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On the functoriality of refined unramified cohomology

Kees Kok, Lin Zhou

TL;DR

This work extends refined unramified cohomology to have functorial pullbacks along morphisms of smooth schemes by a Fulton–style deformation construction, and defines a robust cycle-action framework via cross products and correspondences that is compatible with the Bloch–Ogus spectral sequence. It establishes a projective bundle and blow-up calculus for these refined groups and uses these tools to translate Rost nilpotence into refined unramified cohomology statements, broadening the scope of known RNP results. The paper also computes the refined unramified cohomology for smooth projective linear varieties, obtaining surjectivity of the restriction map and, hence, RNP in characteristic zero in this class. Furthermore, it analyzes varieties with complete decomposition of the diagonal to deduce surjectivity of restriction maps and to derive structural features of $H^p_{q, ext{nr}}(X,M(n))$ over algebraically closed fields. Overall, these developments deepen the interaction between cycle-theoretic methods and refined cohomological invariants, with consequences for nilpotence phenomena and diagonal decompositions in algebraic geometry.

Abstract

In this paper, we generalise the construction of the functorial pullback of refined unramified cohomology between smooth schemes, by following the ideas of Fulton's intersection theory and Rost's cycle modules. We also define standard actions of algebraic cycles on the refined unramified cohomology groups of smooth proper schemes avoiding Chow's moving lemma, which coincide with Schreieder's constructions for smooth projective schemes. As applications, we prove the projective bundle and blow-up formulas for refined unramified cohomology groups and we reduce the Rost nilpotence principle in characteristic zero to a statement concerning certain refined unramified cohomology groups. Moreover, we compute the refined unramified cohomology for smooth proper linear varieties and show that Rost's nilpotence principle holds for these varieties in characteristic zero.

On the functoriality of refined unramified cohomology

TL;DR

This work extends refined unramified cohomology to have functorial pullbacks along morphisms of smooth schemes by a Fulton–style deformation construction, and defines a robust cycle-action framework via cross products and correspondences that is compatible with the Bloch–Ogus spectral sequence. It establishes a projective bundle and blow-up calculus for these refined groups and uses these tools to translate Rost nilpotence into refined unramified cohomology statements, broadening the scope of known RNP results. The paper also computes the refined unramified cohomology for smooth projective linear varieties, obtaining surjectivity of the restriction map and, hence, RNP in characteristic zero in this class. Furthermore, it analyzes varieties with complete decomposition of the diagonal to deduce surjectivity of restriction maps and to derive structural features of over algebraically closed fields. Overall, these developments deepen the interaction between cycle-theoretic methods and refined cohomological invariants, with consequences for nilpotence phenomena and diagonal decompositions in algebraic geometry.

Abstract

In this paper, we generalise the construction of the functorial pullback of refined unramified cohomology between smooth schemes, by following the ideas of Fulton's intersection theory and Rost's cycle modules. We also define standard actions of algebraic cycles on the refined unramified cohomology groups of smooth proper schemes avoiding Chow's moving lemma, which coincide with Schreieder's constructions for smooth projective schemes. As applications, we prove the projective bundle and blow-up formulas for refined unramified cohomology groups and we reduce the Rost nilpotence principle in characteristic zero to a statement concerning certain refined unramified cohomology groups. Moreover, we compute the refined unramified cohomology for smooth proper linear varieties and show that Rost's nilpotence principle holds for these varieties in characteristic zero.
Paper Structure (13 sections, 38 theorems, 70 equations)

This paper contains 13 sections, 38 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Notations and Conventions
  4. Pullback on Refined Unramified Cohomology
  5. Borel--Moore Homology and Étale Cohomology with Supports
  6. General Pullbacks along Smooth Schemes
  7. Action of Cycles
  8. Cross Product
  9. Correspondence Action
  10. Applications
  11. Projective Bundle and Blow-Up Formula
  12. Rost Nilpotence Principle
  13. Varieties with Complete Decomposition of the Diagonal

Key Result

Theorem 1.1

Let $f\colon X\to Y$ be a morphism between smooth schemes, there is a functorial pull-back map $f^\ast\colon H^p_{q,\mathrm{nr}}(Y,n)\to H^p_{q,\mathrm{nr}}(X,n)$, which coincides with Schreieder's construction in schreieder2022moving when $X$ and $Y$ are smooth quasi-projective schemes admitting a

Theorems & Definitions (84)

  • Theorem 1.1: cf. \ref{['lem-the same pullback']} and \ref{['thm-functoriality']}
  • Theorem 1.2: cf. \ref{['prop-actions of corr']}, \ref{['rem-compatible action']}
  • Lemma 1.3: = \ref{['lem-projection formula II']}
  • Proposition 1.4: cf. \ref{['prop-projective bundle formula']}
  • Proposition 1.5: cf. \ref{['prop-blow-up bundle']}
  • Proposition 1.6: = \ref{['prop-RNP']}
  • Proposition 1.7: cf. \ref{['prop-surjective']}, \ref{['lem-computation for algebraically closed field']}
  • Definition 2.1: cf. BO74
  • Proposition 2.2: schreieder2020refined
  • Proposition 2.3: schreieder2022moving
  • ...and 74 more