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

TL;DR

This paper establishes a surjectivity result for restriction maps of zero-cycles on smooth projective schemes over henselian local rings, connecting it to the $p$-adic cycle class map under certain conjectural assumptions.

Contribution

It introduces an idelic approach to relate zero-cycle restriction maps to the $p$-adic cycle class map, assuming the Gersten conjecture for Milnor K-theory.

Findings

Surjectivity of the restriction map from one-cycles to thickened zero-cycles.

Connection between the restriction map and the $p$-adic cycle class map.

Conditional results based on the Gersten conjecture.

Abstract

Using an idelic argument and assuming the Gersten conjecture for Milnor K-theory, we show that the restriction map from one-cycles on a smooth projective scheme over a henselian local ring to a pro-system of thickened zero-cycles is surjective. We relate this restriction map to the $p$-adic cycle class map.