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We construct a locally geometric $\infty$-stack $M_{Hod}(X,Perf)$ of perfect complexes with $\lambda$-connection structure on a smooth projective variety $X$. This maps to $A ^1 / G_m$, so it can be considered as the Hodge filtration of its fiber over 1 which is $M_{DR}(X,Perf)$, parametrizing complexes of $D_X$-modules which are $O_X$-perfect. We apply the result of Toen-Vaquie that $Perf(X)$ is locally geometric. The proof of geometricity of the map $M_{Hod}(X,Perf) \rightarrow Perf(X)$ 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.

Geometricity of the Hodge Filtration on the ∞-stack of perfect complexes over XDR