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

DOI10.1007/S00209-022-03122-4arXiv2108.13563WikidataQ114231047 ScholiaQ114231047MaRDI QIDQ6376414

Jinhyun Park

Publication date: 30 August 2021

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 $mgeq2$, 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.

Mathematics Subject Classification ID

Algebraic cycles (14C25) (Equivariant) Chow groups and rings; motives (14C15) Motivic cohomology; motivic homotopy theory (14F42) Higher symbols, Milnor (K)-theory (19D45) Formal power series rings (13F25) Formal neighborhoods in algebraic geometry (14B20)

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