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

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.