Exact description of the right kernel for open surfaces

Establish an explicit description of the right kernel of the Brauer–Manin pairing for an open smooth surface over a local field in terms of the prime-to-residue-characteristic Brauer group of a dense regular open subset of the boundary and its reciprocity-theoretic orthogonal subgroup.

Background

For a smooth projective surface, the paper recalls known descriptions of the right kernel of the Brauer–Manin pairing. For an open surface, the right kernel is bounded between the Brauer group of a compactification and the Brauer group of a regular model of the open surface, but these inclusions can be strict.

The authors state that no explicit description is known and formulate a conjectural short exact sequence relating the right kernel to a subgroup defined from the boundary. The conjecture is later proved under the paper’s stated Tate-conjecture and reduction hypotheses, but remains an explicit conjecture in the general setting described here.