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Lichtenbaum-Tate duality for varieties over \(p\)-adic fields. (English) Zbl 1059.14029

Let \(\varphi: X \to \operatorname{Spec} K\) be a smooth proper variety over a finite extension \(K\) of \(\mathbb{Q}_p\). In the paper under review the author introduces a new homology theory, pseudo-motivic homology \(^1H_i(X,\mathbb{Z}):= \text{Ext}^{-i}_{k_{\text{sm}}}(R\varphi_* \mathbb{G}_m, \mathbb{G}_m)\), and proves that for every \(r \in \mathbb{Z}\) the Yoneda pairing \[ ^1H_r(X, \mathbb{Z}) \times H^{r+2}(X, \mathbb{G}_m) \longrightarrow \text{Br}(K) = \mathbb{Q}/\mathbb{Z} \] is nondegenerate, induces perfect pairings (\(A\hat{\;}\) denotes the completion of \(A\) with respect to the profinite topology) \[ ^1H_r(X, \mathbb{Z}) \times H^{r+2}(X, \mathbb{G}_m)\sphat \longrightarrow \mathbb{Q}/\mathbb{Z} \;\;\text{for}\;\;r=-2,-1, \]
\[ ^1H_r(X, \mathbb{Z})\hat{\;} \times H^{r+2}(X, \mathbb{G}_m) \longrightarrow \mathbb{Q}/\mathbb{Z} \;\;\text{for}\;\;r=0,1, \] and \[ ^1H_r(X, \mathbb{Z}) \times H^{r+2}(X, \mathbb{G}_m) \longrightarrow \mathbb{Q}/\mathbb{Z} \;\;\text{for}\;\;r\geq 2. \] Then using the Poincaré duality \(H^i(C, \mathbb{G}_m) \overset {\sim} {\rightarrow} {}^1H_{i-1}(C, \mathbb{Z})\), due to P. Deligne, for a smooth projective curve \(C\) over a field of characteristic zero the author deduces the Lichtenbaum-Tate duality \[ H^i(C, \mathbb{G}_m) \times H^{3-i}(C, \mathbb{G}_m) \longrightarrow \mathbb{Q}/ \mathbb{Z} \] for every \(i\in \mathbb{Z}\). Thus the pairings considered in the paper are good generalizations to higher dimensions of Lichtenbaum’s pairing between the Picard group and the Brauer group of a nonsingular projective curve \(C\) defined over a finite extension of the \(p\)-adic field \(\mathbb{Q}_p\). The author proves also that for a smooth projective irreducible variety \(X\) over a \(p\)-adic field \(K\) the kernel of the canonical map \( \text{Br}(K) \to \text{Br}(K(X))\) is a finite cyclic group of order \(PsI(X)\) (the author calls \(PsI(X)\) pseudo-index of \(X\)) dual to the cokernel of the degree map \(^1H_0(X, \mathbb{Z})\to \mathbb{Z}\). This result may be regarded as a generalization of Roquette result on index of a curve over \(p\)-adic field. If \(I(X)\) is the index of \(X\) (the index of the image of the degree map CH\(_0(X) \to \mathbb{Z}\)), and \(P(X)\) is the period of \(X\) (the order of the cokernel of the degree map \(\text{Alb}^*(X) \to \mathbb{Z}\)), then \(P(X)| PsI(X)| I(X)\). As a corollary of the obtained results the author proves that for a principal homogeneous space \(X\) of an abelian variety over a \(p\)-adic field \(K\) the natural map \(\text{Br} (K) \to \text{Br}(K(X))\) from the Brauer group of \(K\) to the Brauer group of the function field of \(X\) is injective if and only if \(X\) is a trivial principal homogeneous space.

MSC:

14G20 Local ground fields in algebraic geometry
14F22 Brauer groups of schemes
19E15 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)
11G25 Varieties over finite and local fields
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14F42 Motivic cohomology; motivic homotopy theory
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[1] Artin A., Lect. Notes Math. pp 305– (1973)
[2] [Bl] S. Bloch, Lectures on algebraic cycles, Duke University Mathematics Department, 1980.
[3] Br L. S, Bull. Amer. Math. Soc. 75 pp 1249– (1969)
[4] Br L. S, Invent. Math. 9 pp 15– (1969)
[5] [CTS] J.L. Colliot-Thelene and S. Saito, Zero-cycles sur les varietes p-adiques et groupe de Brauer, Int. Math. Res. Not. (1996), 151-160. · Zbl 0878.14006
[6] Deligne, Lect. Notes Math. pp 569– (1977)
[7] [Gr1] A. Grothendieck, Fondements de la geometrie algebrique, Extraits du Seminaire Bourbaki, 1957-1962, Secretariat mathematique, Paris 1962.
[8] [Gr2] A. Grothendieck, Le groupe de Brauer I, II, III, in: Dix exposes sur la cohomologie des schemas, Adv. Stud. Pure Math. 3, North-Holland, Masson (1968), 46-188.
[9] Jannsen, Math. Ann. 280 pp 207– (1988)
[10] [K] B. Kahn, A sheaf-theoretic reformulation of the Tate conjecture, preprint 1998, http://arxiv.org/abs/ math.AG/9801017.
[11] [La] G. Laumon, Homologie etale, Exp. VIII, in: Seminaire de Geometrie Analytique, A. Douady and J.L. Verdier, eds., Asterisque 36-37 (1974-1975), 163-188.
[12] Invent. Math. 7 pp 120– (1969)
[13] [L2] S. Lichtenbaum, Values of zeta-functions at nonnegative integers, in: Number theory, H. Jager, ed. (Noordwijkerhout 1983), Lect. Notes Math. 1068, Springer-Verlag (1984), 127-138.
[14] [M1] J. S. Milne, Eatale cohomology, Princeton University Press, 1980.
[15] [M2] J. S. Milne, Arithmetic duality theorems, Perspect. math. 1, Academic Press, 1986.
[16] [Mum] D. Mumford, Abelian Varieties, Tata Inst. Fundam. Res. Stud. Math.5, Oxford University Press, 1970.
[17] Mur J. P, Sci. Publ. Math. 23 pp 5– (1964)
[18] Oort, Lect. Notes Math. pp 15– (1966)
[19] Parimala V., Invent. Math. 122 pp 83– (1995)
[20] [Ra] N. Ramachandran, Duality of Albanese and Picard 1-motives, K-Theory 22 (2001), 271-301. · Zbl 0983.14003
[21] [Ro] A. A. Rojtman, The torsion of the group of 0-cycles modulo rational equivalence, Ann. Math. (2) 111 (1980), 553-569. · Zbl 0504.14006
[22] Ro P, Nagoya Math. J. 27 pp 625– (1966)
[23] [Sa] S. Saito, A global duality theorem for varieties over global fields, in: Algebraic K-theory: connections with geometry and topology, J. F. Jardine, V. P. Snaith, eds. (Lake Louise 1987), Kluwer Acad. Publ. (1989), 425-444.
[24] Se J., Lect. Notes Math. pp 5– (1994)
[25] Manuscr. Math. 98 pp 409– (1999)
[26] [vH2] J. van Hamel, The Brauer-Manin obstruction for zero-cycles on Severi-Brauer fibrations over curves, J. London Math. Soc. (2) 68 (2003), 317-337. · Zbl 1083.14024
[27] Documenta Math. 8 pp 125– (2003)
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