Cohomology of De Rham complex with respect to different sites.

Question

Sheaf cohomology of algebraic De Rham complex gives the algebraic De Rham cohomology groups. This coincides with the singular cohomology if for example the variety is defined over $\mathbb{C}$. I was wondering whether taking sheaf cohomology of the De Rham complex with respect to different sites other than the Zariski topology gives any interesting cohomology groups? Especially I couldn't figure out what would be the cohomology groups in the case of etale topology?

Answer 1

The de Rham cohomology does not change if you replace the Zariski topology by the étale topology, at least for smooth schemes. More generally, for arbitrary schemes, the Hodge-completed derived de Rham cohomology does not change as long as the Grothendieck topology in question is coarser than the fpqc topology. The point is that, given a base ring $R$, the functor $\operatorname{Alg}_{R}\to \widehat{\operatorname{DF}}(R),A\mapsto\widehat{\operatorname{dR}}_{A/R}$ satisfies fpqc descent. This is (essentially) proved in Example 5.10 of

B. Bhatt, M. Morrow, P. Scholze, Topological Hochschild homology and integral $p$-adic Hodge theory.

Indeed, recall that the Hodge-completed derived de Rham cohomology is the completion of the derived de Rham cohomology with respect to the Hodge filtration. In order to see that the functor $A\mapsto\widehat{\operatorname{dR}}_{A/R}$ satisfies fpqc descent, it suffices to check that the associated graded pieces $A\mapsto\bigwedge_A^iL_{A/R}$ satisfies fpqc descent, which is Theorem 3.1 of loc. cit.