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definition of linear differential operators

An importance of the sheaves of jets is that they govern linear differential operators.

DEFINITION 9.8   Let $ \varphi:X\to S$ be a separated morphism of schemes. Let $ \mathcal{F},\mathcal{G}$ be quasi coherent sheaves on $ X$ . Then a linear differential operator of $ n$ -th order from $ \mathcal{F}$ to $ \mathcal{G}$ on $ X$ relative to $ S$ is a composition of an element of

$\displaystyle \operatorname{Hom}_{\mathcal{O}_X} (\mathcal J_n(\mathcal{F}),\mathcal{G}).
$

with the Taylor expansion.



2007-12-11