Brauer Groups Part II

Continuing from Part I, we delve deeper into Brauer groups. In this post, we'll compute examples of Brauer groups of fields, relate them to cohomology, discuss period-index, and prove several relevant theorems including Skolem–Noether theorem and Wedderburn's theorem. We will also generalize the Brauer group construction to rings. These will correspond generally to ⟦cite:Poo17⟧ from chapters 2 and 3, but I will not follow them precisely, and will add some additional details and proofs. We follow previous notation, and fix $K$ a field.

Cohomological Interpretation of the Brauer Group

For each $r\ge 1$, there is injection

$$\frac{\{\textrm{Azumaya }K\textrm{-algebras of dimension }r^2\}}{\textrm{isomorphism}}\hookrightarrow \mathrm{H}^1(K,\mathrm{PGL}_r)$$

Proof.

Let $A$ be an Azumaya $K$-algebra of dimension $r^2$. Choose an isomorphism of $K$-algebras $\varphi:\mathrm{M}_r(K^{\mathrm{sep}})\xrightarrow{\sim} A\otimes_K K^{\mathrm{sep}}$. For each $\sigma\in \mathrm{Gal}(K^{\mathrm{sep}}/K)$, define $\xi_{\sigma}=\varphi^{-1}\circ {}^{\sigma}\!\varphi\in\mathrm{Aut}_{K^{\mathrm{sep}}}(\mathrm{M}_r(K^{\mathrm{sep}}))\cong\mathrm{PGL}_r(K^{\mathrm{sep}})$ where the last isomorphism follows from Skolem–Noether theorem. It is not hard to verify $\xi_{\sigma\tau}=\xi_{\sigma}\cdot \sigma(\xi_{\tau})$, i.e. $\xi$ is a $1$-cocycle. This defines a cohomology class since changing $\varphi$ is equivalent to composing it with an automorphism of $\mathrm{M}_r(K^{\mathrm{sep}})$, which changes $\xi_{\sigma}$ by a coboundary. Thus we have defined a map from the set of isomorphism classes of Azumaya $K$-algebras of dimension $r^2$ to $\mathrm{H}^1(K,\mathrm{PGL}_r)$. It is straightforward to check that this map is injective via descent.

The aforementioned injection is actually a bijection, but we will not prove this here. Indeed, one can show that an Azumaya algebra of dimension $r^2$ is the same as a Galois twist of the matrix algebra $\mathrm{M}_r(K)$.

There is an isomorphism of abelian groups

$$\mathrm{Br}(K)\xrightarrow{\sim} \mathrm{H}^2(K,\mathbb{G}_m)$$

Proof.

Taking cohomology of the short exact sequence of algebraic groups over $K^{\mathrm{sep}}$

$$1\to \mathbb{G}_m\to \mathrm{GL}_r\to \mathrm{PGL}_r\to 1$$

gives a long exact sequence

$$\cdots \to \mathrm{H}^1(K,\mathrm{GL}_r)\to \mathrm{H}^1(K,\mathrm{PGL}_r)\to \mathrm{H}^2(K,\mathbb{G}_m)\to \mathrm{H}^2(K,\mathrm{GL}_r)\to \cdots$$

Composing the injection from ⟦ref:prop-azumaya-to-pglr⟧ with the map $\mathrm{H}^1(K,\mathrm{PGL}_r)\to \mathrm{H}^2(K,\mathbb{G}_m)$, we get a map $\mathrm{Br}(K)\rightarrow \mathrm{H}^2(K,\mathbb{G}_m)$. To see that this map is injective note that by Hilbert's Theorem 90, $\mathrm{H}^1(K,\mathrm{GL}_r)=0$. To see that it is surjective, given a cohomology class $\alpha\in \mathrm{H}^2(K,\mathbb{G}_m)$, we construct an Azumaya algebra $A$ such that its image in $\mathrm{H}^2(K,\mathbb{G}_m)$ is $\alpha$. First note that we can reduce to the case of a finite Galois extension $L\mid K$, where we only need to consider a class in $\mathrm{H}^2(\mathrm{Gal}(L\mid K),L^\times)$ that inflates to $\alpha$. Choose a representative $2$-cocycle $c:\mathrm{Gal}(L\mid K)^2\to L^\times$, i.e. $c(\sigma, \tau) c(\sigma \tau, \rho)=\sigma(c(\tau, \rho)) c(\sigma, \tau \rho)$ for all $\sigma, \tau, \rho\in \mathrm{Gal}(L\mid K)$, we define the crossed product algebra $A=\bigoplus_{\sigma\in \mathrm{Gal}(L\mid K)}Lu_{\sigma}$ with multiplication defined by $u_{\sigma}\ell=\sigma(\ell)u_{\sigma}$ and $u_{\sigma}u_{\tau}=c(\sigma,\tau)u_{\sigma\tau}$, which one checks is a central simple algebra over $K$ that splits over $L$. One can check that the image of $A$ in $\mathrm{H}^2(K,\mathbb{G}_m)$ is precisely $\alpha$. The map is additive by routine check.

