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.)
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!