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

The aim of this paper is to study $p\text{-}$adic Hodge theory for non-commutative algebraic varieties. Specifically, we propose a conjecture and prove several results related to the relationship between $K(1)\text{-}$local $K\text{-}$theory and topological periodic homology in this non-commutative setting.

Cohomology theories such as de Rham cohomology $R{\mathrm{\Gamma }}_{\mathrm{dR}}(-/K)\text{,}$ Hodge cohomology $R{\mathrm{\Gamma }}_{\mathrm{Zar}}(-,{\mathrm{\Omega }}_{-/K}^{\ast })\text{,}$ $l\text{-}$adic cohomology $R{\mathrm{\Gamma }}_{\mathrm{ét}}(-,{\mathbb{Z}}_l)$ and crystalline cohomology $R{\mathrm{\Gamma }}_{\mathrm{cry}}(-/W(k))$ are important tools in the study of algebraic geometry and arithmetic geometry. On the other hand, homological invariants such as (topological) periodic homology, (topological) cyclic homology and $K\text{-}$theory play an important role in the study of $C^{\ast }\text{-}$algebra, Lie algebra and non-commutative geometry. There are deep and subtle links between cohomology invariants and homological invariants. One of the most well-known examples is the Atiyah–Hirzebruch spectral sequence. For a finite-dimensional CW-complex $M\text{,}$ Atiyah–Hirzebruch [4] prove that there is a spectral sequence

$E_2^{i,j}=\{\begin{array}{cc}{H}_{\mathrm{Sing}}^i(M,\mathbb{Z}),\hfill & j\text{\hspace{0.17em}\hspace{0.17em}even}\hfill \\ 0,\hfill & j\text{\hspace{0.17em}\hspace{0.17em}odd}\hfill \end{array}⟹K_{j-i}^{\mathrm{top}}(M).$

 

The Thomason spectral sequence is an arithmetic-geometrical analog of the Atiyah–Hirzebruch spectral sequence. For a smooth variety $X$ over a field of characteristic $0\text{,}$ Thomason [36] shows that there exists a spectral sequence

$E_2^{i,j}=\{\begin{array}{cc}{H}_{\mathrm{ét}}^i(X,{\mathbb{Z}}_p(l)),\hfill & j=2l\hfill \\ 0,\hfill & j\text{\hspace{0.17em}\hspace{0.17em}odd}\hfill \end{array}⟹{\pi }_{j-i}{L}_{K(1)}K(X),$

 

and this spectral sequence degenerates after tensoring ${\mathbb{Q}}_p\text{.}$ Besides, Hesselholt [21] shows a close relation between $p\text{-}$adic cohomology theory and topological cyclic homology.

Bondal–Kapranov [10], Orlov [34] and Kontsevich [27] introduce non-commutative algebraic geometry in which a dg-category (or a stable $\mathrm{\infty }\text{-}$category) is studied as a non-commutative space. Nowadays, non-commutative algebraic geometry plays an important role in research on mirror symmetry, mathematical physics and algebraic geometry. Besides, homological invariants are, in general, well defined for dg-categories and stable $\mathrm{\infty }\text{-}$categories. It has been known that some comparison theorems between cohomology groups can be formulated naturally for dg-categories and stable $\mathrm{\infty }\text{-}$categories: instead of cohomology theories, one can consider homological invariants. For a smooth proper dg-category $𝒯$ over $ℂ\text{,}$ Kaledin [25] proves that there is an isomorphism ${\mathrm{HP}}_n(𝒯/ℂ)\simeq {⨁}_{i\in \mathbb{Z}}{\mathrm{HH}}_{n+2i}(𝒯/ℂ)\text{,}$ which has been conjectured by Kontsevich–Soibelman [28] and can be regarded as a non-commutative Hodge decomposition via Connes [12], Feigin–Tsygan [18] and Hochschild–Kostant–Rosenberg [24]. Besides, Blanc [9] conjectured that there is an equivalence ${\mathrm{Ch}}^{\mathrm{top}}{\wedge }_{𝕊}Hℂ:K_{\mathrm{top}}(𝒯){\wedge }_{𝕊}Hℂ\to \mathrm{HP}(𝒯/ℂ)\text{,}$ which can be regarded as the non-commutative de Rham comparison theorem, and in some cases, the equivalence is proved by A. A. Khan [26]. For a smooth proper stable $\mathrm{\infty }\text{-}$category $𝒯$ over $W\text{,}$ Scholze proves that there is an isomorphism of $W\text{-}$modules ${\pi }_n\mathrm{TP}({𝒯}_k;{\mathbb{Z}}_p)\simeq {\pi }_i\mathrm{HP}(𝒯/W)$ (it has been unpublished yet), and Petrov–Vologodsky obtain the same result for any stable $\mathrm{\infty }\text{-}$category [35].

