topos theory

# Contents

## Idea

The étale site of a scheme is an analog of the category of open subsets of a topological space. The corresponding cohomology is étale cohomology.

The étale topology has similar cohomological properties to complex analytic topology?, and in particular it is much finer for cohomological purposes than the Zariski topology.

## Definition

###### Definition

Let $X$ be a scheme.

The big étale site ${\mathrm{Sch}}_{/X,\mathrm{et}}$ of $X$ is the over category ${\mathrm{Sch}}_{/X}$ of schemes over $X$ equipped with the coverage given by étale covers (after forgetting the maps to $X$).

The small étale site $i:{X}_{\mathrm{et}}↪\mathrm{Sch}/X,\mathrm{et}$ is the full subcategory of ${\mathrm{Sch}}_{/X}$on the étale morphisms $U\to X$.

The abelian sheaf cohomology of the étale site is called étale cohomology.

## Properties

### Cohomology

###### Proposition

The inverse image restriction functor ${i}^{*}\mathrm{Sh}\left({\mathrm{Sch}}_{/X,\mathrm{et}},\mathrm{Ab}\right)\to \mathrm{Sh}\left({X}_{\mathrm{et}},\mathrm{Ab}\right)$ on the categories of sheaves with values in Ab

###### Corollary

For $X$ a scheme and $F\in \mathrm{Sh}\left({\mathrm{Sch}}_{/X,\mathrm{et}},\mathrm{Ab}\right)$ we have that the etale cohomology of $X$ with coefficients in $F$ may be computed on the small site:

${H}^{p}\left({X}_{\mathrm{et}},F{\mid }_{\mathrm{et}}\right)\simeq {H}^{p}\left(X,F\right)\phantom{\rule{thinmathspace}{0ex}}.$H^p(X_{et}, F|_{et}) \simeq H^p(X,F) \,.

This appears for instance in (deJong, prop. 3.4).

### Derived geometry

The derived geometry of the étale site is the étale (∞,1)-site. The precise statement is at derived étale geometry.

fpqc-site $\to$ fppf-site $\to$ syntomic site $\to$ étale site $\to$ Nisnevich site $\to$ Zariski site

## References

The classical references are

• Pierre Deligne et al., Cohomologie étale , Lecture Notes in Mathematics, no. 569, Springer-Verlag, 1977.
• James Milne, Etale cohomology, Princeton Mathematical Series 33, 1980. xiii+323 pp.

A detailed survey is in chapter 34 of

Lecture notes are

Revised on April 3, 2012 14:51:31 by Tim Porter (95.147.236.147)