The Stacks project

61.1 Introduction

The material in this chapter and more can be found in the preprint [BS].

The goal of this chapter is to introduce the pro-étale topology and to develop the basic theory of cohomology of abelian sheaves in this topology. A secondary goal is to show how using the pro-étale topology simplifies the introduction of $\ell $-adic cohomology in algebraic geometry.

Here is a brief overview of the history of $\ell $-adic étale cohomology as we have understood it. In [Exposés V and VI, SGA5] Grothendieck et al developed a theory for dealing with $\ell $-adic sheaves as inverse systems of sheaves of $\mathbf{Z}/\ell ^ n\mathbf{Z}$-modules. In his second paper on the Weil conjectures ([WeilII]) Deligne introduced a derived category of $\ell $-adic sheaves as a certain 2-limit of categories of complexes of sheaves of $\mathbf{Z}/\ell ^ n\mathbf{Z}$-modules on the étale site of a scheme $X$. This approach is used in the paper by Beilinson, Bernstein, and Deligne ([BBD]) as the basis for their beautiful theory of perverse sheaves. In a paper entitled “Continuous Étale Cohomology” ([Jannsen]) Uwe Jannsen discusses an important variant of the cohomology of a $\ell $-adic sheaf on a variety over a field. His paper is followed up by a paper of Torsten Ekedahl ([Ekedahl]) who discusses the adic formalism needed to work comfortably with derived categories defined as limits.

It turns out that, working with the pro-étale site of a scheme, one can avoid some of the technicalities these authors encountered. This comes at the expense of having to work with non-Noetherian schemes, even when one is only interested in working with $\ell $-adic sheaves and cohomology of such on varieties over an algebraically closed field.

A very important and remarkable feature of the (small) pro-étale site of a scheme is that it has enough quasi-compact w-contractible objects. The existence of these objects implies a number of useful and (perhaps) unusual consequences for the derived category of abelian sheaves and for inverse systems of sheaves. This is exactly the feature that will allow us to handle the intricacies of working with $\ell $-adic sheaves, but as we will see it has a number of other benefits as well.