Dimension of central simple algebra over a global field "built using class field theory".

Question

If $F$ is a global field then a standard exact sequence relating the Brauer groups of $F$ and its completions is the following:

$$0\to Br(F)\to\oplus_v Br(F_v)\to\mathbf{Q}/\mathbf{Z}\to 0.$$

The last non-trivial map here is "sum", with each local $Br(F_v)$ canonically injecting into $\mathbf{Q}/\mathbf{Z}$ by local class field theory.

In particular I can build a class of $Br(F)$ by writing down a finite number of elements $c_v\in Br(F_v)\subseteq \mathbf{Q}/\mathbf{Z}$, one for each element of a finite set $S$ of places of $v$, rigging it so that the sum $\sum_vc_v$ is zero in $\mathbf{Q}/\mathbf{Z}$.

This element of the global Brauer group gives rise to an equivalence class of central simple algebras over $F$, and if my understanding is correct this equivalence class will contain precisely one division algebra $D$ (and all the other elements of the equiv class will be $M_n(D)$ for $n=1,2,3,\ldots$).

My naive question: is the dimension of $D$ equal to $m^2$, with $m$ the lcm of the denominators of the $c_v$? I just realised that I've always assumed that this was the case, and I'd also always assumed in the local case that the dimension of the division algebra $D_v$ associated to $c_v$ was the square of the denominator of $c_v$. But it's only now, in writing notes on this stuff, that I realise I have no reference for it. Is it true??

Answer 1

To paraphrase Igor Pak: OK, this I know. It is remarkable how difficult it is to track down a reference which gives an actual proof for this fact (moreover applicable to all global fields). The notes of Pete Clark don't give a proof or a reference for a proof, and its omission in Cassels-Frohlich is an uncorrected error. :)

But here is a reference: Theorem 3.6 in the notes on Honda-Tate theory on Kirsten Eisentraeger's webpage. The assertion is even stronger: one can find a cyclic splitting field of the expected minimal degree. A moment's reflection leads one to realize what is actually going on: in the non-archimedean local theory we know that one can always arrange the splitting field to be the unramified one of the expected minimal degree (already in Serre's Local Fields, and part of the story of the "local invariant"), so in particular it is cyclic in that case. Taking into account the real case, and using the exactness at the left of the global-to-local sequence for Brauer groups, the global problem reduces to making a global cyclic extension inducing specified local ones at finitely many places and having a predicted degree which is lcm of local degrees (in the local theory the degree is actually all that really matters, not the unramifiedness).

Enter Grunwald-Wang... and since all that matters locally is the degree, if we don't care about global cyclicity but just global degree and some local degrees then weak approximation & Krasner's Lemma suffice to do the job (so for the question as asked, in which there's no cyclicity, the global problem is actually very elementary once the local case is settled!). Note that in Cassels-Frohlich the global cyclic splitting field is addressed, but not its degree (since Grunwald-Wang is not addressed in Cassels-Frohlich).

Historically the existence of a global cyclic splitting field, moreover of the expected degree, was regarded as one of the real triumphs of global class field theory, and the early attempts at class field theory by the German school were intimately tied up with this problem of the cyclic splitting field. This is why it was such a shock to Artin when Wang discovered that Grunwald's proof of local-to-global for cyclic extensions was not true (but fortunately Wang's fix was sufficient); see Roquette's historical notes on CFT.

Finally, to put this in perspective, it should be noted (as remarked in Eisentraeger's notes) that there are examples of complex function fields in transcendence degree 3 admitting nontrivial 2-torsion Brauer classes not represented by a quaternion division algebra! (The appearance of trdeg 3 is reasonable, as the period-index problem for surfaces over an algebraically closed field was proved by deJong.)

Answer 2

Isn't this the period-index problem in the Brauer group of a number field? In which case the answer is yes, it's true -- see e.g. Fact 4(c) on page 4 of these notes by Pete Clark (as well as Example 1.1.2 and the definition following it, both on the same page):

http://alpha.math.uga.edu/~pete/periodindexnotes.pdf

Answer 3

I think you mean "lcm of the denom. of the $c_v$" rather than "gcd".

Colliot-Thélène, in

• Algèbres simples centrales sur les corps de fonctions de deux variables, Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque, no. 307 (2006), Exposé no. 949, pp. 379-413. (EuDML),

attributes the result that "exponent = index" in the Brauer group of a global (and local) field to Brauer-Hasse-Noether and Albert and gives references to:

• R. BRAUER, H. HASSE et E. NOETHER, Beweis eines Hauptsatzes in der Theorie der Algebren, J. reine angew. Math. (Crelle) 167 (1932) 399–404. (EuDML)

• M. DEURING, Algebren, Zweite, korrigierte Auflage. Ergebnisse der Mathematik und ihrer Grenzgebiete 41, Springer-Verlag, Berlin-New York 1968. doi:10.1007/978-3-642-85533-7

and

• P. ROQUETTE, The Brauer-Hasse-Noether theorem in historical perspective. Schriften der Math.-Phys. Klasse der Heidelberger Akademie der Wissenschaften 15 (2004) doi:10.1007/b138384 (author pdf)

The result seems to be proved in Reiner's book (Theorem (32.19):

• I. REINER, Maximal orders, Corrected reprint of the 1975 original. With a foreword by M. J. Taylor. London Mathematical Society Monographs. New Series 28, The Clarendon Press, Oxford University Press, Oxford, 2003. (publisher page)

Answer 4

The fact that the Schur index of an element of the Brauer group of a number field equals its order in the Brauer group (and also is a cyclic algebra) is Theorem 18.6 in Richard Pierce's Associative Algebras (GTM88 Springer-Verlag 1982). The proof uses the Grunwald-Wang theorem.