Motivic cohomology of fat points in Milnor range

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.