An exceptional review of a paper by Bhatt and Scholze on étale topology

Another great review.  Here Pieter Belmans reviews a paper by Bhatt and Scholze on étale topology.  Before describing the authors’ work, Belmans tells us where étale topology comes from and why some news ideas might be necessary.  He then gives a quick description of what Bhatt and Scholze are doing and why it is a good thing.  Once the history and context are in place, Belmans goes through the contents of the paper, with plenty of comments to help the reader.  He concludes by giving a reference to the Stacks Project, where you can find out lots more about pro-étale cohomology.

Belmans is a mathematician and a coder. He is very involved in the Stacks Project.   On a smaller scale, he wrote a handy python script that helps MathSciNet users obtain BibTeX versions of the references in a LaTeX file.  This is an update to a clunky shell script to do the same thing.

MR3379634
Bhatt, Bhargav(1-MI); Scholze, Peter(D-BONN)
The pro-étale topology for schemes. (English, French summary)
Astérisque No. 369 (2015), 99–201. ISBN: 978-2-85629-805-3
14F05 (14F20 14F35 14H30 18B25)

One of the reasons for the introduction of the étale topology was the definition of the $\ell$-adic Weil cohomology theory. In trying to mimic the approach from algebraic topology by using constant sheaves on varieties over some (algebraically closed) field $k$, the need for a topology finer than the Zariski topology arises, and the étale topology is a reasonable candidate for this. Unfortunately one easily shows that things only work as intended for torsion sheaves, while the goal is to obtain coefficients in a field (of characteristic zero), such as $\overline{\Bbb{Q}}_{\ell}$, where $\ell$ is prime to the characteristic of $k$. So taking sheaf cohomology of the constant sheaf associated to $\overline{\Bbb{Q}}_\ell$ does not yield satisfactory results. Nevertheless, it is possible to rectify the situation, by defining $\ell$-adic cohomology as the inverse limit of the cohomology of $\Bbb{Z}/\ell^n\Bbb{Z}$ in the étale topology, and tensoring it with $\overline{\Bbb{Q}}_\ell$. It can be shown that this indeed satisfies the axioms for a Weil cohomology theory.

The price one pays for this approach is that one is not working directly with étale sheaves of $\Bbb{Z}_\ell$– or $\overline{\Bbb{Q}}_\ell$-modules, but rather with pro-sheaves of $\Bbb{Z}/\ell^n\Bbb{Z}$-modules. Therefore the usual yoga of setting up sheaf cohomology does not work: one does not get abelian categories and injective resolutions for free. Also checking that the functors constructed in this ad hoc fashion indeed satisfy the axioms for a Weil cohomology theory is hard, because they are not derived functors as such. This becomes only more problematic when one tries to study $\ell$-adic sheaves in a relative setting, where one would like to have an appropriate triangulated category that behaves like the derived category of sheaves. These were obtained by Deligne and later T. Ekedahl [in The Grothendieck Festschrift, Vol. II, 197–218, Progr. Math., 87, Birkhäuser Boston, Boston, MA, 1990; MR1106899] along the lines of Grothendieck’s ad hoc definition, whilst U. Jannsen’s continuous étale cohomology [Math. Ann. 280 (1988), no. 2, 207–245; MR0929536] gives a theory of $\ell$-adic cohomology for non-algebraically closed base fields, where Mittag-Leffler-type conditions are not readily available.

The paper under review shows how one can overcome these difficulties by changing the underlying site. Indeed, the main result is that $\ell$-adic cohomology truly is sheaf cohomology in the pro-étale topos, and that the derived category one obtains naturally in this way is indeed equivalent to the triangulated category constructed in an ad hoc fashion. The reason for this is that this new site has better behaviour with respect to inverse limits, thereby eliminating many of the technicalities one encounters in the classical approach.

In section 2 the theory of weakly étale and pro-étale morphisms of rings is introduced in a detailed fashion. A weakly étale morphism of rings (also called absolutely flat) is a flat morphism whose diagonal is also flat. A pro-étale morphism is an inductive limit of étale morphisms of rings. One easily shows that étale implies pro-étale implies weakly étale implies formally étale. So these two new notions are weakenings of the finiteness conditions for étale morphisms, and various properties are proven.

