local description of differential operators

Differential operators are defined locally. Thus we may restrict ourselves to the affine case and look them carefully by the language of algebras and modules.

LEMMA 9.9   Let $X=\operatorname{Spec}(B)\to \operatorname{Spec}(A)=Y$ be a morphism of affine scheme. Then $X\times_Y X=\operatorname{Spec}(B\otimes_A B)$ . $\Delta_{X/Y}$ corresponds to an ideal $I_{B/A}$ of $B\otimes_A B$ generated by

$$\{ f\otimes 1 -1\otimes f; f\in B\}$$

PROOF.. In $B\otimes_A B/I_{B/A}$ , every class $[\sum_j f_j\otimes g_j]$ of an element $\sum_j f_j\otimes g_j\in B\otimes_A B$ is equal to

$$[\sum_j f_j\otimes g_j] = \sum_j [f_j\otimes 1] [1\otimes g_j] = \sum_j [1\otimes f_j] [1\otimes g_j] = [1\otimes \sum_j f_j g_j]$$

$\qedsymbol$

LEMMA 9.10 (criterion for being a differential operator)   Let $X=\operatorname{Spec}(B)\to \operatorname{Spec}(A)=Y$ be an morphism of schemes. Let $M,N$ be $B$ -modules. Then an $n$ -th order differential operator $P$ from $\mathcal{F}=\mathcal{O}_X \otimes_ B M$ to $\mathcal{G}=\mathcal{O}_X \otimes_ B N$ is identified with an $A$ -linear homomorphism

$$\Gamma(P):M\to N.$$

An $A$ -linear homomorphism $\phi:M\to N$ corresponds to an $n$ -th order differential operator if and only if for any elements $f_1,f_2,\dots, f_n, f_{n+1}$ of $A$ and for any element $m\in M$ , a relation

$$(\mu_N\circ(id_B\otimes \phi))((\prod_{i=1}^n(f_i\otimes 1 - 1\otimes f_i))m)$$

$$(= \sum_{I\subset \{1,2,3,\dots,n+1\}}(-1)^{\vert I\vert} f^{\complement I}\phi( f^I))$$

$$=0$$

holds.

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COROLLARY 9.11   A first order differential operator $P: \mathcal{O}_X \to \mathcal{G}$ on a scheme $X$ relative to $S$ corresponds to an $\mathcal{O}_S$ -module homomorphism $P: \mathcal{O}_X \to \mathcal{G}$ such that for any local section $f,g\in \mathcal{O}_S$ , we have

$$P(fg)=fP(g)+gP(f)-P(1)fg.$$

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Using the Lemma of criterion for being a differential operator, We deduce the following useful lemma.

LEMMA 9.12   Let $X=\operatorname{Spec}(B)\to \operatorname{Spec}(A)=Y$ be an morphism of schemes. $A$ -linear homomorphism $\phi:M\to N$ corresponds to an $n$ -th order differential operator if and only if for any $f\in B$ , the ``commutator''

$$[\phi,f]=\phi(f\bullet)-f\phi(\bullet)$$

corresponds to an $n-1$ -th order differential operator.

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COROLLARY 9.13   A composition of an $n$ -th order differential operator $P:\mathcal{F}\to \mathcal{G}$ and an $m$ -th order differential operator $Q:\mathcal{G}\to \mathcal{H}$ is a differential operator of $(n+m)$ -th order.

PROOF.. We note that for any local regular function $f$ , w

$$[QP,f]=[Q,f]P+Q[P,f]$$

holds. Then we may easily verify the statement by using induction.

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DEFINITION 9.14   For any separable scheme $X$ over $S$ , we denote the sheaf of $n$ -th linear differential operators on $X$ from a quasi coherent sheaf $\mathcal{F}$ to a quasi coherent sheaf $\mathcal{G}$ relative to $S$ by

$$\mathcal{D}iff_{X/S}^n(\mathcal{F},\mathcal{G}).$$

The inductive limit

$$\mathcal{D}iff_{X/S}(\mathcal{F},\mathcal{G})=\varinjlim_{n} \mathcal{D}iff_{X/S}^n(\mathcal{F},\mathcal{G})$$

is called the sheaf of linear differential operators on $X$ relative to $S$ .

We use the following abbreviational symbols.

$$\mathcal{D}iff_{X/S}^n(\mathcal{F})= \mathcal{D}iff_{X/S}(\mathca... ...al{F})^n,\qquad \mathcal{D}iff_{X/S}^n= \mathcal{D}iff_{X/S}^n(\mathcal{O}_X),$$

$$\mathcal{D}iff_{X/S}(\mathcal{F})= \mathcal{D}iff_{X/S}(\mathcal{F},\mathcal{F}),\qquad \mathcal{D}iff_{X/S}= \mathcal{D}iff_{X/S}(\mathcal{O}_X).$$

Note that $\mathcal{D}iff_{X/S}$ is a sheaf of algebras over $X$ . It is an important example of an object which is a ``non-commutative algebras glued together''.

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