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

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 $m\ge 2$, 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.

arXiv commentv3: 38 pages. some referee comments reflected. A version of it was accepted to appear in Mathematische Zeitschrift. Due to copyright reason, this is NOT the final accepted version, but one of the earlier versions