Examples of Brauer Groups of Fields

Let $K$ be a field, then

If $\mathrm{char}(K)\nmid n$, then $\mathrm H^1(K, \mu_n)=K^\times/K^{\times n}$.
If $\mathrm{char}(K)\nmid n$, then $\mathrm H^2(K, \mu_n)\cong \mathrm{Br}(K)\left[n\right]$.
For any Galois extension $L\mid K$, we have $\mathrm H^2(\mathrm{Gal}(L\mid K), L^\times)=\mathrm{Ker}(\mathrm{Br}(K)\rightarrow\mathrm{Br}(L))$.

Proof.

Take short exact sequence $$1\to \mu_n\to \mathbb{G}_m\xrightarrow{(\cdot)^n} \mathbb{G}_m\to 1$$ and take cohomology to get long exact sequence $$\cdots \to \mathrm H^0(K, \mathbb{G}_m)\xrightarrow{(\cdot)^n} \mathrm H^0(K, \mathbb{G}_m)\to \mathrm H^1(K, \mu_n)\to \mathrm H^1(K, \mathbb{G}_m)\to \cdots$$ then use Hilbert's Theorem 90 to conclude.
Same idea as above, taking the same short exact sequence and taking cohomology to get long exact sequence $$\cdots \to \mathrm H^1(K, \mathbb{G}_m)\xrightarrow{(\cdot)^n} \mathrm H^1(K, \mathbb{G}_m)\to \mathrm H^2(K, \mu_n)\to \mathrm H^2(K, \mathbb{G}_m)\xrightarrow{(\cdot)^n} \mathrm H^2(K, \mathbb{G}_m) \to \cdots$$ then use Hilbert's Theorem 90 to conclude.
Skipped

Here are some examples of Brauer groups of fields:

If $K$ is algebraically closed, $\mathrm{Br}(K)=0$,
If $K$ is finite, then $\mathrm{Br}(K)=0$,
If $K=\mathbb{R}$, then $\mathrm{Br}(\mathbb{R})\cong \mathbb{Z}/2\mathbb{Z}$,
If $K$ is a non-archimedean local field, then $\mathrm{Br}(K)\cong \mathbb{Q}/\mathbb{Z}$,
(Albert–Brauer–Hasse–Noether) If $K$ is a global field, then there is an exact sequence $$0\to \mathrm{Br}(K)\to \bigoplus_v \mathrm{Br}(K_v)\xrightarrow{\sum \mathrm{inv}_v} \mathbb{Q}/\mathbb{Z}\to 0$$ where the sum is over all places $v$ of $K$.

Proof.

