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Towards the $p$-adic Hodge theory for non-commutative algebraic varieties

Keiho Matsumoto

Abstract

We construct a K-theory version of Bhatt-Morrow-Scholze's Breuil-Kisin cohomology theory for $\sO_K$-linear idempotent-complete, small smooth proper stable infinity-categories, where $K$ is a discretely valued extension of $\Q_p$ with perfect residue field. As a corollary, under the assumption that $K(1)$-local K theory satisfies the Künneth formula for $\sO_K$-linear idempotent-complete, small smooth proper stable $\infty$-categories, we prove a comparison theorem between $K(1)$-local K theory of the generic fiber and topological cyclic periodic homology theory of the special fiber with $\Bcry$-coefficients, and $p$-adic Galois representations of $K(1)$-local K theory for $\sO_K$-linear idempotent-complete, small smooth proper stable $\infty$-categories are semi-stable. We also provide an alternative K-theoretical proof of the semi-stability of p-adic Galois representations of the p-adic étale cohomology group of smooth proper varieties over $K$ with good reduction. is a short, This is a short preliminary version of the work that was later expanded in 2309.13654.

Towards the $p$-adic Hodge theory for non-commutative algebraic varieties

Abstract

We construct a K-theory version of Bhatt-Morrow-Scholze's Breuil-Kisin cohomology theory for -linear idempotent-complete, small smooth proper stable infinity-categories, where is a discretely valued extension of with perfect residue field. As a corollary, under the assumption that -local K theory satisfies the Künneth formula for -linear idempotent-complete, small smooth proper stable -categories, we prove a comparison theorem between -local K theory of the generic fiber and topological cyclic periodic homology theory of the special fiber with -coefficients, and -adic Galois representations of -local K theory for -linear idempotent-complete, small smooth proper stable -categories are semi-stable. We also provide an alternative K-theoretical proof of the semi-stability of p-adic Galois representations of the p-adic étale cohomology group of smooth proper varieties over with good reduction. is a short, This is a short preliminary version of the work that was later expanded in 2309.13654.
Paper Structure (12 sections, 28 theorems, 91 equations)

This paper contains 12 sections, 28 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Non-commutative version of Breuil-Kisin cohomology
  4. Breuil-Kisin modules and Breuil-Kisin cohomology theory $R\Gamma_{\mathfrak{S}}$
  5. Perfect modules and Künneth formula
  6. Breuil-Kisin module of stable $\infty$-categories
  7. Comparison between $\operatorname{TC}^{-}(\mathcal{T}/\mathbb{S}[z];\mathbb{Z}_p)$ and $\operatorname{TP}(\mathcal{T}_{\mathcal{O}_\mathcal{C}};\mathbb{Z}_p)$
  8. Comparison between $\operatorname{TC}^{-}(\mathcal{T}/\mathbb{S}[z];\mathbb{Z}_p)$ and $\operatorname{TP}(\mathcal{T}_{k};\mathbb{Z}_p)$
  9. Comparison theorem between $\operatorname{TC}^{-}(\mathcal{T}/\mathbb{S}[z];\mathbb{Z}_p)$ and ${L_{K(1)}} K(\mathcal{T}_{\mathcal{C}})$
  10. Non-commutative version of Bhatt-Morrow-Scholze's comparison theorem
  11. Galois representation of K(1)-local K theory
  12. Breuil-Kisin $G_K$-module

Key Result

Lemma 1.2

We assume $X_\mathcal{C}$ is smooth over $\mathcal{C}$. $H^i_{{\operatorname{\acute{e}t}}}(X_\mathcal{C},\mathbb{Q}_p)$ is crystalline (resp. semi-stable) for any $i$ if and only if $\pi_j{L_{K(1)}} K(X_\mathcal{C})$ is crystalline (resp. semi-stable) for $j=0,1$.

Theorems & Definitions (60)

  • Lemma 1.2
  • proof
  • Conjecture 1.3: The lattice conjecture, Blanc
  • Conjecture 1.4: Non-commutative version of crystalline comparison theorem Falcrys
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Remark 1.8
  • Conjecture 1.9
  • Proposition 1.10
  • ...and 50 more