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We present a conjectural formula describing the cokernel of the Albanese mapof zero-cycles of smooth projective varieties $X$ over $p$-adic fields in termsof the N\'eron-Severi group and provide a proof under additional assumptions onan integral model of $X$. The proof depends on a non-degeneracy result ofBrauer-Manin pairing due to Saito-Sato and on Gabber-de Jong's comparisonresult of cohomological- and Azumaya-Brauer groups. We will also mention thelocal-global problem of the Albanese-cokernel; the abelian group on the "localside" turns out to be a finite group.

A higher dimensional generalization of Lichtenbaum duality in terms of the Albanese map