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Les suites spectrales de Hodge-Tate

Ahmed Abbes, Michel Gros

Abstract

This book presents two important results in p-adic Hodge theory following the approach initiated by Faltings, namely (i) his main p-adic comparison theorem, and (ii) the Hodge-Tate spectral sequence. We establish for each of these results two versions, an absolute one and a relative one. While the absolute statements can reasonably be considered as well understood, particularly after their extension to rigid varieties by Scholze, Faltings' initial approach for the relative variants has remained much less studied. Although we follow the same strategy as that used by Faltings to establish his main p-adic comparison theorem, part of our proofs is based on new results. The relative Hodge-Tate spectral sequence is new in this approach.

Les suites spectrales de Hodge-Tate

Abstract

This book presents two important results in p-adic Hodge theory following the approach initiated by Faltings, namely (i) his main p-adic comparison theorem, and (ii) the Hodge-Tate spectral sequence. We establish for each of these results two versions, an absolute one and a relative one. While the absolute statements can reasonably be considered as well understood, particularly after their extension to rigid varieties by Scholze, Faltings' initial approach for the relative variants has remained much less studied. Although we follow the same strategy as that used by Faltings to establish his main p-adic comparison theorem, part of our proofs is based on new results. The relative Hodge-Tate spectral sequence is new in this approach.

Paper Structure

This paper contains 53 sections, 185 theorems, 1950 equations.

Table of Contents

  1. La suite spectrale de Hodge-Tate relative -- un survol
  2. Introduction
  3. La suite spectrale de Hodge-Tate relative: sections globales au-dessus d'un petit schéma affine
  4. La suite spectrale de Hodge-Tate relative: localisation
  5. La suite spectrale de Hodge-Tate relative
  6. Les principaux théorèmes de comparaison p-adiques de Faltings
  7. Topos de Faltings relatif
  8. Préliminaires
  9. Notations et conventions
  10. Schémas K(pi,1)
  11. Compléments sur la connexité
  12. Compléments de géométrie logarithmique
  13. Rappel sur une construction de Fontaine-Grothendieck
  14. Faisceaux de alpha-modules
  15. Conditions de alpha-finitude
  16. ...and 38 more sections

Key Result

Proposition 1.4.2

Pour tout faisceau abélien de torsion, localement constant et constructible $F$ de $X_{{\overline{\eta}},{\rm \acute{e}t}}$, on a ${\rm R}^i\psi_*(F)=0$ pour tout $i\geq 1$.

Theorems & Definitions (185)

  • Proposition 1.4.2: cf. \ref{['acycloc2']}
  • Proposition 1.4.6: cf. \ref{['sshtr11']}
  • Proposition 1.6.3: cf. \ref{['tfr30']}
  • Proposition 1.6.7: cf. \ref{['eccr51']}
  • Proposition 1.6.10: cf. \ref{['ktfr27']}
  • Proposition 1.6.12: cf. \ref{['ktfr30']}
  • Proposition 2.2.1
  • Proposition 2.2.5
  • Proposition 2.2.7
  • Proposition 2.2.10: sga4 XI 3.3
  • ...and 175 more