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

Consider a complete discrete valuation ring O with quotient field F and finite residue field. Then the inclusion map O -> F induces a map (K) over cap (M)(*) O -> (K) over cap (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 for O. This result is new in the p-adic case. (C) 2017 Elsevier B.V. All rights reserved