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The Brauer group and the second Abel-Jacobi map for 0-cycles on algebraic varieties. (English) Zbl 1095.14008

The author defines a higher Abel–Jacobi map \[ \Delta^2_n: F^2\text{ CH}_0(X)\longrightarrow \text{ Ext}^2(\text{ Br}(\bar{X}),\bar{k}^\times)\otimes\mathbb{Q}, \] in line with Beilinson’s conjectures. Here \(k\) is a ground field, \(X\) is a smooth projective scheme, \(F^2\text{ CH}_0(X)\) denotes the group of all 0-cycles of degree zero and mapping to zero under the first Abel–Jacobi map, \(\text{ Br}(X)\) is the Brauer group, and extensions are understood to be extensions of Galois modules. The map is defined as follows: Given a 0-cycle \(Z\) in \(F^2\text{ CH}_0(X)\), there is an exact sequence of Galois modules \[ 0\rightarrow {\text{ Pic}}(\bar{X},\bar{Z})\rightarrow \prod{\text{ Pic}}(\bar{C}) \rightarrow \tilde{\text{Br}}(\bar{X},\bar{Z}) \rightarrow{\text{ Br}}(\bar{X})\rightarrow 0. \] Here \({\text{ Pic}}(\bar{X},\bar{Z})\) denotes the group of line bundles endowed with a trivialization on the 0-cycle, the product runs over all curves on \(\bar{X}\), and the term \(\tilde{\text{ Br}}(\bar{X},\bar{Z})\) denotes the group of equivalence classes of a quadrupel \((P,\xi,f,\varphi)\), where \(P\) is a Severi–Brauer variety over \(\bar{X}\), and \(\xi\) is a family of trivializations of \(P\) restricted to all curves \(\bar{C}\), and \(f\) is a trivialization of \(P\) restricted to the 0-cycle, and \(\varphi\) is a compatibility isomorphism between these trivializations. Pushing this Yoneda 2-extension forward along a norm map \({\text{ Pic}}(\bar{X},\bar{Z})\rightarrow \bar{k}^\times\), one obtains the desired extension class \(\Delta^2_n(Z)\). Large parts of this carefully written paper compare this map with \(l\)-adic Abel–Jacobi maps [U. Jannsen, in: Motives. Proc. sum. res. conf. motives, Washington, Seattle, 1991. Proc. Symp. Pure Math. 55, 245–302 (1994; Zbl 0811.14004)], and Green’s description for surfaces [M. Green, Doc. Math., J. DMV, Extra Vol. ICM Berlin 1998, II, 267–276 (1998; Zbl 0914.14002)]

MSC:

14C15 (Equivariant) Chow groups and rings; motives
14F22 Brauer groups of schemes
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