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Logarithmic prismatic cohomology, motivic sheaves, and comparison theorems

Federico Binda, Tommy Lundemo, Alberto Merici, Doosung Park

Abstract

We prove that (logarithmic) prismatic and (logarithmic) syntomic cohomology are representable in the category of logarithmic motives. As an application, we obtain Gysin maps for prismatic and syntomic cohomology, and we explicitly identify their cofibers. We also prove a smooth blow-up formula and we compute prismatic and syntomic cohomology of Grassmannians. In the second part of the paper, we develop a descent technique inspired by the work of Nizioł~ on log $K$-theory. Using the resulting \emph{saturated descent}, we prove de Rham and crystalline comparison theorems for log prismatic cohomology, and the existence of Gysin maps for $A_{\inf}$-cohomology.

Logarithmic prismatic cohomology, motivic sheaves, and comparison theorems

Abstract

We prove that (logarithmic) prismatic and (logarithmic) syntomic cohomology are representable in the category of logarithmic motives. As an application, we obtain Gysin maps for prismatic and syntomic cohomology, and we explicitly identify their cofibers. We also prove a smooth blow-up formula and we compute prismatic and syntomic cohomology of Grassmannians. In the second part of the paper, we develop a descent technique inspired by the work of Nizioł~ on log -theory. Using the resulting \emph{saturated descent}, we prove de Rham and crystalline comparison theorems for log prismatic cohomology, and the existence of Gysin maps for -cohomology.
Paper Structure (21 sections, 50 theorems, 234 equations)

This paper contains 21 sections, 50 theorems, 234 equations.

Table of Contents

  1. Introduction
  2. Motivic realizations
  3. Saturated descent and comparison theorems
  4. A log version of Breuil-Kisin cohomology
  5. Future perspectives
  6. Log motives and abstract representability results
  7. Motivic sheaves
  8. Generalities on spectra
  9. Log motivic spectra
  10. First examples: de Rham and crystalline motivic spectra
  11. Prismatic and syntomic realizations
  12. The first Chern class
  13. Syntomic cohomology
  14. Consequences of motivic representability
  15. Saturated descent and crystalline comparison
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $S\in \operatorname{QSyn}$. Let $X$ be a smooth scheme over $S$ and $Z\subseteq X$ a closed subscheme of relative codimension $d$, such that $Z\to X\to S$ is smooth. Let ${\mathbf{Bl}}_Z(X)$ denote the blow-up of $X$ in $Z$ and $E$ be the exceptional divisor, so that $({\mathbf{Bl}}_Z(X),E)\in \ whose homotopy cofibers are respectively given as If $S\in \mathrm{QSyn}_R$ with $R$ perfectoid, w

Theorems & Definitions (130)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4: Theorems \ref{['thm:crys_comp_triv']} and \ref{['thm:crys_comp_logpoint']}
  • Theorem 1.5: Theorem \ref{['thm:main-sat-descent-prism']} and Example \ref{['ex:log-smooth-descent']}
  • Theorem 1.6: Theorems \ref{['thm:BK-horizontal']} and \ref{['thm:BK-semistable']}
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • ...and 120 more