By ⟦ref:thm-brauer-cohomology⟧, $\mathrm{Br}(K)\cong \mathrm{H}^2(K,\mathbb{G}_m)$. Since $K$ is algebraically closed, $\mathrm{Gal}(K^{\mathrm{sep}}\mid K)=0$, so $\mathrm{H}^2(K,\mathbb{G}_m)=0$.
Any central division algebra over a finite field is finite, hence by Wedderburn’s little theorem is a field, hence every Azumaya algebra over a finite field is some $\mathrm M_n(K)\sim K$, so the Brauer group is trivial.
The only Azumaya algebras over $\mathbb{R}$ are $\mathbb{R}$ and the Hamiltonian quaternions $\mathbb{H}$ so the Brauer group is $\mathbb{Z}/2\mathbb{Z}$.
It suffice to show for each $n\ge 1$ we have $\mathrm{Br}(K)[n]\cong \mathbb{Z}/n\mathbb{Z}$, so that $$\mathrm{Br}(K)=\lim_{\longrightarrow}\mathrm{Br}(K)[n]\cong\lim_{\longrightarrow}\mathbb{Z}/n\mathbb{Z}=\lim_{\longrightarrow} \frac{1}{n}\mathbb Z/\mathbb Z=\mathbb Q/\mathbb Z$$ By ⟦ref:prop-brauer-cohomology-mun⟧ part (ii), we have $\mathrm{Br}(K)[n]\cong \mathrm H^2(K, \mu_n)$. By Tate local duality from local class field theory, we have $\mathrm H^2(K, \mu_n)\cong \mathbb{Z}/n\mathbb{Z}$ as required.
Skipped

Let $G$ be a profinite group and $p$ a prime. The $p$-cohomological dimension $\mathrm{cd}_p(G)$ of $G$ is the smallest $n\in\mathbb N\cup\{\infty\}$ such that for all torsion $G$-module $A$, $\mathrm H^i(G,A)[p]=0$ for all $i>n$. The strict $p$-cohomological dimension $\mathrm{scd}_p(G)$ of $G$ is defined as the same way without the condition of torsion for $A$. The cohomological dimension $\mathrm{cd}(G)$ (resp. strict cohomological dimension $\mathrm{scd}(G)$) is defined as $\sup_p \mathrm{cd}_p(G)$ (resp. $\sup_p \mathrm{scd}_p(G)$).

For any profinite $G$, we have $\mathrm{cd}(G)\le\mathrm{scd}(G)\le \mathrm{cd}(G)+1$

If $\mathrm{cd}(K)\le 1$ then $\mathrm H^2(G_K,\mu_\infty)=0$ implies $\mathrm{Br}(K)=0$. If $\mathrm{cd}(K)= 2$ then $\mathrm H^i(G_K,\mu_\infty)=0$ for all $i\ge 3$.

If $K$ is algebraically closed then $\mathrm{cd}(K)=0$. If $K$ is finite then $\mathrm{cd}(K)=1$. If $K$ is a local field or global field then $\mathrm{cd}(K)=2$.

A field $K$ is $C_r$ for some integer $r\ge 0$ if every homogeneous polynomial of degree $d$ in $n$ variables with $n>d^r$ has a nontrivial zero in $K$. The adjective quasi-algebraically closed means $C_1$.

If $K$ is a $C_r$ field, $L\mid K$ an extension,

If $L$ is algebraic over $K$, then $L$ is $C_r$.
If $L$ is transcendental of trasncendence degree $s$ over $K$, then then $L$ is $C_{r+s}$.

(Chevalley–Warning) If $K$ is a finite field, then $K$ is $C_1$.

If $K$ is a $C_1$ field, then $\mathrm{Br}(K)=0$.

Period and Index

The index of a finite dimensional central division algebra $D$ over a field $K$ is $\sqrt{[D:K]}\in\mathbb Z_{\ge 1}$, and the index of the Azumaya algebra $\mathrm M_r(D)$ is defined to be that of $D$. The period of an Azumaya algebra $A$ is the order of its class in $\mathrm{Br}(K)$.

For any Azumaya algebra $A$ over a field $K$, the period of $A$ divides the index of $A$.

Proof.

We reduce to central division algebras. Let $D$ be a central division algebra over $K$ of index $n$. A basic result in the theory of central simple algebras is that $D$ contains a maximal subfield $L$ that is a degree $n$ separable extension of $K$ that splits $D$, i.e. $D\otimes_K L\cong \mathrm M_n(L)$. Thus the class of $D$ in $\mathrm{Br}(K)$ maps to zero in $\mathrm{Br}(L)$, by the restriction-corestriction identity

$$\mathrm{cor}_{L\mid K}\circ \mathrm{res}_{L\mid K}([D])=[L:K][D]=n[D]$$

we see that the period of $D$ divides $n$, as required.

Period and index have the same prime factors, and over certain fields such as local and global fields, they are equal. However, in general they can differ. The relationship between period and index is an active area of research in number theory.

