Derivations of differential operators

Question

For a smooth affine variety $\operatorname{Spec} A$ over a ring $R$ we have an algebra of differential operators $\mathcal{D}_A$ (here I mean not the Grothendieck differential operators but PD-ones). Is it possible to describe $\operatorname{Der} \mathcal{D}_A$? As far as I understand, over $\mathbb{C}$ there is an explicit relation with $\Omega_{A}^1$ but I am mostly interested in the case of $R = \mathbb{Z}/p^n \mathbb{Z}$ and $A = \mathbb{Z}/p^n \mathbb{Z} [x]$. How to describe it in this case?

Answer 1

In case $R$ is a field $k$ of characteristic $p$, I can answer this question.

For any $k$-algebra $D$, set $HH^1(D) = Der_R(D)/Inn(D)$ to be the space of outer derivations. Thus we have a short exact sequence

$$ 0\to Inn(D) \to Der_R(D) \to HH^1(D)\to 0.$$

If $D = \mathcal D_A$ for a smooth $k$-algebra $A$, then the center of $D$ is just $k$, so $Inn(D) = D/k$. The outer derivations $HH^1(D)$ are as follows:

Theorem: $HH^1(\mathcal D_A) = \varprojlim\limits_r (A / A^{(p^r)}) / A$, where $A^{(p^r)}$ is the $k$-span of $\{a^{p^r} \mid a \in A\}$.

In case $A = k[x]$, you can think of this limit as the space of power series $\sum_i c_i x^{\alpha_i}$ where for any $r \geq 0$, all but finitely many $\alpha_i$ are divisible by $p^r$; then one goes modulo polynomials. An example of a nontrivial such series is $x + x^p + x^{p^2} + x^{p^3} + \cdots$. Taking the commutator with this series gives a derivation because every differential operator on $k[x]$ commutes with some $x^{p^r}$.

The displayed theorem appears in my recent preprint "Hochschild cohomology of differential operators in positive characteristic," https://arxiv.org/abs/2303.07373. See especially Example 5.1.