The Derived Ring of Differential Operators

By reading a standard formula for the ring of Grothendieck differential operators in a derived way, we construct a derived (sheaf of) ring of Grothendieck differential operators for Noetherian schemes $X$ separated and finite-type over a base $S$, when the map $X\to S$ is finite tor-amplitude. Using this ring of differential operators, we (re-)develop the theory of $D$-modules from scratch and show an equivalence of categories between $D$-modules using our definition and crystals over the infinitesimal site.