Special and general types
Traditional étale cohomology (e.g. Deligne 77) is the abelian sheaf cohomology for sheaves on the étale site of a scheme – which is an analog of the category of open subsets of a topological space , or rather the analog of the category of étale spaces over , with finite fibers.
A certain inverse limit over étale cohomology groups for different coefficients yields ℓ-adic cohomology, which is a Weil cohomology theory.
More generally, there is étale generalized cohomology theory with coefficients in sheaves of spectra on the étale site (Jardine 97). Still more generally, there is étale generalized cohomology on the étale (∞,1)-site (Antieau-Gepner 12, Lurie).
Given a scheme of finite type, the small étale site is the category whose objects are étale morphisms and whose morphisms are morphisms of schemes completing triangles: (notice that the morphisms between étale morphisms are automatically étale). This category naturally carries a Grothendieck topology that makes it a site, the étale site.
For an abelian sheaf on , the étale cohomology of with coefficients in is the abelian sheaf cohomology with respect to this site.
The following are some basic properties of étale cohomology groups for various standard choices of coefficients.
Relation to Zariski cohomology
This is originally due to (Grothendieck, SGA 4 (Chapter VII, p355)). Reviews include (Tamme, II 1.3).
With coefficients in coherent modules
For a quasi-coherent sheaf of -modules and the induced étale sheaf (by the discussion at étale topos – Quasicohetent sheaves), then the edge morphism
of the Leray spectral sequence of remark 2 is an isomorphism for all , itentifying the abelian sheaf cohomology on the Zariski site with coefficients in with the étale cohomology with coefficients in .
Moreover, for affine we have
This is due to (Grothendieck, FGA 1). See also for instance (Tamme, II (4.1.2)).
By the discussion at edge morphism it suffices to show that
By the discussion at direct image (also at abelian sheaf cohomology), is the sheaf on the Zariski topology which is the sheafification of the presheaf given by
hence it is sufficient that this vanishes, or rather, by locality (sheafification) it suffices to show this vanishes for an affine algebraic variety.
By the existence of cofinal affine étale covers the full subcategory with the induced coverage is a dense subsite of affines. Therefore it suffices to show the statement there. Moreover, by the finiteness condition on étale morphisms every cover of may be refined by a finite cover, hence by an affine covering map
It follows (by a discussion such as e.g. at Sweedler coring) that the corresponding Cech cohomology complex
is of the form
known as the Amitsur complex.
Since is a faithfully flat morphism it follows by the descent theorem that this is exact, hence that the cohomology indeed vanishes.
With coefficients in a cyclic group
Under the given assumptions, the Artin-Schreier sequence (see there) induces a long exact sequence in cohomology of the form
where is the Frobenius endomorphism. By prop. 2 the terms of the form vanish, and so from exactness we find an isomorphism
hence the claimed isomorphism
By the same argument all the higher cohomology groups vanish, as claimed.
With coefficients in the multiplicative group
the étale cohomology groups with coefficients in the multiplicative group in the first few degrees go by special names:
With coefficients in groups of roots of unity
(Tamme, II, 4.4)
The following are the main theorems characterizing properties of étale cohomology. Together these theorems imply that étale cohomology, in its variant as l-adic cohomology, is a Weil cohomology theory.
Proper base change theorem
(Milne, section 17)
Comparison theorem: Relation to singular cohomology
(Milne, section 21)
(Milne, section 22)
(Milne, section 23)
(Milne, section 24)
Lefschetz fixed-point formula
Künneth formula + cycle map + Poincaré duality Lefschetz fixed-point formula
(Milne, section 25)
History, motivation and original accounts
Étale cohomology was conceived by Artin, Deligne, Grothendieck and Verdier in 1963. It was used by Deligne to prove the Weil conjectures. Some useful (and also funny) remarks on this are in the beginning of
The classical references include SGA, esp.
- Pierre Deligne et al., Cohomologie étale , Lecture Notes in Mathematics 569, Springer-Verlag, 1977.
- Barry Mazur, Notes on étale cohomology of number fields, Annales scientifiques de l’École Normale Supérieure, Sér. 4, 6 no. 4 (1973), p. 521-552 (Numdam, pdf)
Reviews and modern accounts
A modern textbook, though largely based on the material in SGA is
- Lei Fu, Étale cohomology theory, Nankai Tracts in Math. 13, World Sci. 2011; (toc pdf; Preface pdf; chap. 1 Descent theory pdf)
Lecture notes include
Evan Jenkins, Étale cohomology seminar (web)
Donu Arapura, An introduction to Étale cohomology (pdf)
Antoine Ducros, Étale cohomology of schemes and analytic spaces, (pdf)
Edgar Costa, Étale cohomology (pdf)
Thomas H. Geisser, Weil-etale motivic cohomology, K-th archive
Discussion of generalized cohomology theory on the étale site but with coefficients in sheaves of spectra is in
- Rick Jardine, Generalized Étale cohomology theories, 1997 Progress in mathematics volume 146
Discussion of generalized étale cohomology over the étale (∞,1)-site (hence in higher topos theory/higher algebra) is in