Local acyclic fibrations and the de Rham complex

TL;DR

This paper reinterprets algebraic de Rham cohomology for singular complex varieties using sheaf cohomology in the h-topology, extending to the algebraic de Rham complex via hypercover techniques.

Contribution

It introduces a new approach to algebraic de Rham cohomology using local acyclic fibrations and hypercovers in the h-topology, broadening the scope to singular varieties.

Findings

Extended Cech cohomology to local acyclic fibrations

Reinterpreted de Rham cohomology in the h-topology

Applied hypercover techniques to simplicial presheaves

Abstract

We reinterpret algebraic de Rham cohomology for a possibly singular complex variety X as sheaf cohomology in the site of smooth schemes over X with Voevodsky's h-topology. Our results extend to the algebraic de Rham complex as well. Our main technique is to extend Cech cohomology of hypercovers to arbitrary local acyclic fibrations of simplicial presheaves.