In the study of $p\text{-}$adic cohomology theories, crystalline comparison theory [16] states that for a smooth proper variety $X$ over ${𝒪}_K\text{,}$ the $p\text{-}$adic étale cohomology $H_{\mathrm{ét}}^i(X_{𝒞},{\mathbb{Q}}_p){\otimes }_{{\mathbb{Q}}_p}{B}_{\mathrm{crys}}$ is isomorphic to $H_{\mathrm{cry}}^i(X_k/W){\otimes }_W{B}_{\mathrm{crys}}\text{,}$ and the isomorphism is compatible with $G_K\text{-}$action, Frobenius endomorphism and filtration. We study a non-commutative version of the crystalline comparison theorem. For a commutative ring $R\text{,}$ we will refer to $R\text{-}$linear idempotent-complete, small stable $\mathrm{\infty }\text{-}$categories simply as $R\text{-}$linear categories. For an ${𝒪}_K\text{-}$linear category $𝒯\text{,}$ $G_K$ acts continuously on ${\mathbb{Z}}_p\text{-}$module ${\pi }_i{L}_{K(1)}K({𝒯}_{𝒞})\text{,}$ and there is a Frobenius operator $\mathrm{Fr}:{\pi }_i\mathrm{TP}({𝒯}_k;{\mathbb{Z}}_p)\left[\frac{1}{p}\right]\stackrel{can}{\simeq }{\pi }_i{\mathrm{TC}}^{-}({𝒯}_k;{\mathbb{Z}}_p)\left[\frac{1}{p}\right]\stackrel{𝜑}{\to }{\pi }_i\mathrm{TP}({𝒯}_k;{\mathbb{Z}}_p)\left[\frac{1}{p}\right]$ (see [2]). Inspired by Petrov and Vologodsky’s work on non-commutative crystalline cohomology theory [35], and the study of motivic filtration of the $K(1)\text{-}$local $K\text{-}$theory [36] and the topological periodic homology [8], we predict the following conjecture.

For a smooth proper variety $X$ over ${𝒪}_K\text{,}$ by [36] one has a $G_K\text{-}$equivariant isomorphism ${\pi }_i{L}_{K(1)}K({𝒯}_{𝒞}){\otimes }_{{\mathbb{Z}}_p}{\mathbb{Q}}_p\simeq {⨁}_{n\in \mathbb{Z}}{H}_{\mathrm{ét}}^{i+2n}(X_{𝒞},{\mathbb{Z}}_p(n)){\otimes }_{{\mathbb{Z}}_p}{\mathbb{Q}}_p\text{,}$ and similarly by [7] one has an isomorphism ${\pi }_i\mathrm{TP}({𝒯}_k;{\mathbb{Z}}_p)\left[\frac{1}{p}\right]\simeq {⨁}_{n\in \mathbb{Z}}{H}_{\mathrm{cry}}^{i+2n}\left(X_k/W\left[\frac{1}{p}\right]\right)(n)$ of isocrystals, thus the conjecture holds for $𝒯=\mathrm{perf}(X)$ via crystalline comparison theorem [37]. In this paper, we approach this conjecture via $K\text{-}$theoretical version of Bhatt–Morrow–Scholze’s comparison theorem [8]. We will use the language of stable $\mathrm{\infty }\text{-}$categories, following Lurie [31].

In many cases, a stable $\mathrm{\infty }\text{-}$category is known to admit a geometric realization. For example, the derived Fukaya category of a symplectic manifold is known or expected to admit a geometric realization from the study of mirror symmetry. Besides, Lunts–Bergh–Schnürer [5] proved that the stable infinity category of perfect complexes on a smooth proper Deligne–Mumford admits a geometric realization.

