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 Kähler differential forms.

arXiv commentv1: 24 pages/ v2: 29 pages. A few errors fixed. Some basic functoriality properties requested were added/ v3: 30 pages. Some improvements. Not the final accepted version due to copyright reason. A version was accepted to appear in Doc. Math