'Strong' density of differential operators in linear operators

Question

Let $A$ be a commutative, regular, finitely generated $k$-algebra, where $k$ is a field of characteristic 0.

Then Grothendieck proved that in this case the algebra of differential operators on $A$ is simply the subalgebra of $End_k(A)$ generated by multiplication by $A$ and $Der_k(A)$, $k$-linear derivations. Denote this algebra by $D(A)$.

I have read that $D(A)\subseteq End_k(A)$ is 'strongly dense', i.e. given any finite-dimensional subspace $M\subseteq A$ and any linear operator $L\in End_k(A)$, there exists a differential operator $D\in D(A)$ such that $L|_M=D|_M$. Does anyone know that proof of this fact?

My thought process has been to try and work locally first: we know that $Spec(A)$ has a cover by affines $Spec(A_f)$ such that $D(A_f)\cong A_f[\frac{\partial}{\partial x_1},\dots,\frac{\partial}{\partial x_n}]$, where $x_1,\dots,x_n\in A_f$ are a system of parameters at all closed points in $Spec(A_f)$. Here the derivations $\frac{\partial}{\partial x_i}$ are only required to satisfy $x_j\mapsto \delta_{ij}$ (if I understand things correctly). So it's easy to find a differential operator which does anything you like to a subspace generated by monomials in the $x_i$, but I don't see what to do about other elements.

Answer 1

I think it may be easier if you do not choose the open cover in advance, so that the cover may depend on $M$. Equivalently, let us work with local rings first.

Here is a sketch of the argument. I am keeping your notation: $A$ is smooth over a characteristic zero field $k$, $D(A)$ is the algebra of differential operators, and $M\subset A$ is a finite-dimensional subspace. Consider the evaluation map $$ev:D(A)\to Hom_k(M,A).$$ We want to show that $ev$ is surjective.

Note that $ev$ is a morphism of $A$-modules, so it is enough to check its surjectivity locally. (Explicitly, this step involves looking at partitions of unity.) That is to say, for every point $x\in Spec(A)$, we consider the local ring $A_x$, an we need to prove surjectivity of the map $$ev_{A_x}:D(A_x)=A_x\otimes_A D(A)\to Hom_k(M,A_x).$$ Moreover, $Hom_k(M,A)$ is a finitely generated $A$-module (in fact, it is free of finite rank), so it suffices to consider closed points $x\in Spec(A)$.

Let now $x\in Spec(A)$ be a closed point, and we want to prove surjectivity of $ev_{A_x}$. By Nakayama's Lemma, it suffices to check that the composition $$D(A_x)\to Hom_k(M,A_x)\to Hom_k(M,k_x)$$ is surjective, here $k_x$ is the residue field of $x$. However, for this statement you can replace $A_x$ by its completion, which turns it into a claim about Taylor series.

P.S. It may be psychologically easier to extend scalars from $k$ to its algebraic closure at the very beginning... But if you resist the temptation, you get to think about things like $k$-linear differential operators on the ring of Taylor series with coefficients in a finite extension $k_x\supset k$...