The Stacks project

Central simple algebras are classified by Galois cohomology of PGL.

Lemma 59.61.5. Let $K$ be a field and let $K^{sep}$ be a separable algebraic closure. Then the set of isomorphism classes of central simple algebras of degree $d$ over $K$ is in bijection with the non-abelian cohomology $H^1(\text{Gal}(K^{sep}/K), \text{PGL}_ d(K^{sep}))$.

Sketch of proof.. The Skolem-Noether theorem (see Brauer Groups, Theorem 11.6.1) implies that for any field $L$ the group $\text{Aut}_{L\text{-Algebras}}(\text{Mat}_ d(L))$ equals $\text{PGL}_ d(L)$. By Theorem 59.61.1, we see that central simple algebras of degree $d$ correspond to forms of the $K$-algebra $\text{Mat}_ d(K)$. Combined we see that isomorphism classes of degree $d$ central simple algebras correspond to elements of $H^1(\text{Gal}(K^{sep}/K), \text{PGL}_ d(K^{sep}))$. For more details on twisting, see for example [SilvermanEllipticCurves]. $\square$

Comments (1)

Comment #3247 by Giulio on

Suggested slogan: Central simple algebras are classified by Galois cohomology of PGL.

There are also:

  • 5 comment(s) on Section 59.61: Brauer groups

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 03R6. Beware of the difference between the letter 'O' and the digit '0'.