Milnor K-theory of complete discrete valuation rings with finite residue fields

Abstract

Consider a complete discrete valuation ring $\mathcal{O}$ with quotient field $F$ and finite residue field. Then the inclusion map $\mathcal{O}\hookrightarrow F$ induces a map ${\hat{\mathrm{K}}}_{\ast }^{\mathrm{M}}\mathcal{O}\to {\hat{\mathrm{K}}}_{\ast }^{\mathrm{M}}F$ on improved Milnor K-theory. We show that this map is an isomorphism in degrees bigger or equal to 3. This implies the Gersten conjecture for improved Milnor K-theory. This result is new in the $p$-adic case.

18 pages. An edited version will appear in J. Pure Appl. Algebra (2017)

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