In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures.
This has many applications, such as the proof of the Weil conjectures and the construction of representations of finite groups of Lie type.
Further contact with classical theory was found in the shape of the Grothendieck version of the Brauer group; this was applied in short order to diophantine geometry, by Yuri Manin.
Grothendieck's use of these universes (whose existence cannot be proved in Zermelo–Fraenkel set theory) led to some speculation that étale cohomology and its applications (such as the proof of Fermat's Last Theorem) require axioms beyond ZFC.
However, in practice étale cohomology is used mainly in the case of constructible sheaves over schemes of finite type over the integers, and this needs no deep axioms of set theory: with care the necessary objects can be constructed without using any uncountable sets, and this can be done in ZFC, and even in much weaker theories.
Étale cohomology quickly found other applications, for example Deligne and George Lusztig used it to construct representations of finite groups of Lie type; see Deligne–Lusztig theory.
In the case of cohomology of coherent sheaves, Serre showed that one could get a satisfactory theory just by using the Zariski topology of the algebraic variety, and in the case of complex varieties this gives the same cohomology groups (for coherent sheaves) as the much finer complex topology.
However, for constant sheaves such as the sheaf of integers this does not work: the cohomology groups defined using the Zariski topology are badly behaved.
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.
The intersection of two open sets of a topological space corresponds to the pullback of two étale maps to X.
A presheaf on a topological space X is a contravariant functor from the category of open subsets to sets.
In applications to algebraic geometry over a finite field Fq with characteristic p, the main objective was to find a replacement for the singular cohomology groups with integer (or rational) coefficients, which are not available in the same way as for geometry of an algebraic variety over the complex number field.
Here Zℓ denotes the ℓ-adic integers, but the definition is by means of the system of 'constant' sheaves with the finite coefficients Z/ℓkZ.
When V is a non-singular algebraic curve of genus g, H1 is a free Zℓ-module of rank 2g, dual to the Tate module of the Jacobian variety of V. Since the first Betti number of a Riemann surface of genus g is 2g, this is isomorphic to the usual singular cohomology with Zℓ coefficients for complex algebraic curves.
It also shows one reason why the condition ℓ ≠ p is required: when ℓ = p the rank of the Tate module is at most g. Torsion subgroups can occur, and were applied by Michael Artin and David Mumford to geometric questions[citation needed].
To remove any torsion subgroup from the ℓ-adic cohomology groups and get cohomology groups that are vector spaces over fields of characteristic 0 one defines This notation is misleading: the symbol Qℓ on the left represents neither an étale sheaf nor an ℓ-adic sheaf.
In general the ℓ-adic cohomology groups of a variety tend to have similar properties to the singular cohomology groups of complex varieties, except that they are modules over the ℓ-adic integers (or numbers) rather than the integers (or rationals).
This step works for varieties X of any dimension (with points replaced by codimension 1 subvarieties), not just curves.
If μn is the sheaf of n-th roots of unity and n and the characteristic of the field k are coprime integers, then: where Picn(X) is group of n-torsion points of Pic(X).
This follows from the previous results using the long exact sequence of the Kummer exact sequence of étale sheaves and inserting the known values In particular we get an exact sequence If n is divisible by p this argument breaks down because p-th roots of unity behave strangely over fields of characteristic p. In the Zariski topology the Kummer sequence is not exact on the right, as a non-vanishing function does not usually have an n-th root locally for the Zariski topology, so this is one place where the use of the étale topology rather than the Zariski topology is essential.
with compact support vanish if q > 2n, and if in addition X is affine of finite type over a separably closed field the cohomology groups
More generally if f is a separated morphism of finite type from X to S (with X and S Noetherian) then the higher direct images with compact support Rqf!
are defined by for any torsion sheaf F. Here j is any open immersion of X into a scheme Y with a proper morphism g to S (with f = gj), and as before the definition does not depend on the choice of j and Y. Cohomology with compact support is the special case of this with S a point.
is the same as the usual higher direct image with compact support (for the complex topology) for torsion sheaves.
If X is a smooth algebraic variety of dimension N and n is coprime to the characteristic then there is a trace map and the bilinear form Tr(a ∪ b) with values in Z/nZ identifies each of the groups and with the dual of the other.
The points of X that are defined over Fpn are those fixed by Fn, where F is the Frobenius automorphism in characteristic p. The étale cohomology Betti numbers of X in dimensions 0, 1, 2 are 1, 2g, and 1 respectively.