A Uniform Construction of Cohomology Theories of Varieties via Fundamental Groupoids

Time: 13:30, October 29

Place: Room 612, Building A6

Online Zoom link: 

https://zoom.us/j/96772606174?pwd=nMziWbVbk31HaB31xjJbZp2Cl1f6e2.1

Meeting ID: 967 7260 6174
Passcode: 123456

Abstract: We prove that the cohomology of a variety is fully recovered from the diagram of fundamental groupoids of its Zariski open subsets in various cohomological 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. Our sheaves assign each Zariski open subset a module over its fundamental groupoid. Each choice of a fundamental groupoid functor and suitable abelian coefficient category yields a cohomology theory on varieties, providing a uniform construction of several cohomology theories.

The topological fundamental groupoid with abelian groups recovers singular cohomology. The étale fundamental groupoid with discrete abelian groups recovers étale cohomology. The pro-algebraic fundamental groupoid with vector spaces recovers algebraic de Rham cohomology. Most importantly, the pro-algebraic fundamental groupoid with commutative formal groups recovers both étale and algebraic de Rham cohomology in a single sheaf category. 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 recovers both étale and p-adic cohomology.