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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)
##### Keywords:
Brauer groups; algebraic cycles; Tate conjecture
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##### References:
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