Cyclic Algebras

Let $L\mid K$ be a degree $n$ cyclic extension, $a\in K^\times$ and $\sigma$ a generator of $\mathrm{Gal}(L\mid K)$. Let $L[x]_{\sigma}$ be the twisted polynomial ring with relation $x\ell=\sigma(\ell)x$ for all $\ell\in L$. Let $A=L[x]_{\sigma}/(x^n - a)$. It is straightforward to check that $A$ is a central simple algebra over $K$ of dimension $n^2$ that splits over $L$. Azumaya algebras of this form are called cyclic algebras. Generally, we have the following.

Let $a\in K^\times$ and $\chi:\mathrm{G}_K\rightarrow \mathbb Z/n\mathbb Z$. Let $\mathrm G_K$ act on $\mathbb Z/n\mathbb Z$ by $g\cdot x=x+\chi(g)$, which, by a proposition from Part I, corresponds to an étale $K$-algebra $L$, and the automorphism $s\mapsto s+1$ of the $\mathrm G_K$ set $\mathbb Z/n\mathbb Z$ corresponds to an automorphism $\sigma$ of the $K$-algebra $L$. The cyclic algebra associated to the pair $(\chi,a)$ is defined to be the $K$-algebra $(\chi,a)=L[x]_{\sigma}/(x^n - a)$.

Cyclic algebra $(a,\chi)$ have cohomological interpretations. Suppose $\mathrm{char}(K)\nmid n$, then $a$ can be mapped into $K^\times/K^{\times n}\cong \mathrm H^1(K,\mu_n)$ and $\chi\in \mathrm{Hom}(\mathrm G_K,\mathbb Z/n\mathbb Z)=\mathrm H^1(K,\mathbb Z/n\mathbb Z)$. Under the cup product

$$\mathrm H^1(K,\mu_n)\times \mathrm H^1(K,\mathbb Z/n\mathbb Z)\to \mathrm H^2(K,\mu_n)\cong \mathrm{Br}(K)[n]$$

the pair $(a,\chi)$ maps to an element in $\mathrm H^2(K,\mu_n)\cong \mathrm{Br}(K)[n]$ that corresponds to the cyclic algebra $(\chi,a)$ up to a sign.

Let $L$ and $\chi$ be as above. An $A\in\mathbf{Az}_K$ with $[A:K]=[L:K]^2$ is split by $L$ iff $A\cong (\chi,a)$ for some $a\in K^\times$.

Let $L$ and $\chi$ as above, $a\in K^\times$, then $(\chi, a)$ is split as $K$-algebra iff $a\in N_{L\mid K}(L^\times)$.

Brauer Groups of Rings

Let $R$ be a ring, a not necessarily commutative $R$-algebra $A$ is an Azumaya algebra over $R$ if

$A$ is finitely generated projective as an $R$-module,
the natural map $A\otimes_R A^{\mathrm{op}}\to \mathrm{End}_R(A)$ by sending $a\otimes b^{\mathrm{op}}\mapsto (x\mapsto axb)$ is an isomorphism of $R$-algebras.

Geometrically, this is the same as saying $\mathrm{Spec}(R)$ is étale-locally isomorphic to a matrix algebra over $R$.

Two Azumaya algebras $A$ and $B$ over $R$ are Brauer equivalent if there exists $m,n\ge 1$ such that $A\otimes_R \mathrm{M}_m(R)\cong B\otimes_R \mathrm{M}_n(R)$ as $R$-algebras. The Brauer group $\mathrm{Br}(R)$ of $R$ is the set of Brauer equivalence classes of Azumaya algebras over $R$ with group operation induced by tensor product, identity element $[R]$ and inverse $[A]^{-1}=[A^{\mathrm{op}}]$.

For rings and generally schemes, their Brauer groups are not necessarily equal to the second cohomology group. Instead one defines so called cohomological Brauer group $\mathrm{Br}'(R)=\mathrm H^2_{ét}(\mathrm{Spec}(R), \mathbb{G}_m)_{\mathrm{tors}}$, there is a canonical injection $\mathrm{Br}(R)\hookrightarrow \mathrm{Br}'(R)$ but it is not necessarily surjective. The same is true for schemes.

References

[Poo17]Bjorn Poonen. Rational Points on Varieties. American Mathematical Society. 2017.