A Short Course on Galois Cohomology - William Stein - University of ...

22.2 Étale Cohomology
Inspired by Serre’s introduction of homological techniques into Algebraic
Geometry, Grothendieck introduced the idea of étale cohomology in 1958.
He was partly motivated by a desire to more deeply understand and prove
Weil’s conjectures, which imply the function field analogue of the Riemann
Hypothesis (Deligne did indeed prove these conjectures in the 1970s using
étale cohomology). Étale cohomology gives a precise meaning to
H q
ét (X, F),
where X is a scheme and F is a “sheaf for the étale site on X”, which is a
generalization of the notion of module (and also related to the notion of sheaf
for the Zariski topology on X). An example of a scheme is X = Spec(K),
where K is a field, in which case étale sheaves are in bijection with GKmodules,
and
H q
ét (X, F) = Hq
ét (Spec(K), F) = Hq (K, F( ¯ K))
is just the usual concrete notion of Galois chomology that we now know and
love.
Thus étale cohomology is an extremely powerful and natural generalization
of Galois cohomology.
Remark 22.4. If you know sheaf cohomology in the context of algebraic geometry
and the Zariski topology on a scheme (e.g., chapter 3 of Hartshorne’s
book Algebraic Geometry), then you’ll find the following interesting:
“The reason that the Zariski topology does not work well is that
it is too coarse: it has too few open sets. There seems to be
no good way to fix this by using a finer topology on a general
algebraic variety. Grothendieck’s key insight was to realize that
there is no reason why the more general open sets should be
subsets of the algebraic variety: the definition of a sheaf works
perfectly well for any category, not just the category of open
subsets of a space. He defined étale cohomology by replacing
the category of open subsets of a space by the category of étale
mappings to a space: roughly speaking, these can be thought of
as open subsets of finite unbranched covers of the space. These
turn out (after a lot of work) to give just enough extra open sets
that one can get reasonable cohomology groups...”
– the Wikipedia article on étale cohomology
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