Kato–Saito local-to-global conjecture for zero-cycles

Let $K$ be a number field, let $X$ be a smooth projective geometrically integral variety over $K$, and let $P_K$ be the set of places of $K$. Write $X_{K_v}$ for the base change to the completion $K_v$, and let $A_0$ denote the Chow group of zero-cycles of degree zero. Kato–Saito conjecture. The complex

$$\varprojlim_n A_0(X)/n\rightarrow \prod_{v\in P_K}\varprojlim_n A_0(X_{K_v})/n\rightarrow \operatorname{Hom}(\operatorname{Br}(X)/\operatorname{Br}(K),\mathbb{Q}/\mathbb{Z})$$

is exact.

This is a local-to-global principle for zero-cycles governed by the Brauer–Manin obstruction; the paper verifies the conjecture for the higher-cycle setting it studies.