On the relative Gersten conjecture for Milnor K-theory in the smooth case

We show that the Gersten complex for the (improved) Milnor K-sheaf on a smooth scheme over an excellent discrete valuation ring is exact except at the first place and that exactness at the first place may be checked at the discrete valuation ring associated to the the generic point of the special fibre. This complements results of Gillet-Levine for K-theory, Geisser for motivic cohomology and Schmidt-Strunk and the author for étale cohomology.