Motivic cohomology of fat points in Milnor range via formal and rigid geometries

TL;DR

This paper develops a formal scheme-based cycle model for motivic cohomology of fat points over a field, computes their Milnor range cycle class groups, and proves their isomorphism to Milnor K-groups using rigid geometry and the Gersten conjecture.

Contribution

It introduces a new formal scheme approach to motivic cohomology of fat points and generalizes known theorems relating cycle class groups to Milnor K-theory.

Findings

Cycle class groups are isomorphic to Milnor K-groups for fat points.

The approach uses rigid analytic geometry and the Gersten conjecture.

Generalizes results of Nesterenko-Suslin and Totaro.

Abstract

We present a formal scheme based cycle model for the motivic cohomology of the fat points defined by the truncated polynomial rings $k[t]/(t^m)$ with $m \geq 2$, in one variable over a field $k$. We compute their Milnor range cycle class groups when the field has sufficiently many elements. With some aids from rigid analytic geometry and the Gersten conjecture for the Milnor $K$-theory resolved by M. Kerz, we prove that the resulting cycle class groups are isomorphic to the Milnor $K$-groups of the truncated polynomial rings, generalizing a theorem of Nesterenko-Suslin and Totaro.