Geometricity of the Hodge Filtration on the ∞-stack of Perfect Complexes over Xdr

We construct a locally geometric ∞-stack MHod(X,Perf) of perfect complexes with λ-connection structure on a smooth projective variety X. This maps to A1/Gm, so it can be considered as the Hodge filtration of its fiber over 1 which is MDR(X,Perf), parametrizing complexes of DX-modules which are OX -perfect. We apply the result of Toen-Vaquié that Perf(X) is locally geometric. The proof of geometricity of the map MHod(X,Perf) → Perf(X) uses a Hochschild-like notion of weak complexes of modules over a sheaf of rings of differential operators. We prove a strictification result for these weak complexes, and also a strictification result for complexes of sheaves of O-modules over the big crystalline site.