Why are central simple algebras classified by cohomology?

Question

In their article on the Brauer group Wikipedia writes:

Since all central simple algebras over a field $K$ become isomorphic to the matrix algebra over a separable closure of $K$, the set of isomorphism classes of central simple algebras of degree $n$ over $K$ can be identified with the Galois cohomology set $H^1(K, \mathrm{PGL}(n))$.

I understand all the words here, but the reasoning goes too quick for me to follow it. Could someone explain why this is true?

I believe $H^1(K, \mathrm{PGL}(n))$ is really the group cohomology $H^1(G, \mathrm{PGL}(n,K))$ where $G$ is the Galois group of the separable closure of the field $K$. The group cohomology $H^1(G,M)$ is explained here. Perhaps it's not explained in sufficient generality, since they seem to define $H^1(G,M)$ only where $M$ is an abelian group acted on by a group $G$. But I think the same thing should work for any set acted on by $G$, and I know how $PGL(n,K)$ is acted on by the Galois group $G$.

I guess I need to see how a

• central simple algebra $A$ over a field $K$ that becomes isomorphic to an $n \times n$ matrix algebra when tensored with the separable closure of $A$

gives rise to a

• 1-cocycle $c_A \colon G \to M$

and why isomorphic algebras of this sort give cocycles that differ by a coboundary. (Also how to go back.)

Answer 1

A good reference for what I'm about to explain is Berhuy's text "An Introduction to Galois Cohomology and Its Applications".

First, we want to set up the conditions for Galois descent. The statement of Galois descent (p.106, Berhuy) is (basically) that if we have a nice functor $\mathbf{F}$ from fields extensions over $k$, denoted $\mathscr{C}_k$, to $\mathbf{Sets}$, and an action of some groups $G(K)$ on $\mathbf{F}(K)$ for all $K/k$ (satisfying certain conditions), we have for every $a\in \mathbf{F}(\Omega)$ a bijection for every field extension $K/k$ and Galois extension $\Omega/K$

\begin{equation} \mathbf{F}_a(\Omega/K)\xrightarrow{\sim}\ker\left[H^1(\mathcal{G}_\Omega,\mathbf{Stab}_G(a)(\Omega))\to H^1(\mathcal{G}_\Omega,G(\Omega))\right] \end{equation}

where $\mathbf{F}_a$ is the set of $G(K)$-orbits in the orbit $G(\Omega)\cdot a$ after taking the Galois invariance functor.

In our case, letting $M_n(K)$ denote the ring of $n\times n$ matrices over $K$, we let $\mathbf{F}(\Omega/K)$ be the set of all central simple $K$-algebras with underlying vector space $M_n(\Omega)$ after tensoring with $\Omega$. We construct an action on this set by seeing how automorphisms of the vector space structure change the algebra structure, so $G(\Omega)=GL(M_n(\Omega))$ acting on $\mathbf{F}(\Omega/K)$. Then, we see that the stabilizer of this action with respect to some central simple algebra $A\in\mathbf{F}(K)$ is exactly the set $\textrm{Aut}_{K-alg}(A)$.

Next, note that $PGL_n(K)$ can be thought of as $\textrm{Aut}_{K-alg}(M_n(K))$ and that $\mathbf{F}_A(\Omega/K)$ is the set of isomorphism classes of central simple $K$-algebras, which are isomorphic to $M_n(\Omega)$ after tensoring with $\Omega$.

Finally, we construct the exact sequence

\begin{equation} 0\to PGL_n(\Omega)\to GL(M_n(\Omega))\to GL(M_n(\Omega))\cdot (A\otimes\Omega)\to 0$ \end{equation}

Getting the long exact sequence,

\begin{equation} \cdots\to H^1(\mathcal{G}_\Omega,PGL_n(\Omega))\to H^1(\mathcal{G}_\Omega,GL(M_n(\Omega)))\to\cdots \end{equation}

By Hilbert's Theorem 90 $H^1(\mathcal{G}_\Omega,GL(M_n(\Omega)))=0$. Thus, by Galois descent, we get the desired result.

Apologies for any errors, still learning this stuff myself!

Answer 2

I think the best reference is the book [G-S] by Gille & Szamuely, "Central Simple Algebras and Galois Cohomology", Cambridge Univ. Press, 2017.

Over a field $k$, recall that a (finite dimensional) $k$-algebra $A$ is called simple if it has no (two-sided) ideal other than $0$ and $A$. Moreover it is called $central$ if its centre equals $k$. A finite-dimensional $k$-algebra $A$ is central simple iff there exists an integer $n>0$ and a finite Galois extension $K/k$ s.t. $A\otimes_k K$ is isomorphic to the matrix ring $M_n(K)$. Such a field extension $K/k$ is called a splitting field for $k$. Let us denote by $CSA_K(n)$ the pointed set of $k$-isomorphisms classes of central simple $k$-algebras of degree $n$ split by $K$, the base point being the class of the matrix algebra $M_n(k)$.

A first cohomological description of $CSA_K(n)$ is as follows. Recall that for a group $G$ acting on another group (not necessarily abelian) group $A$, a 1-cocycle of $G$ with values in $A$ is a map $s\in G \to a_s\in A$ satisfying $a_{st}=a_s.s(a_t)$, and two 1-cocycles $a_s$ and $b_s$ are equivalent if there exists $c\in A$ s.t. $a_s=c^{-1}b_s s(c)$ (I repeat the definitions because here $A$ is not commutative). This is an equivalence relation between 1-cocycles, and the first cohomology $set$ $H^1(G,A)$ is the set of the equivalence classes of 1-cocycles, pointed by the trival cocycle $s\to Id_A$. The main result is the existence of a base point preserving bijection between $CSA_K(n)$ and $H^1(G, PGL_n(K))$, where $G=Gal(K/k))$. See [G-S], thms. 2.3.3 and 2.4.3. This should answer your questions, but unfortunately, the proof of thm. 2.4.3 relies on thm. 2.3.3, which is rather elaborate and cannot be summarized here.

For $H^i(G,A)$ to be a group (with the usual natural definition) you need $A$ to be abelian. In the non abelian case, the tensor product induces natural associative and commutative products $CSA_K(m)\times CSA_K(n)$ $\to CSA_K(mn)$, which can be naturally "transported" to the other side to define analogous products $H^1(G, PGL_m(K))\otimes H^1(G, PGL_n(K))\to H^1(G, PGL_{mn}(K))$. Fixing a separable closure $k_s$ of $k$ and writing $G_k=Gal(k_s/k)$, one can then take inductive limits to get an isomorphism of abelian groups $Br(k)$ (by definition) $\cong H^1(G_k, PGL_{\infty})$. But this is rather cumbersome. The second definition of $Br(k)$ via Brauer-equivalence classes of central simple $k$-algebras is more supple and yields natural isomorphisms of abelian groups $Br(K/k)\cong H^2(G, K^*)$ and $Br(k)\cong H^2(G_k,{k_s}^*)$. Anyway, these are the versions of Brauer groups which are used in class-field theory.