$-torsors. It provides an explicit description: extensions correspond to matrices of the form $P_f^{-1} M P_f + (P_f^{-1})' P_f$ with $M \in \mathfrak g_{ar{F}}$, ensuring a non-empty set of extensions. This framework generalizes known results for separable or central simple algebras and connects derivation extension to twisted forms and differential torsor theory, broadening applicability to a wide class of $F$-algebras.
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 $B$.