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

We construct a K-theory version of Bhatt-Morrow-Scholze’s Breuil-Kisin cohomology theory for ${\text{sO}}_K$-linear idempotent-complete, small smooth proper stable infinity-categories, where $K$ is a discretely valued extension of ${\text{Q}}_p$ with perfect residue field. As a corollary, under the assumption that $K(1)$-local K theory satisfies the Künneth formula for ${\text{sO}}_K$-linear idempotent-complete, small smooth proper stable $\mathrm{\infty }$-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 $\text{Bcry}$-coefficients, and $p$-adic Galois representations of $K(1)$-local K theory for ${\text{sO}}_K$-linear idempotent-complete, small smooth proper stable $\mathrm{\infty }$-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.