First, we fix a sequence of elements ${\zeta }_n\in \overline{K}$ inductively such that ${\zeta }_0=\mathrm{\hspace{0.17em}1}$ and ${({\zeta }_{n+1})}^p={\zeta }_n\text{.}$ Given this sequence, let $\epsilon =({\zeta }_0,{\zeta }_1,{\zeta }_2,\text{…})\in {\underleftarrow{\lim }}_{\mathrm{Frob}}{𝒪}_{ℂ}/p$ be the corresponding element in the tilt, and let $[\epsilon ]\in {𝔸}_{\inf }$ be its Teichmüller lift. Let $𝔖=W⟦z⟧\text{,}$ and let $\tilde{\theta }:𝔖\to {𝒪}_K$ be the usual map whose kernel is generated by Eisenstein polynomial $E$ of $\pi \text{.}$ Let $\varphi :𝔖\to A_{\inf }$ be the $W\text{-}$linear map that sends $z$ to $[{\pi }^{\mathrm{♭}}]\text{.}$ We write $\xi =\varphi (E)\text{.}$ We write $\mu =[\epsilon ]-1\text{.}$ Let $\phi :𝔖\to 𝔖$ be a Frobenius endomorphism which is Frobenius on $W$ and sends $z$ to $z^p\text{.}$ We prove a non-commutative version of Bhatt–Morrow–Scholze’s Breuil–Kisin cohomology theory $R{\mathrm{\Gamma }}_{𝔖}(-)$ [8].

Bhatt–Morrow–Scholze’s Breuil–Kisin cohomology theory [8, Theorem 1.2] implies the crystalline comparison theorem. On the other hand, Theorem 1.9 does not imply Conjecture 1.2. This difference arises as follows. On Breuil–Kisin cohomology theory, there is a $G_K\text{-}$equivariant isomorphism (see [7, Theorem 1.8 (iii)])

$R{\mathrm{\Gamma }}_{A_{\inf }}(X_{{𝒪}_{𝒞}}){\otimes }_{A_{\inf }}{A}_{\mathrm{cry}}\simeq R{\mathrm{\Gamma }}_{\mathrm{cry}}(X_{{𝒪}_{𝒞}/p}/A_{\mathrm{cry}}).$

 

By combining (1.3) with [7, Theorem 1.8 (iv)], we obtain a canonical $(G_K,\phi )\text{-}$equivariant isomorphism

$R{\mathrm{\Gamma }}_{\mathrm{ét}}(X_{𝒞},{\mathbb{Z}}_p){\otimes }_{{\mathbb{Z}}_p}{A}_{\mathrm{cry}}\left[\frac{1}{p\mu }\right]\simeq R{\mathrm{\Gamma }}_{\mathrm{cry}}(X_{{𝒪}_{𝒞}/p}/A_{\mathrm{cry}})\left[\frac{1}{p\mu }\right].$

 

The isomorphism induces the crystalline comparison theorem (see [7, Theorem 14.4]). On the other hand, the non-commutative analog of (1.3) becomes the following $G_K\text{-}$equivariant isomorphism:

${\pi }_i\mathrm{TP}({𝒯}_{{𝒪}_{𝒞}};{\mathbb{Z}}_p){\otimes }_{A_{\inf }}\widehat{A_{\mathrm{cry}}}\simeq {\pi }_i\mathrm{TP}({𝒯}_{{𝒪}_{𝒞}/p};{\mathbb{Z}}_p),$

 

where $\widehat{A_{\mathrm{cry}}}$ is the completion of $A_{\mathrm{cry}}$ with respect to the Nygaard filtration (see [8, Definition 8.9]). The problem is that $\mu$ is a $0\text{-}$divisor in $\widehat{A_{\mathrm{cry}}}$ (see [6, Corollaries 2.11 and 2.12]). Thus the non-commutative analog of (1.4) becomes the trivial equation. Therefore, we will study ${\pi }_i{\mathrm{TC}}^{-}(𝒯/𝕊[z];{\mathbb{Z}}_p)$ in more detail. In Section 3, we will show that the dual Breuil–Kisin module ${\pi }_i{\mathrm{TC}}^{-}{(𝒯/𝕊[z];{\mathbb{Z}}_p)}^{\vee }$ admits a Breuil–Kisin $G_K\text{-}$module structure in the sense of Gao [19]. In Section 4, using Du–Liu’s work on $(\phi ,\widehat{G})\text{-}$module [14], we will prove the following.

We prove the following as a corollary.