Invariance of base for derived differential operators over arbitrary F-finite algebras

Determine whether the canonical invariance-of-base quasi-isomorphism between the derived differential operator complexes \(\mathsf R D_{R\mid k}(M,M)\) and \(\mathsf R D_{S\mid k}(M,M)\) holds when \(R\) is an arbitrary \(F\)-finite algebra over a perfect field \(k\) of characteristic \(p>0\), \(S\to R\) is a surjection from a regular \(F\)-finite \(k\)-algebra, and \(M\) is an \(R\)-complex; establish this to enable a dg-algebra model for \(\mathsf R D_{R\mid k}(M,M)\) in this generality.

Background

For essentially finite-type algebras, the paper obtains an invariance-of-base quasi-isomorphism by presenting $R$ as a quotient of an essentially smooth algebra $S$, with the proof relying on the noetherianity of $S\otimes_k S$. In characteristic $p>0$, the authors note that Gabber’s result supplies a surjection from a regular $F$-finite algebra $S$ onto an arbitrary $F$-finite $k$-algebra $R$, and Kunz’s theorem allows the derived differential operators over $S$ to be represented using Frobenius-stage Hom complexes.

The unresolved issue is whether the invariance-of-base comparison remains valid without the essentially finite-type hypothesis. A positive answer would extend the dg-algebra modeling framework from the paper’s principal setting to arbitrary $F$-finite algebras and arbitrary $R$-complexes.