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

Keiho Matsumoto
Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG), K-Theory and Homology (math.KT)
2023-09-23 16:00:00
Let $\sT$ be an $\sO_K$-linear idempotent-complete, small smooth proper stable $\infty$-category, where $K$ is a finite extension of $\Q_p$. We give a Breuil-Kisin module structure on the topological negative cyclic homology $\pi_i\TCn(\sT/\bS[z];\Z_p)$, 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 $\Z_p[G_K]$-module $T_{\Ainf}(\pi_i\TCn(\sT/\bS[z];\Z_p)^{\vee})$ is a $\Z_p$-lattice of a crystalline representation. As a corollary, if the generic fibre of $\sT$ 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 $\Bcry$-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.
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