Motivic cohomology of fat points in Milnor range

TL;DR

This paper develops a new algebraic-cycle model for motivic cohomology of truncated polynomial rings, computes these groups in characteristic zero, and links them to Milnor K-theory and differential forms.

Contribution

It introduces a novel approach using deformation theory and non-archimedean analysis for motivic cohomology of fat points, distinct from existing methods.

Findings

Computed motivic cohomology groups for truncated polynomials in characteristic zero.

Established these groups as Milnor K-groups of the algebra.

Identified the relative part with absolute Kähler differential forms.

Abstract

We introduce a new algebraic-cycle model for the motivic cohomology theory of truncated polynomials $k[t]/(t^m)$ in one variable. This approach uses ideas from the deformation theory and non-archimedean analysis, and is distinct from the approaches via cycles with modulus. We compute the groups in the Milnor range when the base field is of characteristic $0$, and prove that they give the Milnor $K$-groups of $k[t]/(t^m)$, whose relative part is the sum of the absolute K\"ahler differential forms.