Crystalline representations and $p$-adic Hodge theory for non-commutative algebraic varieties

TL;DR

This paper develops a non-commutative $p$-adic Hodge theory framework for smooth proper stable $"infinity$-categories, establishing crystalline Galois representations and comparison theorems linking $K$-theory and topological cyclic homology.

Contribution

It introduces a novel approach to $p$-adic Hodge theory for non-commutative algebraic varieties using topological cyclic homology and Breuil-Kisin modules, extending classical results.

Findings

Proves the $T_{A_{inf}}$ module is a crystalline Galois representation.

Establishes a comparison theorem between $K(1)$-local $K$-theory and topological cyclic homology.

Shows the $p$-adic $K$-theory of the generic fiber is a crystalline representation.

Abstract

Let $\mathcal{T}$ be an $\mathcal{O}_K$-linear idempotent-complete, small smooth proper stable $\infty$-category, where $K$ is a finite extension of $\mathbb{Q}_p$. We give a Breuil-Kisin module structure on the topological negative cyclic homology $\pi_i{\rm TC}^-(\mathcal{T}/\mathbb{S}[z];\mathbb{Z}_p)$, and prove a $K$-theory version of Bhatt-Morrow-Scholze's comparison theorems. Moreover, using Gao's Breuil-Kisin $G_K$-module theory and Du-Liu's $(\varphi,\hat{G})$-module theory, we prove the $\mathbb{Z}_p[G_K]$-module $T_{A_{\rm inf}}(\pi_i{\rm TC}^-(\mathcal{T}/\mathbb{S}[z];\mathbb{Z}_p)^{\vee})$ is a $\mathbb{Z}_p$-lattice of a crystalline representation. As a corollary, if the generic fibre of $\mathcal{T}$ admits a geometric realization in the sense of Orlov, we prove a comparison theorem between $K(1)$-local $K$ theory of the generic fibre and topological cyclic periodic homology theory of the special fibre with $B_{\rm crys}$-coefficients, in particular, we prove the $p$-adic representation of the $K(1)$-local $K$-theory of the generic fibre is a crystalline representation, this can be regarded as a non-commutative analogue of $p$-adic Hodge theory for smooth proper varieties proved by Tsuji and Faltings. This is the full version of arXiv:2305.00292, containing additional details and results.