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Galois descent for higher Brauer groups. (English) Zbl 07260772
We say that the Tate conjecture in codimension \(m\) holds for a smooth variety \(X\) over a field \(k\) if the image of the cycle class map \[\text{CH}^m \left( \overline{X} \right) \otimes \mathbb{Q}_l \rightarrow H^{2m}_{\text{ét}} \left( \overline{X}, \mathbb{Q}_l(m) \right)\] coincides with the union \[\bigcup_{U} H^{2m}_{\text{ét}} \left( \overline{X}, \mathbb{Q}_l(m) \right)^U,\] where \(U\) ranges over the open subgroups of the absolute Galois group \(\text{Gal}\left( \overline{k} /k \right)\). The Tate conjecture for divisors (\(m=1\)) is equivalent to the finiteness of the \(l\)-primary torsion of the Brauer group \[\text{Br} \left( X \right)[l^\infty]^{\text{Gal}\left( \overline{k} /k \right)} \cong H^{2}_{\text{ét}} \left( \overline{X}, \mathbb{G}_m \right)[l^\infty]^{\text{Gal}\left( \overline{k} /k \right)}\] (see for instance [J. Tate, Adv. Stud. Pure Math. 3, 189–214 (1968); Sém. Bourbaki 1965/66, Exp. No. 306, 415–440 (1966; Zbl 0199.55604)] or [J. S. Milne, Ann. Math. (2) 102, 517–533 (1975; Zbl 0343.14005)]).
The definition of the Brauer group may be generalized to higher dimension by setting \(\text{Br}^m (X)\) to be the étale motivic cohomology group \(H^{2m+1}_L \left( X, \mathbb{Z} (m) \right)\). Suppose that \(k\) is a finitely generated field of characteristic zero. Analogously to the results for the Tate conjecture for divisors, it is proved in the present paper that the Tate conjecture in codimension \(m\) is equivalent to the finiteness of the \(l\)-primary torsion \(\text{Br}^m \left( \overline{X}\right) \left[l^\infty\right]^{\text{Gal}\left( \overline{k} /k \right)}\). After establishing preliminary results and background on higher Brauer groups in Sections 1 and 2, the proof of this statement is provided in Section 3.1 using similar methods to the case \(m=1\) as proved in [A. N. Skorobogatov and Y. G. Zarhin, J. Algebr. Geom. 17, No. 3, 481–502 (2008; Zbl 1157.14008)].
It is expected that the failure of Galois descent for (higher) Brauer classes is at worst finite. This finiteness was established in the case \(m=1\), over fields of characteristic \(0\) in [J.-L. Colliot-Thélène and A. N. Skorobogatov, J. Reine Angew. Math. 682, 141–165 (2013; Zbl 1317.14042)]. In the present paper, the finiteness is proved for higher Brauer groups. Let \(X\) be a smooth projective variety over a finitely generated field \(k\). Assume that \(k\) is of characteristic \(0\) or that \(X\) satisfies the standard conjectures (conjectures \(B\), \(C\), and \(D\) in [S. L. Kleiman, Dix Exposes Cohomologie Schemas, Advanced Studies Pure Math. 3, 359-386 (1968; Zbl 0198.25902)]). The author proves that the cokernel of the map \[\text{Br}^m \left(X\right) \left[ \tfrac1p \right] \rightarrow \text{Br}^m \left( \overline{X} \right)^{\text{Gal}\left( \overline{k}/k \right)} \left[ \tfrac1p \right]\] is finite, where \(p\) is the exponential characteristic of \(k\). The proof of this main result relies on the degeneracy of the Hochschild-Serre spectral sequence. Using the preliminary results established in Sections 1 and 2, the proof of this statement is given in Section 3.2.
MSC:
14F20 Étale and other Grothendieck topologies and (co)homologies
14F22 Brauer groups of schemes
19E15 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)
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