Status of a conjecture in Grothendieck's "Crystals and the de Rham Cohomology of Schemes"

Question

Let $X/\mathbb{C}$ be a scheme over the complex numbers. In "Crystals and the de Rham cohomology of schemes," Grothendieck constructs the infinitesimal ringed site $(X_{\operatorname{inf}}, \mathcal{O})$ of $X/\mathbb{C}$ and proves that when $X$ is a smooth $\mathbb{C}$-scheme, there is a canonical isomorphism $$ H^*(X_{\operatorname{inf}}, \mathcal{O}) \cong H^*_{\operatorname{sing}}(X^{\operatorname{an}}; \mathbb{C}) $$ between the cohomology of the structure sheaf of the infinitesimal site and the singular cohomology of the underlying complex analytic space.

Grothendieck goes on to conjecture (Conjecture 4.2) that the above isomorphism holds for any scheme which is locally of finite type over $\mathbb{C}$ (not necessarily smooth).

What is the status of Grothendieck's conjecture?

Answer 1

I think that this is solved in

Hartshorne, Robin. On the de Rham cohomology of algebraic varieties. Publications Mathématiques de l'IHÉS, Volume 45 (1975), pp. 5-99. http://www.numdam.org/item/PMIHES_1975__45__5_0/

Since he adopts different terminologies, let me explain a bit.

Recall that the infinitesimal site of $X$ consists of an open subscheme $U\to X$ along with a nilpotent thickening $U\to T$. Note that the nilpotent thickening $U\to T$ induces a homeomorphism $\DeclareMathOperator\an{an}\lvert U^{\an}\rvert\to\lvert T^{\an}\rvert$ of underlying topological spaces of analytifications, therefore the map $R\Gamma(T^{\an};\mathbb C)\to\mathcal O(T)$ of (derived) abelian sheaves on the infinitesimal site induces a map $R\Gamma((-)^{\an};\mathbb C)\to Rv_*(\mathcal O)$ of (derived) abelian Zariski sheaves, where $\DeclareMathOperator\inf{inf}v\colon X_{\inf}\to X_{\operatorname{Zar}}$ is the canonical map of sites. It suffices to show that this is a quasi-isomorphism, and thus we reduce to check that, for every affine open subscheme $U\to X$, the map $R\Gamma(U^{\an};\mathbb C)\to R\Gamma(U_{\inf};\mathcal O)$ is a quasi-isomorphism.

This is the main result of Hartshorne's paper above: we can embed $U$ as a closed subscheme of a smooth affine $\mathbb C$-scheme $Y$, and a Čech–Alexander computation tells us that $R\Gamma(U_{\inf};\mathcal O)$ is represented by Hartshorne's algebraic de Rham complex associated to $U\to Y$ (a similar result in crystalline cohomology can be found in [Stacks Projects, Tag 07LG]).

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Let me mention that, this result could be strengthened a bit to incorporate coefficient systems. More precisely, in [Scholze, Geometrization of the real local Langlands], an analytic Riemann–Hilbert is established. In Chapter II, a proof for the smooth case is explained, but this generalizes to singular cases essentially by considering stacks on totally disconnected $\mathbb C_{\operatorname{gas}}$-algebras as in Section V.3 (cf. Prop V.3.8).