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The sheaf of differential 1-forms as the universal derivation

Let $ X$ be a separated $ S$ -scheme. We define derivative

$\displaystyle d: \mathcal{O}_X \to \Omega_{X/S}^1
$

as follows

$\displaystyle \mathcal{O}_X \overset{jet_1}{\to} \mathcal J_{1} \to \Omega_{X/S}^1
$

PROPOSITION 9.19   For any sheaf homomorphism $ \varphi: \Omega^1_{X/S},\mathcal{F})$ ,

$\displaystyle f\mapsto \varphi(df)
$

is a derivation from $ \mathcal{O}_X \to \mathcal{F}$ relative to $ S$ . This assignment yields a isomorphism of $ \mathcal{O}_X$ -module

$\displaystyle \mathscr{D}er_{X/S}(\mathcal{O}_X,\mathcal{F}) \cong \mathscr{H}om(\Omega^1_{X/S},\mathcal{F}).
$



2007-12-11