The lack of a suitable resolution of singularities comes up often in work on étale cohomology from the 1960s and 70s, And I think even the latest version of Milne's lecture notes says "It is likely that de Jong’s resolution theorem (Smoothness, semi-stability and alterations. Inst. Hautes Etudes Sci. Publ. Math. No. 83 (1996), 51–93) will allow many improvements" to both theorems and proofs. Is there currently some comprehensive treatment of étale cohomology that uses this?

In other words has anyone worked on realizing the hope that Milne expressed, that basic questions already prominent in the SGA could be usefully "re-thought" by greater use of Artin's algebraic spaces, de Jong's removal of singularities, or other new ideas?

Ten days after posting the question, the community wiki answer makes me think the situation is as follows, though I would love to learn otherwise: On one hand, étale cohomology finds so many concrete applications treating the general theory as a "black box," that there is not much pressure to rethink the general theorems. Proofs can cite theorems from the books by Freitag and Kiehl or Milne. Those books in turn give many general theorems without proof, citing SGA for the proofs.

On the other hand, new fundamental ideas keep coming in. Notably Artin's algebraic spaces, plus de Jong's theorem, made Milne say "the structure of the subject needs to be re-thought" (http://www.jmilne.org/math/CourseNotes/LEC.pdf page 6). More recently Ofer Gabber's work suggests the same but up to now that work has been presented on the base of the SGA and not as a way to reorganize the SGA.

Because such results are still coming in, and raise many possibilities, comprehensive treatments of étale cohomology even through Lei Fu, Etale Cohomology Theory (Nankai Tracts in Mathematics) are still substantially based on the SGA, including SGA 4.5. SGA 4.5 did revise several major proofs from SGA 4. But no one has yet attempted a thoroughgoing synthesis with the latest ideas.

I will mention de Jong's http://stacks.math.columbia.edu/download/etale-cohomology.pdf does not seem to reach the issues where resolution of singularities would matter.

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    $\begingroup$ What comes to my mind is the introduction of "Travaux de Gabber sur l'uniformization locale [...]", that you can find [here][1], where there are some examples where Gabber refinement of de Jong's alterations method is used to establish results for non projective morphisms (see, in the link Theorem 1 and Theorem 5, for example). It does not qualify as "comprehensive treatment of étale cohomolgy", of course. [1]: cmls.polytechnique.fr/perso/orgogozo/travaux_de_Gabber/… $\endgroup$
    – Giulia
    Apr 8, 2016 at 12:13

1 Answer 1


I am turning Giulia's comment into a CW answer. At over 400 pages, I think that "Travaux de Gabber", published in Astérisque, ought to count as a "comprehensive treatment of [some aspects of ] étale cohomology" using the method of alterations. The style is very SGA, and indeed there is a nonempty intersection among the set of authors.


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