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

Abstract

In this paper, we study $p$-adic Hodge theory for non-commutative algebraic varieties. Firstly, we propose a conjecture that for $K$ a complete discretely valued nonarchimedean extension $K$ of ${\mathbb{Q}}_p$ with perfect residue field $k$ and $\mathcal{T}$ an ${\mathcal{O}}_K$-linear idempotent-complete, small smooth proper stable $\infty$-category, there exists a $B_{\mathrm{crys}}$-coefficient isomorphism preserving additional structures between the $K(1)$-local $K$-theory on the generic fiber of $\mathcal{T}$ and the topological periodic cyclic homology on the special fiber. This conjecture can be regarded as a non-commutative analog of the crystalline comparison theorem. We then proceed to prove the following results: the topological negative cyclic homology ${\pi }_i{\mathrm{TC}}^{-}(\mathcal{T}/\mathbb{S}[z];{\mathbb{Z}}_p)$ admits a Breuil–Kisin module structure, and the non-commutative analog of Bhatt–Morrow–Scholze’s comparison theorems holds. Additionally, we demonstrate that the ${\mathbb{Z}}_p[G_K]$-module obtained from the topological negative cyclic homology is a ${\mathbb{Z}}_p$-lattice of a crystalline representation. Finally, we show that when the generic fiber of $\mathcal{T}$ admits a geometric realization in the sense of Orlov, the non-commutative analog of the crystalline comparison theorem proposed by the author holds.