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-Vaquie 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