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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
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