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

Jinhyun Park

Published 2021-08-31Version 1

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, 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. We deduce that their relative parts are isomorphic to the big de Rham-Witt forms of k. In particular, this provides another cycle-theoretic description of them after a result of K. R\"{u}lling (J. Algebraic Geom., 2007).