Syntomification and crystalline local systems

Technical Summary: Syntomification and Crystalline Local Systems

Problem and Context
The paper addresses the relationship between the geometry of $p$-adic formal schemes and the representation theory of $p$-adic Galois groups. Specifically, it seeks to generalize the Bhatt-Scholze equivalence between prismatic $F$-crystals and crystalline Galois representations (established for a point, i.e., $\text{Spf } \mathcal{O}_K$) to the setting of smooth $p$-adic formal schemes $X$. The central object of study is the syntomic stack $X_{\text{Syn}}$ associated to $X$, introduced by Bhatt and Lurie. The paper investigates the étale realization functor $\mathcal{T}_{\text{ét}}: \text{Perf}(X_{\text{Syn}}) \to D^{(b)}_{\text{lisse}}(X_\eta, \mathbb{Z}_p)$, which maps perfect complexes on the syntomic stack to derived categories of local systems on the rigid generic fiber $X_\eta$.

The primary challenges addressed are:

1. Characterizing the essential image of $\mathcal{T}_{\text{ét}}$ on the category of perfect complexes (and its isogeny category) in terms of crystalline local systems.
2. Determining whether the equivalence $\text{Perf}(\mathbb{Z}_p^{\text{Syn}})[1/p] \simeq D^b(\text{Rep}^{\text{cris}}_{\mathbb{Q}_p}(G_{\mathbb{Q}_p}))$ generalizes to higher-dimensional $X$.
3. Providing an explicit algebraic description of the category $\text{Perf}(X_{\text{Syn}})[1/p]$ in the proper case, analogous to the description of crystalline representations via weakly admissible filtered $F$-isocrystals.

Methodology
The author employs a combination of stacky geometry, descent theory, and $p$-adic Hodge theory techniques:

• Stacky Framework: The paper utilizes the formal stacks $X_{\text{Syn}}$, $X_{\text{Nyg}}$ (Nygaard), and $X_\Delta$ (prismatic). The structure of $X_{\text{Syn}}$ is analyzed via pushouts and coequalizers involving these stacks.
• Reflexive Sheaves: A key technical innovation is the introduction of reflexive sheaves ($\text{Refl}(X_{\text{Syn}})$) on the syntomic stack. The author defines these as coherent sheaves $E$ where the natural map $E \to E^{\vee\vee}$ is an isomorphism. This subcategory is shown to be the correct domain for the étale realization functor to induce an equivalence.
• Kernel Analysis: The paper analyzes the kernel of the étale realization functor. By adapting methods from Bhatt-Lurie and utilizing graded connections on Nygaard-associated graded objects, it is shown that the kernel consists of coherent $F$-gauges that are $(p, v_{1,X})$-power torsion.
• Descent and Covers: The proofs rely heavily on flat-local surjections (specifically using Rees stacks associated with Breuil-Kisin prisms) to reduce global questions on $X$ to local questions on affine schemes where explicit computations with Breuil-Kisin modules and $q$-connections are feasible.
• Beilinson Fiber Square: In the proper case, the paper constructs a Beilinson fiber square relating the syntomic cohomology of $X$ to its crystalline cohomology (over the special fiber) and de Rham cohomology (over the generic fiber). This square is used to define the target category of admissible filtered $F$-isocrystals.

Key Contributions and Results

1. Equivalence for Reflexive Sheaves (Theorem A):
   The paper establishes that for a smooth and quasicompact $p$-adic formal scheme $X/\text{Spf } \mathcal{O}_K$, the étale realization functor induces an equivalence of categories:
   $$\mathcal{T}_{\text{ét}}: \text{Refl}(X_{\text{Syn}}) \xrightarrow{\sim} \text{Loc}^{\text{cris}}_{\mathbb{Z}_p}(X_\eta)$$
   where $\text{Loc}^{\text{cris}}_{\mathbb{Z}_p}(X_\eta)$ is the category of pro-étale crystalline $\mathbb{Z}_p$-local systems. This result characterizes the essential image of the functor $\Pi_X$ (constructed by Guo-Li) as precisely the reflexive sheaves.

2. Isogeny Category and Crystalline Cohomology (Corollary 1.3):
   Upon inverting $p$, the functor induces an equivalence on the hearts of the respective t-structures:
   $$\text{Coh}(X_{\text{Syn}})[1/p] \xrightarrow{\sim} \text{Loc}^{\text{cris}}_{\mathbb{Q}_p}(X_\eta)$$
   The essential image of $\mathcal{T}_{\text{ét}}[1/p]$ on the full derived category is contained in the subcategory of objects with crystalline cohomology sheaves, though the paper notes that inclusions are generally strict (i.e., not every complex with crystalline cohomology sheaves is in the essential image).

3. Failure of Naive Generalization (Example 4.7):
   The paper demonstrates that the equivalence $\text{Perf}(X_{\text{Syn}})[1/p] \simeq D^b(\text{Loc}^{\text{cris}}_{\mathbb{Q}_p}(X_\eta))$ does not hold in general for higher-dimensional $X$ (e.g., $X = \mathbb{P}^1_{\mathbb{Z}_p}$). This is due to a failure of full faithfulness, evidenced by non-vanishing syntomic cohomology groups $H^2_{\text{Syn}}$ which do not correspond to extensions in the category of crystalline local systems.

4. Equivalence for Proper Schemes (Theorem B):
   For $X$ smooth and proper over $\text{Spf } \mathcal{O}_K$, the paper proves an equivalence:
   $$\text{Perf}(X_{\text{Syn}})[1/p] \xrightarrow{\sim} \text{Perf}^{\text{adm}}_{f\text{Isoc}_\varphi}(X)$$
   Here, $\text{Perf}^{\text{adm}}_{f\text{Isoc}_\varphi}(X)$ is the category of perfect complexes of admissible filtered $F$-isocrystals. This category is defined via a Beilinson fiber square involving the crystalline site of the special fiber and the de Rham stack of the generic fiber. The "admissible" condition ensures that the cohomology sheaves correspond to crystalline local systems.

5. Cohomological Computations:
   The paper provides a cohomological calculation showing that for $X = \text{Spf } \mathcal{O}_K$, the syntomic cohomology $H^i_{\text{Syn}}(\mathcal{O}_K, E)[1/p]$ vanishes for $i \geq 2$ and is isomorphic to the Bloch-Kato Selmer group (or extension groups in crystalline representations) for $i=1$. This vanishing is crucial for establishing the equivalence in the point case and the proper case.

Significance and Claims
The paper claims to resolve the question of whether the Bhatt-Lurie syntomic stack provides a derived Riemann-Hilbert correspondence for $p$-adic formal schemes.

• It confirms that while the naive derived equivalence fails for general $X$, a precise characterization exists via reflexive sheaves.
• It successfully generalizes the equivalence between syntomic perfect complexes and crystalline representations to the isogeny category for smooth proper schemes, but requires the target category to be refined to admissible filtered $F$-isocrystals rather than just the derived category of crystalline local systems.
• The work bridges the gap between the prismatic/stacky approach (Bhatt-Scholze, Bhatt-Lurie) and the classical theory of filtered $F$-isocrystals (Colmez-Fontaine), providing a unified framework for studying crystalline local systems on $p$-adic formal schemes.

The author explicitly notes that the results rely on $X$ being smooth and quasicompact (or proper) and that the interpretation of isogeny categories requires care when $X$ is not quasicompact. The paper does not claim to solve the problem for non-smooth schemes or to provide a description of the essential image without inverting $p$ (which is left as future work).