# Usage of étale cohomology in algebraic geometry

I'm a student interested in arithmetic geometry, and this implies I use étale cohomology a lot. Regarding its definition, étale cohomology is a purely algebro-geometric object. However, almost every material I found on étale cohomology focus on its number-theoretic applications, such as the Weil conjectures and Galois representations. So, this is my question:

Are there some applications of étale cohomology on pure algebro-geometric problems?

Here, "pure algebro-geometric problems" means some problems of algebraic geometry without number-theoretic flavors, such as birational geometry, the minimal model program, classifying algebraic varieties (curves, surfaces, etc..), especially over algebraically closed fields.

Since étale cohomology coincides with singular cohomology over $$\mathbb{C}$$, there must exist such problems over the complex numbers. Hence, I am looking for applications of étale cohomology which are also useful over algebraically closed fields which are not $$\mathbb{C}$$.

• Well, one can use étale cohomology to solve geometric problems using arithmetic methods. The idea is that a complex algebraic variety is in fact defined by polynomial equations with coefficients which are in a ring of finite type $R$ over $\mathbb Z$. Moereover, what is true for our variety is in fact true on a dense open subscheme of $R$. Playing with étale local systems, we see that many properties over $\mathbb C$ can be detected by pulling back over a closed point of $Spec (R)$. In other words, many theorems in algebraic geometry over finite fields produce theorems in complex geometry. Jan 7 '20 at 11:13
• Explicit instances of the principle I skteched above are Deligne's proof of the relative hard Lefschetz theorem (thm. 6.2.5 in Astérisque 100). The main tools are the version over finite fields (which uses the theory of weights) and Lemma 6.2.6 in loc. cit. which explains how to relate complex geometry and geometry over finite fields using suitable derived categories of étale l-adic sheaves. Jan 7 '20 at 11:16
• It's not algebraic geometry, but an important application of $\ell$-adic cohomology which is not "number theory" is the representation theory of classical groups over finite fields, a.k.a., Deligne-Lusztig theory (en.wikipedia.org/wiki/Deligne%E2%80%93Lusztig_theory) Jan 7 '20 at 21:30

There are many, for example, Artin's proof in nonzero characteristic of Castelnuovo's criterion for the rationality of a surface, and a proof that the Neron-Severi group is finitely generated. Both of these are in Chapter V of Milne's book (3.25, 3.30).

The Betti numbers of many (complex) moduli spaces have been computed by counting points over finite fields, using the Weil conjectures, as proved by Deligne, and comparison theorems for étale and singular cohomology. The first example of which I am aware is Lothar Göttsche's calculation of the Betti numbers of Hilbert schemes of points on a smooth projective surface.

• The work of Harder and Narasimhan on the moduli of vector bundles on curves predates Gottsche's work by about 20 years.
– naf
Jan 16 '20 at 11:43
• Right, I had forgotten about that. Jan 16 '20 at 19:53

Another interesting (at least to me) application of étale cohomology is the following. The étale cohomological dimension of the complement of a variety $$V$$ gives a lower bound on the arithmetical rank of $$V$$, i.e., the minimum number of equations needed to define $$V$$ set-theoretically. For example, Bruns and Schwänzl [BS90] used this to prove that over an algebraically closed field the determinantal variety defined by $$t$$-minors of a generic $$m\times n$$ matrix has arithmetical rank $$mn-t^2+1$$.

[BS90] Bruns, Winfried, Schwänzl, Roland - The number of equations defining a determinantal variety. Bull. London Math. Soc. 22 (1990), no. 5, 439–445.

Borel has a nice \'etale cohomological proof of Matsushima's criterion: A homogeneous space of a connected reductive group is affine if and only if the isotropic group is reductive.

(The paper is https://doi.org/10.1007/BF01194008)