Dg models for derived differential operators

We show that for any algebra $R$ that is essentially of finite type over a field $k$, the derived functors of Smith-Van den Bergh admit a dg enhancement. In particular, for any $R$-module $M$ one can associate a dg algebra of differential operators $E_M$ whose cohomology is a nilpotent thickening of the (classical) ring of $k$-linear differential operators on $M$. We show that these dg algebras fit into a framework of Koszul duality due to Dwyer-Greenlees and specializes to a derived ring of differential operators studied by Yang when $M=R$ and $k=\mathbb{C}$. In particular, upon fixing a presentation $R=S\mathrm{/}I$ with $S$ essentially smooth, we show a derived Kashiwara’s equivalence, namely that whenever ${\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}}_S(M)=\mathrm{V}(I)$ then there is an equivalence of triangulated categories between the derived category of dg $E_M$-modules and the derived category of $D$-modules on $S$ that are set-theoretically supported on $I$. Finally, in the case of positive characteristic, we show that the dg algebra $E_M$ admits a description in terms of the Frobenius endomorphism akin to the classical description of differential operators, and the proof employs simplicial techniques to obtain a result of independent interest in commutative algebra.