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

Keywords:

higher Abel-Jacobi maps

Citations:

Zbl 0811.14004; Zbl 0914.14002
Full Text:

References:

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