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derivations

LEMMA 9.16   For any separated scheme $ X$ over $ S$ , and for any quasi coherent sheaf $ \mathcal{F}$ on $ X$ , An inclusion

$\displaystyle \mathcal{F}\cong \mathcal{D}iff_{X/S}^0(\mathcal{O}_X, \mathcal{F}) \to
\mathcal{D}iff_{X/S}^1(\mathcal{O}_X, \mathcal{F})
$

Admits a section. Namely, ``evaluation by $ 1$ .

$\displaystyle \mathcal{F}\cong \mathcal{D}iff_{X/S}^1(\mathcal{O}_X, \mathcal{F}) \overset{eval_1}{\to}
\mathcal{F}
$

DEFINITION 9.17   The kernel of the evaluation map in the lemma above is called the sheaf of derivations on $ X$ relative to $ S$ . We denote it by $ \mathscr{D}er_{X/S}(\mathcal{O}_X,\mathcal{F})$ .

It is easy to see that

LEMMA 9.18   $ \mathscr{D}er_{X/S}(\mathcal{O}_X,\mathcal{F})$ is a quasi coherent sheaf on $ X$ . Its section consists of $ \mathcal{O}_S$ -linear maps $ D:\mathcal{O}_X\to \mathcal{F}$ which satisfy

$\displaystyle D(fg)=fD(g)+gD(f)
$

for any local regular functions $ f,g$ .



2007-12-11