# Differential operators of finite order

$(X,\mathcal{O}_X)$ is a complex manifold or an algebraic variety over $\mathbb{C}$ with his structure sheaf. Let $\delta : X \to X \times X$ the diagonal morphism and $\mathcal{J} \subseteq \mathcal{O}_X$ the ideal sheaf of $\delta(X)$ i.e:$$\mathcal{J}(V)=\{f \in \mathcal{O}_X(V) \ | \ f(V \cap \delta(X))=0 \}$$.

We have also two canonical projection $p_1,p_2 : X \times X \to X$ such that $\delta^{-1} p_j^{-1}\mathcal{O}_X=\mathcal{O}_X$. Let $\alpha_j$ denote the canonical morphism $p_j^{-1}\mathcal{O}_X \to \mathcal{O}_{X \times X}$ and $\beta_j$ the induced morphism $$\mathcal{O}_X=\delta^{-1} p_j^{-1}\mathcal{O}_X \to \delta^{-1}\mathcal{O}_{X \times X}$$.

Let now consider two $\mathcal{O}_X$ modules $K,L$.Let us consider $$P= \delta^{-1}(\dfrac{\mathcal{O}_{X \times X}}{\mathcal{J}^2})\otimes K$$, where the tensor product is taken over $\mathcal{O}_X$, looking at $\dfrac{\mathcal{O}_{X \times X}}{\mathcal{J}^2}$ as an $\mathcal{O}_X$ module with the structure induced by $\beta_2$.Besides, we consider $P$ as an $\mathcal{O}_X$ module ,by the module structure of $\delta^{-1}(\dfrac{\mathcal{O}_{X \times X}}{\mathcal{J}^2})$ induced by $\beta_1$.

We define $F_1(K,L)$ as the subsheaf of $Hom_{\mathbb{C}}(K,L)$ characterised by $$F_1(K,L)=\{P \ | \ Pf-fP \in Hom_{\mathcal{O}_X}(K,L)\}$$.

Now , we define $$\gamma: Hom_{\mathcal{O}_X}(P,L) \to F_1(K,L)$$ $$\gamma(\psi)(s)=\psi(1 \otimes s)$$.

It is straightforward calculation that this map is well defined, but I would like to show that this is an isomorphism. I do not manage to show this is injective, so that if $\psi(1 \otimes s)=0$ for every $s$ ,the map is null and I have no idea.