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

Keiho Matsumoto

Paper

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

Abstract

Let

be an

-linear idempotent-complete, small smooth proper stable

-category, where

is a finite extension of

. We give a Breuil-Kisin module structure on the topological negative cyclic homology

, and prove a

-theory version of Bhatt-Morrow-Scholze's comparison theorems. Moreover, using Gao's Breuil-Kisin

-module theory and Du-Liu's

-module theory, we prove the

-module

is a

-lattice of a crystalline representation. As a corollary, if the generic fibre of

admits a geometric realization in the sense of Orlov, we prove a comparison theorem between

-local

theory of the generic fibre and topological cyclic periodic homology theory of the special fibre with

-coefficients, in particular, we prove the

-adic representation of the

-local

-theory of the generic fibre is a crystalline representation, this can be regarded as a non-commutative analogue of

-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.