Brauer Groups Part I

Today we study the Brauer group of a field, which classifies central simple algebras over the field. They eventually generalize to schemes and control rational points on a variety. I will mostly follow ⟦cite:Poo17⟧ with some other sources as supplements.

We fix some notation and terminology. We fix a field $K$ with algebraic closure $\overline{K}$, separable closure $K^{\mathrm{sep}}$, and perfect closure $K^{\mathrm{perf}}$. Denote the absolute Galois group as $\mathrm{G}_K=\mathrm{Gal}(K^{\mathrm{sep}}/K)$. If $K$ is a global field, Let $\Omega_K$ be the set of its places, $S_{\infty}\subset L$ the set of archimedean places. And $S\subseteq \Omega_K$ is said to be an admissible set of places of $K$ if it contains $S_\infty$. For an admissible set of places $S$ of $K$, denote by $\mathbb Z_{K, S}$ the $S$-integers of $K$ with $\mathbb Z_K=\mathbb Z_{K,S_{\infty}}$ the usual ring of integers. For $S\subseteq \Omega_K$ a finite set of places, denote the ring of $S$-adeles $\mathbb A_{K,S}$ and the usual adeles $\mathbb A_K$. Denote $\mathbb I_{K,S}$ the $S$-ideles and $\mathbb I_K$ the usual ideles.

Galois Theory of Étale Algebras

The problem with field extensions $L \mid K$ is that if one has an extension $M\mid K$ then the base change $L\otimes_K M$ is not a field but a $M$-algebra. To solve this issue, we introduce the notion of étale algebras in ⟦ref:def-etale⟧, which comes from étale maps i.e. local homeomorphisms in topology, indeed the notion of étale morphism $\mathrm{Spec}(L)\rightarrow \mathrm{Spec}(K)$ is thought of as an analogy for that.

A $K$-algebra $L$ is said to be étale if any of the following equivalent conditions are satisfied

$L\otimes_K K^{\mathrm{sep}}\cong (K^{\mathrm{sep}})^n$ for some $n$,
$L$ is a direct product of finite separable extensions of $K$,
the induced map $\mathrm{Spec}(L)\rightarrow \mathrm{Spec}(K)$ is finite and étale.

Étaleness is stable under base change. For $M\mid K$ a field extension and $L$ an étale $K$-algebra, we have $L\otimes_K M$ is an étale $M$-algebra. With étale algebras, the following is Grothendieck's generalization of Galois theory.

There is an equivalence of categories between $\{\textrm{finite }\mathrm G_K\textrm{-sets}\}^{\mathrm{op}}$ and $\{\textrm{étale }K\textrm{-algebras}\}$ given by the functors that take $S$, a finite $\mathrm G_K$-set, to the étale $K$-algebra $\mathrm{Hom}_{\mathrm G_K\textrm{-sets}}(S,K^{\mathrm{sep}})$, and conversely take an étale $K$-algebra $L$ to the finite $\mathrm G_K$-set $\mathrm{Hom}_{K\textrm{-Alg}}(L,K^{\mathrm{sep}})$.

Proof. See ⟦cite:GR71⟧ .

Suppose $S$ is a transitive $\mathrm G_K$-set, then $S=\mathrm G_K/H$ for some open subgroup $H$, so the corresponding étale algebra is $\mathrm{Hom}_{\mathrm G_K\textrm{-sets}}(\mathrm G_K/H, {K}^{\mathrm{sep}})=(K^\mathrm{sep})^H$ the fixed field in classical Galois theory. In general, a finite $\mathrm G_K$-set is the finite disjoint union of transitive ones $\coprod S_i$, so $S$ corresponds to the étale algebra $\prod L_i$ where $L_i$ is the finite separable extension corresponding to $S_i$.

References

[Poo17] Bjorn Poonen, Rational Points on Varieties, American Mathematical Soc., 2017.
[GR71] Alexander Grothendieck, Michèle Raynaud, Revêtements étales et groupe fondamental (SGA I), Lecture Notes in Mathematics 224, Springer, 1971.