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In Hartshorne's paper On the de Rham cohomology of algebraic varieties (Publ. Math. IHES 45 (1975)), he defines the algebraic de Rham cohomology of a singular variety $Z$ in characteristic $0$ by embedding it in a smooth variety $X$ and taking the hypercohomology of the formal completion of $\Omega^\bullet_X$ along $Z$. In the introduction to that paper (section 0.5), he says explicitly that he doesn't construct a Hodge filtration on $\mathrm{H}^n_{\mathrm{dR}}(Z/K)$. Does anyone know if there is anywhere in the literature which constructs this filtration? I particularly want this Hodge filtration to be defined over the base field $K$, but to agree with the one from Hodge theory when $K=\mathbb{C}$.

(This is the sort of thing which doesn't seem to hard to work out directly, but feels like it should be in the literature somewhere. But I didn't find anything when I searched.)

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Since $K$ has characteristic $0$, you can use the procedure of Deligne [Théorie Hodge III, IHES 1974] or some variation of it such as Guillén et. al. [Hyperresolutions cubiques ..., Springer LN 1335] to produce a smooth (semi)simplicial scheme $\bar X_\bullet$ with a SNC divisor $D_\bullet$, to get the desired filtration on cohomology. In fact, Exp. III of the second reference works this out explicitly.

This might seem more complicated than one might hope for, since Hartshorne's construction does yield an obvious filtration on cohomology, namely $H^i(\widehat{\Omega_X^{\ge p}})$. However, Ogus [The formal Hodge filtration, Invent 1976] shows this naive filtration generally won't agree with coming from Hodge theory.

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    $\begingroup$ Maybe it is relevant to also add that the associated graded of this filtration are given by the hypercohomology of exterior power of the cotangent complex and not Kahler differentials anymore (this is why it won't agree with the naive filtration) $\endgroup$ Commented 18 hours ago

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