Étale cohomology

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In mathematics, the étale cohomology theory of algebraic geometry is a refined construction of homological algebra, introduced in order to attack the Weil conjectures. This eventually proved successful as a strategy, although Dwork managed to prove the rationality part of the conjectures in 1960 using p-adic methods; the remaining conjectures awaited the development of étale cohomology. This theory is an example of a Weil cohomology theory in algebraic geometry, and as such it continues to play an important role in the more general theory of motives.

The formal definition of étale cohomology is as the derived functor of the functor of sections,

F → Γ(F),

for a type of sheaf. The sections of a sheaf can be thought of as Hom(Z,F) where Z is the sheaf that returns the integers as an abelian group; the sheaf F is understood in the sense of a Grothendieck topology. The idea of derived functor here is that the sheaf of sections doesn't respect exact sequences; according to general principles of homological algebra there will be a sequence of functors Hi for i = 0,1, ... that represent the 'compensations' that must be made in order to restore some measure of exactness (long exact sequences arising from short ones). The H0 functor coincides with the section functor Γ.

In these very abstract terms, the existence of such a theory comes down to some properties of étale morphisms in scheme theory, allowing us to use étale coverings as a Grothendieck topology, and some further proofs in homological terms, showing for example that injective resolutions are to be found in the sheaf category. To a very great extent, this attitude masks what is going on.

Some basic intuitions of the theory are these:

  • The étale requirement is the condition that would allow one to apply the implicit function theorem if it were true in algebraic geometry (but it isn't - implicit algebraic functions are called algebroid in older literature).
  • There are certain basic cases, of dimension 0 and 1, and for an abelian variety, where the answers with constant sheaves of coefficients can be predicted (via Galois cohomology and Tate modules).

As it turned out, these base cases in effect determined the theory (perhaps unexpectedly), but the case of a general sheaf on a curve is already complex. 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. The burden and success of the general theory was certainly both to integrate all this information, and to prove general results such as Poincaré duality and the Lefschetz fixed point theorem in this context.

With hindsight, much of the general machinery of topos theory proved unnecessary for a minimal treatment of the étale theory (though applicable to the more subtle crystalline and flat cohomology) — this is Deligne's view as expressed for example in SGA4½. On the other hand, étale cohomology quickly found other applications, for example in representation theory, going beyond the initially planned application.

l-adic cohomology groups

In applications to algebraic geometry over a finite field F, the main objective was to find a replacement for the singular cohomology groups, which are not available in the same way as for geometry of an algebraic variety over the complex number field. The hope, which was generally upheld, was that a replacement would be found in the shape of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell} -adic cohomology. Here Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell} stands for any prime number with

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell} p

where p is the characteristic of F. One considers, for schemes V, the cohomology groups

Hi(V, Z /Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell} kZ)

and defines

Hi(V, ZFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell} )

as their inverse limit. Here ZFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell} denotes the l-adic integers, but the definition is by means of the system of 'constant' sheaves with the finite coefficients Z/Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell} kZ.

The reason that one might guess that this leads to the correct definition, is that in the case that V is a non-singular algebraic curve and i = 1, it can be shown that H1 is a free ZFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell} -module of rank 2g, dual to the Tate module of the Jacobian variety of V, where g is the genus of V. Since the first Betti number of a Riemann surface of genus g is 2g, that value is reassuring. This becomes a kind of 'base case' for inductive study of the general case (that is, i > 1 or V of dimension > 1). It also shows why the condition Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell} p is required: when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell} = p the rank of the Tate module is at most g.

To remove any torsion subgroup from the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell} -adic groups (which can occur, and was applied by Mike Artin and David Mumford to geometric questions) the definition

Hi(V, QFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell} )

with the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell} -adic numbers QFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell} is typically used.

An application to curves

This is how the theory could be applied to the local zeta-function of an algebraic curve.

Theorem. Let X be a curve of genus g defined over the finite field with p elements. Then for every n greater or equal 1 one has

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \#X(\mathbb F_{p^n}) = 1 -\sum_{i=1}^{2g} \alpha_i^n+p^n} ,

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha_i} are certain algebraic numbers satisfying Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\alpha_i|=\sqrt p} .

Notes

  • This agrees with the projective line being a curve of genus 0 and having pn+1 points.
  • We see that number of points on any curve is 'rather close' to that of the projective line.

Idea of proof

According to the Lefschetz fixed point theorem, the number of fixed points of any morphism Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:X\to X} is equals to the sum

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i=0}^{\dim X}(-1)^i\mathrm{Tr} f|_{H^i(X)}} .

This formula is valid for ordinary topological varieties and ordinary topology, but it is wrong for most algebraic topologies. However, this formula does hold for étale cohomology (though this is not so simple to prove).

The points of X that are defined over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb F_{p^n}} 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 resp. 1, 2g, and 1.

According to all of these,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \#X(\mathbb F_{p^n}) = \mathrm{Tr} F^n|_{H^0(X)} -\mathrm{Tr} F^n|_{H^1(X)} +\mathrm{Tr} F^n|_{H^2(X)}} .

This gives the general form of the theorem.

The assertion on the absolute values of the αs requires some deeper argument.

The whole idea fits into the framework of motives: formally [X] = [point]+[line]+[1-part], and [1-part] has something like Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt p} points.