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The de Rham and the syntomic logarithm

Matthias Flach, Achim Krause, Baptiste Morin

Abstract

We define and study an integral refinement of the inverse of the Bloch-Kato exponential map which we call the de Rham logarithm. Our main tool to analyze the de Rham logarithm is the syntomic logarithm, a certain limit construction based on the theory of filtered prismatic cohomology initiated by Antieau, Krause and Nikolaus. We use the syntomic logarithm to prove a version of the Beilinson fibre square for all quasicompact, quasiseparated derived formal schemes. We also use our techniques to prove Conjecture $C_{EP}(\bq_p(n))$ of Fontaine and Perrin-Riou for all local fields $K/\bq_p$ and to compute the correction factor $C(X,n)$ introduced by Flach and Morin in their reformulation of the Bloch-Kato Tamagawa number conjecture for the Zeta function of a smooth projective scheme $X$ over a number ring.

The de Rham and the syntomic logarithm

Abstract

We define and study an integral refinement of the inverse of the Bloch-Kato exponential map which we call the de Rham logarithm. Our main tool to analyze the de Rham logarithm is the syntomic logarithm, a certain limit construction based on the theory of filtered prismatic cohomology initiated by Antieau, Krause and Nikolaus. We use the syntomic logarithm to prove a version of the Beilinson fibre square for all quasicompact, quasiseparated derived formal schemes. We also use our techniques to prove Conjecture of Fontaine and Perrin-Riou for all local fields and to compute the correction factor introduced by Flach and Morin in their reformulation of the Bloch-Kato Tamagawa number conjecture for the Zeta function of a smooth projective scheme over a number ring.
Paper Structure (36 sections, 72 theorems, 411 equations)

This paper contains 36 sections, 72 theorems, 411 equations.

Table of Contents

  1. Introduction
  2. The Bloch-Kato exponential map
  3. The fundamental fibre square
  4. The syntomic logarithm
  5. Organisation of this article
  6. Acknowledgements
  7. Conventions
  8. Prismatic cohomology of graded and filtered rings
  9. Derived de Rham cohomology of graded and filtered rings
  10. Filtrations and derived formal stacks
  11. Prismatic cohomology relative to a $\delta$-ring
  12. Prismatic F-gauges
  13. The syntomic package
  14. Compatibility with the definitions of akn23
  15. Prismatic cohomology of graded and filtered $\delta$-pairs
  16. ...and 21 more sections

Key Result

Theorem 1.1.1

Let $K/\mathbb Q_p$ be a finite extension with discriminant $D_K$ and residue field of cardinality $q$. Then for $n\geq 2$ inside $\mathrm{det}_{\mathbb Q_p}^{-1}R\Gamma(K,\mathbb Q_p(n))\simeq\mathrm{det}_{\mathbb Q_p}H^1(K,\mathbb Q_p(n))$.

Theorems & Definitions (203)

  • Theorem 1.1.1
  • proof
  • Corollary 1.1.1
  • proof
  • Corollary 1.1.2
  • proof
  • Proposition 1.2.1
  • proof
  • Theorem 1.2.1
  • proof
  • ...and 193 more