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

Morten Lüders

Abstract

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 fiber. This complements results of Gillet and Levine for K-theory, Geisser for motivic cohomology and Schmidt and Strunk and the author for étale cohomology.

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

Abstract

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 fiber. This complements results of Gillet and Levine for K-theory, Geisser for motivic cohomology and Schmidt and Strunk and the author for étale cohomology.

Paper Structure

This paper contains 11 sections, 19 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Milnor $K$-theory, Chow groups with coefficients and geometric representation
  3. Nisnevich descent and $\mathbb{A}^1$-invariance for Milnor $K$-theory
  4. Chow groups with coefficients
  5. Geometric presentation
  6. Hyperplanes and linear projections
  7. Noether normalization over a DVR
  8. Gabber's geometric presentation theorem over a DVR
  9. Exactness on the right
  10. Exactness at the first and second place
  11. Gersten conjecture for henselian DVRs with finite residue field

Key Result

Theorem 1.1

(Corollary maincorollary) Let the notation be as above. Assume furthermore that $\mathcal{O}_K$ is excellent. Then the sequence of sheaves is exact. Furthermore, if $\mathcal{O}_{X,\eta}$ is the discrete valuation ring induced by a generic point of the special fibre, then $i$ is injective if the map is injective for all such $\eta$.

Theorems & Definitions (36)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1: Milnor $K$-theory
  • Theorem 2.2
  • Theorem 2.3
  • Definition 2.4
  • Theorem 2.5
  • Proposition 2.6
  • Definition 2.7
  • Lemma 2.8
  • ...and 26 more