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Bloch-Ogus theory for smooth and semi-stable schemes in mixed characteristic

Morten Lüders

Abstract

We study Bloch-Ogus theory and the Gersten conjecture for homology theories with duality satisfying certain properties, in particular for étale cohomology with finite coefficients coprime to the residue characteristic of the base, for smooth and semi-stable schemes in mixed characteristic. We prove the Gersten conjecture in the smooth case and prove a special case in the semi-stable situation. As a corollary of the smooth case we obtain the surjectivity of the Galois symbol map for arbitrary local rings over an excellent discrete valuation ring.

Bloch-Ogus theory for smooth and semi-stable schemes in mixed characteristic

Abstract

We study Bloch-Ogus theory and the Gersten conjecture for homology theories with duality satisfying certain properties, in particular for étale cohomology with finite coefficients coprime to the residue characteristic of the base, for smooth and semi-stable schemes in mixed characteristic. We prove the Gersten conjecture in the smooth case and prove a special case in the semi-stable situation. As a corollary of the smooth case we obtain the surjectivity of the Galois symbol map for arbitrary local rings over an excellent discrete valuation ring.
Paper Structure (11 sections, 12 theorems, 58 equations)

This paper contains 11 sections, 12 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Acknowledgement.
  3. Homology theories
  4. Homology theories with duality
  5. An example: étale cohomology
  6. The Gersten conjecture: the smooth case
  7. General reductions
  8. A key lemma due to Bloch and Ogus
  9. Proof of the smooth case
  10. An application to the Galois symbol map
  11. The Gersten conjecture: the semi-stable case

Key Result

Theorem 1.1

(Thm. theorem_smooth_case) Let $S$ and $X$ be as above and $(H_*,H^*)$ a homology theory with duality satisfying the principal triviality Property property_local_principality and lifting Property property_lifting. If $X$ is smooth over $S$, then the Gersten conjecture holds for $\mathcal{H}$ in the

Theorems & Definitions (33)

  • Theorem 1.1
  • Corollary 1.2: Cor. \ref{['corollary_galois_symbol']}
  • Theorem 1.3
  • Remark 1.4: Functoriality in the semi-stable case
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • ...and 23 more