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On the Hasse principle for the Brauer group of a purely transcendental extension field in one variable over an arbitrary field. (English) Zbl 1259.12004
Tohoku Mathematical Publications 36. Sendai: Tohoku University, Mathematical Institute (Diss.). 57 p. (2012).
Let \(k\) be a field, \(X\) be a complete normal curve over \(k\) with function field \(K\). For any point \(P \in X\), denote by \(O_{X,P}\) the local ring of \(X\) at \(P\).
The main question of this paper is the following local-global conjecture. Suppose that \(k\) is a field which is finitely generated over its prime field, and \(m\) is a positive integer such that \(\gcd(m, \text{char}\,k) =1\). Let \(X\) be a complete normal curve over \(k\) with function field \(K\). For any closed point \(P \in X\), denote by \(\tilde{K}_P\) the quotient field of the henselization of \(O_{X,P}\). Question. If \(p\) is a positive integer, is the local-global map \(H^p(K, \mu_m) \to \prod_{P \in X}H^p(\tilde{K}_P, \mu_m)\) injective ?
This article announces an answer when \(p=1\) and \(k\) is an algebraic number field.
On the other hand, the author provides another proof of a similar result of Harder, i.e. Theorem. For any field \(k\), let \(X\) be the projective line over \(k\) with function field \(k(t)\). Denote by \(\hat{k(t)}_P\) the quotient field of the completion of \(O_{X,P}\) where \(P\) is a closed point of \(X\). Let \(\text{Br}(k(t))\) be the Brauer group of \(k(t)\). Then the local-global map \(\text{Br}(k(t)) \to \prod_{P \in X}\text{Br}(\hat{k(t)}_P)\) is injective (see Theorem 7.27). For G. Harder’s original proof, see his paper [Invent. Math. 6, 107–149 (1968; Zbl 0186.25902)].
The author also shows that the above local-global map is injective when \(k\) is separably closed and \(X\) is a complete normal curve over \(k\) (see Corollary 7.23).
The author uses the cohomological tools to prove these results. He establishes something similar to the Faddeev sequence (see Proposition 7.4) and the maps of this sequence are specified very carefully.
In order to specify these maps, laborious works are done for the edge maps of the Grothendieck spectral sequence (see Section 6).
In Section 5, the author recalls previous results about the Hasse principle.
This paper supplements the cohomological study of Brauer groups in Milne’s book [J. S. Milne, Étale cohomology. Princeton Mathematical Series. 33. Princeton, New Jersey: Princeton University Press. (1980; Zbl 0433.14012)]. These two works (together with A. Grothendieck’s “Le groupe de Brauer, I, II, III” [I: Sémin. Bourbaki 17 (1964/65), Exp. No. 290, 21 p. (1966; Zbl 0186.54702), II: Sém. Bourbaki 1965/66, Exp. No. 297, 21 p. (1966), I–III: Dix Exposés Cohomologie Schémas, Adv. Stud. Pure Math. 3, 46–66, 67-87, 88-188 (1968; Zbl 0193.21503, Zbl 0198.25803, Zbl 0198.25901)]) will become convenient references for people interested in the cohomological approach of Brauer groups.

MSC:
12G05 Galois cohomology
11R58 Arithmetic theory of algebraic function fields
11R34 Galois cohomology
16K50 Brauer groups (algebraic aspects)
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
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