From Sheaf Cohomology to the Algebraic de Rham Theorem

TL;DR

This paper provides an elementary proof of Grothendieck's algebraic de Rham theorem, showing that the singular cohomology of a complex manifold can be computed from algebraic differential forms, using standard tools and classical references.

Contribution

It offers a simplified, accessible proof of the algebraic de Rham theorem relying solely on standard textbooks and classical papers, avoiding complex modern machinery.

Findings

Elementary proof of the algebraic de Rham theorem established.

Singular cohomology can be computed from algebraic differential forms.

Proof relies only on standard textbooks and classical references.

Abstract

Let X be a smooth complex algebraic variety with the Zariski topology, and let Y be the underlying complex manifold with the complex topology. Grothendieck's algebraic de Rham theorem asserts that the singular cohomology of Y with complex coefficients can be computed from the complex of sheaves of algebraic differential forms on X. This article gives an elementary proof of Grothendieck's algebraic de Rham theorem, elementary in the sense that we use only tools from standard textbooks as well as Serre's FAC and GAGA papers.