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arXiv · 2504.05969

Extension of derivations to forms

Manujith K. Michel·Chitrarekha Sahu
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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 $\delta$ such that $(Y,\delta)$ is a differential $\rm G_F-$torsor where $Y$ is the $\rm G_F-$torsor corresponding to $B$.

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Manujith K. Michel, Chitrarekha Sahu. 2025-04-08. Extension of derivations to forms. https://arxiv.org/abs/2504.05969

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