Completions and derived de Rham cohomology
Bhargav Bhatt
TL;DR
The paper proves that for finite type morphisms of noetherian $\mathbb{Q}$-schemes in characteristic 0, Illusie’s Hodge-completed derived de Rham cohomology $\widehat{\mathrm{dR}}_{B/A}$ is canonically equivalent to Hartshorne’s algebraic de Rham cohomology $\Omega^H_{B/A}$, yielding a choice-free description via the completed Amitsur complex and a new derived Hodge filtration. The authors develop cosimplicial and Adams-completion machinery, identify Adams completion with classical $I$-adic completion in the noetherian case, and leverage Quillen’s convergence to compare filtrations on cohomology. They globalize the result to schemes, provide explicit models and an affine Betti-cohomology description, and analyze the derived Hodge filtration, including its relation to Deligne–Du Bois and Hodge–Deligne filtrations and implications for singular varieties (including a Bloch-type counterexample). The framework clarifies when higher $E_1$-differentials appear in the derived Hodge-to-de Rham spectral sequence and yields a robust, global, and elementary description of algebraic de Rham cohomology through completed Čech-type constructions.
Abstract
We show that Illusie's derived de Rham cohomology (Hodge-completed) coincides with Hartshorne's algebraic de Rham cohomology for a finite type map of noetherian schemes in characteristic 0; the case of lci morphisms was a result of Illusie. In particular, the E_1-differentials in the derived Hodge-to-de Rham spectral sequence for singular varieties are often non-zero. Another consequence is a completely elementary description of Hartshorne's algebraic de Rham cohomology: it is computed by the completed Amitsur complex for any variety in characteristic 0.