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Using an idelic argument and assuming the Gersten conjecture for MilnorK-theory, we show that the restriction map from one-cycles on a smoothprojective scheme over a henselian local ring to a pro-system of thickenedzero-cycles is surjective. We relate this restriction map to the $p$-adic cycleclass map.

Algebraization for zero-cycles and the $p$-adic cycle class map