In section 3 the notion of replete and locally weakly contractible topos is introduced. A replete topos is one where surjections are closed under sequential limits. It is precisely this property that makes inverse limits behave just like in the category of sets. One sees by virtue of an easy example that the usual topologies (such as Zariski, étale or fppf) certainly do not satisfy this property, whilst the fpqc topology (with the usual appropriate set-theoretical precautions) defines a replete topos.

The notion of locally weakly contractible topos is a strengthening of repleteness, and ensures for instance that the derived category of abelian objects is compactly generated. In the next section it will be shown that the pro-étale topology defines such a locally weakly contractible topos. Another important property of replete toposes is that their derived categories are left-complete, which ensures unbounded cohomological descent, without any conditions on the cohomological dimension, which are usually needed for the étale site. The remainder of the section is dedicated to studying the behaviour of (derived) completions of rings, first in absolute generality and later specialised to the case of Noetherian constant rings.

In section 4 one is given the definition of the pro-étale topology for schemes: one takes the category of weakly étale schemes over the base, and takes covers from the fpqc topology (suitably taking care of set-theoretical issues, as for the fpqc topology). The reason for using weakly étale morphisms is that being pro-étale is not Zariski local on the target, but the authors show that every weakly étale morphism $f\colon Y\to X$ is Zariski locally on $X$ and locally for the pro-étale topology on$Y$ a pro-étale morphism of rings. In other words: the pro-étale site is an analogue of the small étale site, but the objects are weakly étale whilst the coverings are fpqc coverings. The main result is that the pro-étale site is subcanonical, generated by affines, and that the topos is locally weakly contractible as defined in the previous section.

Section 5 is dedicated to the comparison of the étale and the pro-étale topos. Because every étale map is also weakly étale, we get a morphism of toposes${\nu\colon\textrm{Shv}(X_{\text{pro-ét}})\to\textrm{Shv}(X_{\textrm{ét}})}$. It is shown that $\nu^*$ is fully faithful for sheaves of sets and sheaves of abelian groups, and for bounded below complexes. For unbounded complexes the issue regarding left-completeness appears, and extra care is needed, but one can show that (an appropriate subcategory of) the unbounded derived category from the pro-étale site realises the left completion of the unbounded derived category from the étale site. Finally a comparison with Ekedahl’s and Jannsen’s theory is given.

The main motivation for the work can be found in section 6, where the notions of constructible sheaves on the two toposes are compared. It is shown how the pro-étale approach has completely analogous recollement properties for closed subschemes (resp. descriptions of derived categories supported on locally closed constructible subsets). Then constructibility in the étale topology is recalled, and it is shown how constructible complexes form precisely the compact objects of the derived category of $A$-modules, where $A$ is a ring such that affines have bounded cohomological dimension with respect to this ring. Over an algebraically closed field this condition is satisfied for torsion coefficients, which is precisely what makes the usual theory work as intended. The authors then compare all this to constructibility in the pro-étale topology, in particular on Noetherian schemes. Finally it is shown how one obtains a six-functor formalism and that the machinery for$\overline{\Bbb{Q}}_\ell$-sheaves indeed gives equivalent triangulated categories.

Finally, section 7 develops the theory of fundamental groups in the pro-étale topology. In the étale context one has a profinite fundamental group defined in SGA1 and a prodiscrete fundamental group defined in SGA3. The main result is that these are the profinite (resp. prodiscrete) completions of the pro-étale fundamental group; hence this theory recovers the earlier constructions.

The text is very well written, and contains many instructive examples and references. It gives ample motivation for the approach that is taken, and the resulting machinery is indeed beautiful, with the strong relation to (and various improvements of) the classical theory being the main theme of the work. At the moment it is the main reference text on the definition of the pro-étale topology, besides the development of some of the tools and results in the Stacks Project, tag 0965.

Reviewed by Pieter Belmans