A Uniform Construction of Cohomology Theories of Varieties via Fundamental Groupoids

Abstract

We prove that the cohomology of a variety is fully recovered from the fundamental groupoids of its Zariski open subsets in various settings. This follows from a new sheaf theory that replaces the target category of Zariski sheaves with a fibered category while retaining the Zariski site as the source. Given a fundamental groupoid functor and a suitable abelian coefficient category, this theory produces the corresponding cohomology theory on varieties, providing a uniform construction of several cohomology theories. The topological fundamental groupoid with abelian groups recovers singular cohomology. The \'etale fundamental groupoid with discrete abelian groups recovers \'etale cohomology. The pro-algebraic fundamental groupoid with vector spaces recovers algebraic de Rham cohomology. The pro-algebraic fundamental groupoid with commutative formal groups unifies \'etale and algebraic de Rham cohomology. In each case, we prove a comparison theorem with the corresponding classical theory or theories. We conjecture that the Nori fundamental groupoid with commutative formal groups unifies \'etale and p-adic cohomology.