Pro-Algebraic Site

Abstract

For any type of fundamental groupoid scheme, we construct an algebraic cohomology theory for varieties with coefficients in the base field. This is a minor variant of \'etale cohomology, involving neither de Rham complexes nor hypercohomology. The main idea is to delegate the role of \'etale morphisms to fundamental groupoids, thereby bypassing the Grothendieck topology. To validate this theory, we prove a comparison theorem between the algebraically defined cohomology using the pro-algebraic fundamental groupoid over $\mathbb{C}$ and singular cohomology. Furthermore, our construction naturally extends to other types of fundamental groupoids, providing a uniform foundation for various cohomology theories.