Crystalline representations and $p$-adic Hodge theory for non-commutative algebraic varieties
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866917138177982464 |
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| author | Matsumoto, Keiho |
| author_facet | Matsumoto, Keiho |
| contents | Let $\mathcal{T}$ be an $\mathcal{O}_K$-linear idempotent-complete, small smooth proper stable $\infty$-category, where $K$ is a finite extension of $\mathbb{Q}_p$. We give a Breuil-Kisin module structure on the topological negative cyclic homology $π_i{\rm TC}^-(\mathcal{T}/\mathbb{S}[z];\mathbb{Z}_p)$, 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 $(φ,\hat{G})$-module theory, we prove the $\mathbb{Z}_p[G_K]$-module $T_{A_{\rm inf}}(π_i{\rm TC}^-(\mathcal{T}/\mathbb{S}[z];\mathbb{Z}_p)^{\vee})$ is a $\mathbb{Z}_p$-lattice of a crystalline representation. As a corollary, if the generic fibre of $\mathcal{T}$ 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 $B_{\rm crys}$-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. This is the full version of arXiv:2305.00292, containing additional details and results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_13654 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Crystalline representations and $p$-adic Hodge theory for non-commutative algebraic varieties Matsumoto, Keiho Algebraic Geometry K-Theory and Homology Let $\mathcal{T}$ be an $\mathcal{O}_K$-linear idempotent-complete, small smooth proper stable $\infty$-category, where $K$ is a finite extension of $\mathbb{Q}_p$. We give a Breuil-Kisin module structure on the topological negative cyclic homology $π_i{\rm TC}^-(\mathcal{T}/\mathbb{S}[z];\mathbb{Z}_p)$, 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 $(φ,\hat{G})$-module theory, we prove the $\mathbb{Z}_p[G_K]$-module $T_{A_{\rm inf}}(π_i{\rm TC}^-(\mathcal{T}/\mathbb{S}[z];\mathbb{Z}_p)^{\vee})$ is a $\mathbb{Z}_p$-lattice of a crystalline representation. As a corollary, if the generic fibre of $\mathcal{T}$ 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 $B_{\rm crys}$-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. This is the full version of arXiv:2305.00292, containing additional details and results. |
| title | Crystalline representations and $p$-adic Hodge theory for non-commutative algebraic varieties |
| topic | Algebraic Geometry K-Theory and Homology |
| url | https://arxiv.org/abs/2309.13654 |