The pro-\'etale topology for schemes

TL;DR

This paper introduces the pro-étale topology for schemes, providing a simplified framework for the derived category of constructible ll-adic sheaves and a refined fundamental group that captures all lisse ll-adic sheaves, even on non-normal schemes.

Contribution

It defines the pro-tale topology, enabling a more intuitive and effective approach to ll-adic sheaves and fundamental groups in algebraic geometry.

Findings

The pro-tale site is locally contractible.

A new simplified definition of the derived category of ll-adic sheaves.

A refined fundamental group capturing all lisse ll-adic sheaves.

Abstract

We give a new definition of the derived category of constructible $\ell$-adic sheaves on a scheme, which is as simple as the geometric intuition behind them. Moreover, we define a refined fundamental group of schemes, which is large enough to see all lisse $\ell$-adic sheaves, even on non-normal schemes. To accomplish these tasks, we define and study the pro-\'etale topology, which is a Grothendieck topology on schemes that is closely related to the \'etale topology, and yet better suited for infinite constructions typically encountered in $\ell$-adic cohomology. An essential foundational result is that this site is locally contractible in a well-defined sense.