Divisible Abelian Groups are Brauer Groups

(Translation of an article originally published in Russian in

Uspekhi Mat. Nauk, vol. 40 (1985), no. 2(242), 213–214^{1})

A. S. Merkurjev

2010 Mathematics Subject Classification: 16K50

It is well known that the Brauer group of a field is an abelian torsion group. Examples where the Brauer group of a field can be explicitly computed show that this group is close to being divisible. However, for a long time there was not a single known example of an abelian torsion group A such that A ≄ Br(F) for any field F. First examples of this type were constructed in [3], where it was shown that for p = 2 or 3 the p-component of the Brauer group of any field either is an elementary 2-group or contains a non-trivial divisible subgroup. In [1] Fein and Schacher conjectured that any abelian divisible torsion group is isomorphic to the Brauer group of some field. We will now give a proof of this conjecture.

Theorem. For every abelian divisible torsion group A there exists a field F such that Br(F) ≃ A.

Proof. We will construct, inductively, a tower of fields F_{1} ⊂ F_{2} ⊂ F_{3} ⊂ ⋯ and subgroups A_{i}, B_{i} ⊂ Br(F_{i}) satisfying the following conditions:

1. A is isomorphic to A_{1}.

2. Br(F_{i}) = A_{i} ⊕ B_{i} (i = 1, 2, …).

3. The kernel of the natural homomorphism Br(F_{i}) → Br(F_{i+1}) induced by the inclusion of fields F_{i} ⊂ F_{i+1} is B_{i}. Moreover, this homomorphism restricts to an isomorphism between A_{i} and A_{i+1}.

1Translation given here with the kind permission of Uspekhi Mat. Nauk.

Documenta Mathematica · Extra Volume Merkurjev (2015) 461–463

Let us begin with i = 1. By [2, Theorem 2] there exists a field F_{1} such that A is isomorphic to some subgroup A_{1} of Br(F_{1}). Since A_{1} is divisible, it is a direct summand in Br(F_{1}), i.e., there exists a subgroup B_{1} ⊂ Br(F_{1}) such that Br(F_{1}) = A_{1} ⊕ B_{1}. Now suppose we have constructed the fields F_{1} ⊂ F_{2} ⊂ ⋯ ⊂ F_{n} and subgroups A_{i}, B_{i} ⊂ Br(F_{i}) for i = 1, …, n. By [2, Theorem 1] there exists a field F_{n+1} such that F_{n} ⊂ F_{n+1}, and the kernel of the homomorphism Br(F_{n}) → Br(F_{n+1}) induced by this inclusion is B_{n}. Denote by A_{n+1} the image of A_{n} under this homomorphism, and by B_{n+1} any complement to A_{n+1} in Br(F_{n+1}) (a complement to A_{n+1} exists because A_{n+1} ≃ A is divisible). This completes the construction of the tower of fields F_{1} ⊂ F_{2} ⊂ F_{3} ⊂ · · ·. Now denote the union of the fields F_{i} (i = 1, 2, …) by F. Clearly

Br(F) = lim_{⟶} Br(F_{i}) = lim_{⟶}(A_{i} ⊕ B_{i}) = lim_{⟶} A_{i} ≃ A ,

as desired.

References

[1] B. Fein and M. Schacher, Brauer groups of fields, in Ring theory and algebra, III (Proc. Third Conf., Univ. Oklahoma, Norman, Okla., 1979), 345–356, Lecture Notes in Pure and Appl. Math., 55, Dekker, New York. MR0584617 (81k:12025)

[2] B. Fein and M. Schacher, Relative Brauer groups. I, J. Reine Angew. Math. 321 (1981), 179–194. MR0597988 (82f:12027)

[3] A. S. Merkurjev, Brauer groups of fields, Comm. Algebra 11 (1983), no. 22, 2611–2624. MR0733345 (85f:12006)

Editorial remarks

The above note is a translation of one presented at the October 4, 1983, meeting of the Leningrad Mathematical Society on the occasion of Merkurjev winning the Society’s Young Mathematician Prize. It was originally published in Russian in 1985 and has not previously appeared in English. The prehistory of the note was explained to us by Burt Fein: “Merkurjev made a tour of the US in the early 1980s and visited Oregon State. While he was here, Bill Jacob and I took him to the University of Oregon to give a seminar talk; we also took him to Cafe Zenon to sample their wonderful cream puffs. Over cream puffs I told him about the conjecture from [1] and asked him specifically about whether there was a field with Brauer group ℤ/3. He solved it on the spot, first using K_{2} and then coming up with a more traditional proof that it could not. I wrote up that proof and circulated it to the experts in the field under the title ‘Merkurjev’s Cream Puff Theorem’. That was the start of reference [3] and eventually to the note itself.”

Documenta Mathematica · Extra Volume Merkurjev (2015) 461–463

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