Etale Cohomology of Fields and Galois Cohomology

In Cohomology in Algebraic Geometry we have introduced sheaf cohomology and Cech cohomology as well as the concept of etale morphisms, and the Grothendieck topology (see More Category Theory: The Grothendieck Topos) that it defines. In this post, we give one important application of these ideas, related to the ideas discussed in Galois Groups.

Let $K$ be a field (see Rings, Fields, and Ideals). A field has only two ideals: $(0)$ and $(1)$ , the latter of which is the unit ideal and is therefore the entire field itself as well. Its only prime ideal (which is also a maximal ideal) is $(0)$ ; recall that in algebraic geometry (see Basics of Algebraic Geometry), the “points” of the mathematical object we call a scheme correspond (locally, at least) to the prime ideals of a ring $R$ , and we refer to this set of “points” as $\text{Spec }R$ . Therefore, for the field $K$ , $\text{Spec }K=(0)$ , in other words, $\text{Spec }K$ is made up of a single point.

Now we need to define sheaves on $\text{Spec }K$ . Using ordinary concepts of topology will not be very productive, since our topological space consists only of a single point; therefore, we will not be able to obtain any interesting open covers out of this topological space. However, using the ideas in More Category Theory: The Grothendieck Topos, we can “expand” our idea of open covers. Instead of inclusions of open sets, we will instead make use of etale morphisms, as we have discussed in Cohomology in Algebraic Geometry.

Let $K\rightarrow L$ be an etale morphism. This also means that $L$ is an etale $K$ -algebra (see also The Hom and Tensor Functors for the definition of algebra in our context). It is a theorem that an etale $K$ -algebra is a direct product of finitely many separable field extensions of $K$ (see Algebraic Numbers).

The definition of presheaf and sheaf remains the same, however the sheaf conditions can be restated in our case as the following (perhaps easier to understand) statement, which we copy verbatim from the book Etale Cohomology and the Weil Conjecture by Eberhard Freitag and Reinhardt Kiehl:

The elements $s\in \mathcal{F}(B)$ correspond one-to-one to families of elements

$s_{i}\in\mathcal{F}(B_{i})$ , $i\in I$

having the property

$\text{Image }(s_{i})=\text{Image }(s_{j})$ , in $(B_{i}\otimes_{B}B_{j})$

This condition must also hold for $i=j$ !

A separable closure $\bar{K}$ of $K$ is a separable field extension of $K$ (see Cohomology in Algebraic Geometry) that is a subfield of the algebraic closure of $K$ . The algebraic closure of $K$ is an algebraic extension (see Algebraic Numbers) of $K$ which is algebraically closed, i.e., it contains all the roots of polynomials with coefficients in this algebraic extension. Both the algebraic closure and the separable closure of $K$ are unique up to isomorphism. In the case of the field of rational numbers $\mathbb{Q}$ , the separable closure and the algebraic closure coincide and they are both equal to the field of algebraic numbers.

Given the separable closure $\bar{K}$ of $K$ , we define $\mathcal{F}(\bar{K})$ as the stalk (see Localization) of the sheaf at $\bar{K}$ . It is also written using the language of direct limits (also called an inductive limit):

$\displaystyle \mathcal{F}(\bar{K})=\varinjlim\mathcal{F}(L)$

We digress slightly in order to explain what this means. The language of direct limits and inverse limits (the latter are also called projective limits) are ubiquitous in abstract algebra, algebraic geometry, and algebraic number theory, and are special cases of the notion of limits we have discussed in Even More Category Theory: The Elementary Topos.

A directed set $I$ is an ordered set in which for every pair $i,j$ there exists $k$ such that $i\leq k,j\leq k$ . A direct, resp. inverse system over $I$ is a family $\{A_{i},f_{ij}|i,j\in I,i\leq j\}$ of objects $A_{i}$ and morphisms $f_{ij}: A_{i}\rightarrow A_{j}$ , resp. $f_{ij}: A_{j}\rightarrow A_{i}$ such that

$\displaystyle f_{ii}$ is the identity map of $A_{i}$ , and

$\displaystyle f_{ik}=f_{jk}\circ f_{ij}$ resp. $f_{ik}=f_{ij}\circ f_{jk}$

for all $i\leq j\leq k$ .

The direct limit of a direct system is then defined as the quotient

$\displaystyle \varinjlim_{i\in I} A_{i}=\coprod_{i\in I} A_{i}/\sim$

where two elements $x_{i}\in A_{i}$ and $x_{j}\in A_{j}$ are considered equivalent, $x_{i}\sim x_{j}$ if there exists $k$ such that $f_{ik}(x_{i})=f_{jk}(x_{j})$ .

Meanwhile, the inverse limit of an inverse system is the subset

$\displaystyle \varprojlim_{i\in I} A_{i}=\{(x_{i})_{i\in I}\in \prod_{i\in I}A_{i}|f_{ij}(x_{j})=x_{i}\text{ for }i\leq j \}$

of the product $\displaystyle \prod_{i\in I}A_{i}$ .

The classical definition of stalk, for a sheaf $\mathcal{F}$ can then also be expressed as the direct limit of the direct system given by the sets (or abelian groups, or modules, etc.) $\mathcal{F}(U)$ and the restriction maps $\rho_{UV}: \mathcal{F(U)}\rightarrow \mathcal{F}(V)$ for open sets $V\subseteq U$ . In our case, of course, instead of inclusion maps $V\subseteq U$ we instead have more general maps induced by etale morphisms.

