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Milnor K-theory of complete discrete valuation rings with finite residue fields (1509.01087v3)
Published 3 Sep 2015 in math.KT
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 $\hat{\mathrm{K}}\mathrm{M}_*\mathcal{O} \to \hat{\mathrm{K}}\mathrm{M}_*F$ 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.