Example

Tag 0EEM, permanent. Cite it as [Bourbaki, Tag 0EEM].

Variétés différentielles et analytiques, chapter 2, § 14. Opérateurs différentiels

Let $X$ be an open subset of $K^n$ , let $\xi^1,\ldots,\xi^n$ denote the coordinate functions on $X$ , and identify $\Omega$ with $K_X$ by means of the frame $\omega=d\xi^1\wedge\cdots\wedge d\xi^n$ . If $E=F=K_X$ , one has

\widetilde{E}=\widetilde{F}=\mathscr{L}(K_X;\Omega)=\mathscr{L}(K_X;K_X)=K_X,

and the operation $D\mapsto{}^{t}D$ transforms scalar differential operators into scalar differential operators. One has, for example,

{}^{t}(\Delta^\alpha)=(-1)^{|\alpha|}\Delta^\alpha\qquad\text{for every }\alpha\in\mathbf{N}^n.

If $r$ is infinite, transposition is an anti-automorphism of the algebra $\mathscr{D}^{\infty}_{K_X}(K_X,K_X)$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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