Completions and derived de Rham cohomology

TL;DR

This paper proves the equivalence of Illusie's derived de Rham cohomology with Hartshorne's algebraic de Rham cohomology for finite type maps of noetherian schemes in characteristic 0, providing new insights into spectral sequences and cohomology computations.

Contribution

It establishes the equivalence between derived and algebraic de Rham cohomology in characteristic 0 and offers an elementary description using the completed Amitsur complex.

Findings

Derived de Rham cohomology coincides with algebraic de Rham cohomology in characteristic 0.

E_1-differentials in the Hodge-to-de Rham spectral sequence can be non-zero for singular varieties.

Algebraic de Rham cohomology can be computed via the completed Amitsur complex.

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.