derivations

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

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

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

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

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

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

for any local regular functions $f,g$ .