An example of an etale sheaf over $\text{Spec }K$ is given by the following: Let

$\displaystyle \mathcal{G}_{m}(B)=B^{*}$ where $B^{*}$ is the multiplicative group of the etale $K$ -algebra $B$ .

In this case we have $\mathcal{F}(\bar{K})=\bar{K}^{*}$ , the multiplicative group of the separable closure $\bar{K}$ of $K$ . We note that the multiplicative group of a field $F$ is just the group $F-\{0\}$ , with the law of composition given by multiplication.

In order to make contact with the theory of Galois groups, we now define the concept of $G$ -modules, where $G$ is a group. A left $G$ -module is given by an abelian group $M$ and a map $\rho: G\times M\rightarrow M$ such that

$\displaystyle \rho(e,a)=x$ ,

$\displaystyle \rho (gh,a)=\rho(g,\rho(h,a))$ ,

and

$\displaystyle \rho(g,(ab))=\rho(g,a)\rho(g,b)$ .

Instead of $\rho(g,a)$ we usually just write $g\cdot a$ . A right $G$ -module may be similarly defined, and may be obtained from a left $G$ -module by defining $a\cdot g=g^{-1}\cdot a$ .

The abelian group $\mathcal{F}(\bar{K})$ has the structure of a $G$ -module, where $G$ is the Galois group $\text{Gal}(\bar{K}/K)$ (also written as $G(\bar{K}/K)$ ), the group of field automorphisms of $\bar{K}$ that keep $K$ fixed.

We see now that there is a connection between Galois theory and etale sheaves over a field. More generally, there is a connection between the Etale cohomology of a field and “Galois cohomology“, an important part of algebraic number theory that we now define. Galois cohomology is the derived functor (see More on Chain Complexes and The Hom and Tensor Functors) of the fixed module functor.

First we construct the standard resolution of the the Galois module (a $G$ -module where $G$ is the Galois group of some field extension) $A$ . It is given by $X^{n}(G,A)$ , the abelian group of all functions from the direct product $G^{n+1}$ to $A$ , and the coboundary map

$\displaystyle \partial^{n}: X^{n-1}\rightarrow X^{n}$

given by

$\displaystyle \partial^{n}x(\sigma_{0},...,\sigma{n})=\sum_{i=0}^{n}(-1)^{i}x(\sigma_{0},...,\hat{\sigma_{i}},...,\sigma_{n})$

where $\hat{\sigma_{i}}$ signifies that $\sigma_{i}$ is to be omitted.

We now apply the fixed module functor to obtain the cochain complex

$\displaystyle C^{n}(G,A)=X^{n}(G,A)^{G}$ .

The elements of $C^{n}(G,A)$ are the functions $x: G^{n+1}\rightarrow A$ such that

$\displaystyle x(\sigma\sigma_{0},...,\sigma\sigma_{n})=\sigma x(\sigma_{0},...,\sigma_{n})$

for all $\sigma\in G$ .

The Galois cohomology groups $H^{n}(G,A)$ are then obtained by taking the cohomology of this cochain complex, i.e.

$\displaystyle H^{n}(G,A)=\text{Ker }\partial^{n+1}/\text{Im }\partial^{n}$

Note: We have adopted here the notation of the book Cohomology of Number Fields by Jurgen Neukirch, Alexander Schmidt, and Kay Wingberg. Some references use a different notation; for instance $X_{n}$ may be defined as the abelian group of functions from $G^{n}$ to $A$ instead of from $G^{n+1}$ to $A$ . This results in different notation for the cochain complexes and their boundary operators; however, the Galois cohomology groups themselves will remain the same.

It is a basic result of Galois cohomology that $H^{0}(G,A)$ gives $A^{G}$ , the subset of $A$ such that $\sigma\cdot a=a$ for all $\sigma\in G$ . In other words, $A^{G}$ is the subset of $A$ that is fixed by $G$ .

We have the following connection between Etale cohomology for fields and Galois cohomology:

$\displaystyle H^{n}(K,\mathcal{F})=H^{n}(G,\mathcal{F}(\bar{K}))$

We now mention some other basic results of the theory. In analogy with sheaf cohomology, the group $H^{0}(K,\mathcal{F})$ is just the set of “global sections” $\Gamma(K,\mathcal{F})=\mathcal{F}(K)$ of $\mathcal{F}$ . Letting $\mathcal{F}=\mathcal{G}_{m}$ which we have defined earlier, we have

$\displaystyle H^{0}(K,\mathcal{G}_{m})=\mathcal{G}_{m}(K)=K^{*}$

In the language of Galois cohomology,

$\displaystyle H^{0}(G,\mathcal{G}_{m}(\bar{K}))=(\bar{K}^{*})^{G}=K^{*}$

Meanwhile, for $H^{1}$ , we have the following result, called Hilbert’s Theorem 90:

$\displaystyle H^{1}(K,\mathcal{G}_{m})=H^{1}(G,\bar{K}^{*})=\{1\}$ .

The group $H^{2}(K,\mathcal{G}_{m})=H^{2}(G,\bar{K}^{*})$ is called the Brauer group and also plays an important part in algebraic number theory. The etale cohomology of fields, or equivalently, Galois cohomology, are the topic of famous problems in modern mathematics such as the Milnor conjecture and its generalization, the Bloch-Kato conjecture, which was solved by Vladimir Voevodsky in 2009. They also play an important part in the etale cohomology of more general rings.