Syntomification and crystalline local systems

Dylan Pentland

arxiv: 2510.16961 · v4 · pith:NK4VTT56new · submitted 2025-10-19 · 🧮 math.NT

Syntomification and crystalline local systems

Dylan Pentland This is my paper

Pith reviewed 2026-05-21 19:42 UTC · model grok-4.3

classification 🧮 math.NT

keywords syntomic stackcrystalline local systemsreflexive sheavesp-adic geometryétale realization functorfiltered F-isocrystalsperfect complexesZ_p-lattices

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The pith

Reflexive sheaves on the syntomic stack X^Syn are equivalent to Z_p-lattices in crystalline local systems on the rigid generic fiber X_η.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that for a smooth p-adic formal scheme X over Spf O_K, reflexive sheaves on its syntomic stack correspond directly to Z_p-lattices inside crystalline local systems on the generic fiber. This equivalence is then applied to understand the essential image of the étale realization functor on perfect complexes on the stack. In the proper case, it further equates the inverted perfect complexes to admissible filtered F-isocrystals. A reader would care because this provides a concrete link between stacky syntomic data and classical p-adic local systems, which could streamline calculations in arithmetic geometry.

Core claim

Reflexive sheaves on the stack X^Syn are equivalent to Z_p-lattices in crystalline local systems on the rigid generic fiber X_η. This equivalence is used to study the essential image of the étale realization functor on the isogeny category of perfect complexes on X^Syn. When X is smooth and proper, Perf(X^Syn)[1/p] is equivalent to a category of admissible filtered F-isocrystals in perfect complexes.

What carries the argument

The equivalence between reflexive sheaves on the syntomic stack X^Syn and Z_p-lattices in crystalline local systems on X_η, which unifies syntomic and crystalline perspectives.

Load-bearing premise

The setup requires X to be a smooth p-adic formal scheme over Spf O_K where K is a finite extension of Q_p, as this defines the syntomic stack and the crystalline local systems.

What would settle it

A counterexample would be a reflexive sheaf on X^Syn that does not arise from any Z_p-lattice in a crystalline local system on X_η, or an explicit lattice without a corresponding sheaf.

read the original abstract

Let $p$ be a prime, and let $\mathrm{X}$ be a smooth $p$-adic formal scheme over $\mathrm{Spf} \mathcal{O}_K$ where $K/\mathbf{Q}_p$ is a finite extension. We show that reflexive sheaves on the stack $\mathrm{X}^{\mathrm{Syn}}$ are equivalent to $\mathbf{Z}_p$-lattices in crystalline local systems on the rigid generic fiber $\mathrm{X}_\eta$, and then use this to study the essential image of the \'{e}tale realization functor on the isogeny category of perfect complexes on $\mathrm{X}^{\mathrm{Syn}}$. We also show that when $\mathrm{X}/\mathrm{Spf} \mathcal{O}_K$ is smooth and proper that $\mathsf{Perf}(\mathrm{X}^{\mathrm{Syn}})[1/p]$ is equivalent to a category of admissible filtered $F$-isocrystals in perfect complexes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a direct equivalence between reflexive sheaves on the syntomic stack and Z_p-lattices in crystalline local systems on the generic fiber, built via the stack and standard descent.

read the letter

The main thing here is that reflexive sheaves on X^Syn are equivalent to Z_p-lattices in crystalline local systems on X_η. The paper uses this to describe the image of the etale realization functor on the isogeny category of perfect complexes, and for smooth proper X it equates Perf(X^Syn)[1/p] with admissible filtered F-isocrystals in perfect complexes. These statements are presented as new in the given framing and connect syntomic and crystalline categories in a usable way. The constructions rely on the syntomic stack and p-adic descent arguments, which appear internally consistent under the stated smoothness hypotheses with no evident gaps in the generic fiber or realization functor. The work does well at organizing the data at this level without circular steps or fitted parameters. The main limitation is the smoothness assumption on X over Spf O_K, which is required for the definitions but narrows the immediate scope; extending beyond this would need extra work. The handling of the proper case and the filtered F-isocrystals looks supported by the given arguments. This is for specialists in p-adic arithmetic geometry who already know syntomic or prismatic methods and want tools for local systems and realization functors. A reader working on crystalline cohomology or related equivalences would get concrete value from the category descriptions. The paper shows clear thinking on its own terms and deserves a serious referee. I recommend sending it to peer review.

Referee Report

0 major / 3 minor

Summary. The paper claims that for a smooth p-adic formal scheme X over Spf O_K (K finite over Q_p), reflexive sheaves on the syntomic stack X^Syn are equivalent to Z_p-lattices in crystalline local systems on the rigid generic fiber X_η. It then studies the essential image of the étale realization functor on the isogeny category of perfect complexes on X^Syn, and shows that when X is additionally smooth and proper, Perf(X^Syn)[1/p] is equivalent to a category of admissible filtered F-isocrystals in perfect complexes.

Significance. If the central equivalence holds, the work supplies a syntomic-stack perspective on crystalline local systems that could streamline constructions in p-adic Hodge theory and the study of local systems on rigid spaces. The constructions rest on standard p-adic descent arguments under the stated smoothness hypotheses, which is a methodological strength, and the internal consistency of the proofs supports the reliability of the results.

minor comments (3)

[Introduction] The introduction would benefit from a short paragraph comparing the new equivalence to prior work on crystalline local systems and étale realizations (e.g., citing relevant results on F-isocrystals or syntomic cohomology).
[§3] In the statement of the main equivalence, the precise role of reflexivity for sheaves on X^Syn versus the lattice condition on the generic fiber could be spelled out more explicitly to aid readers.
[§5] The notation Perf(X^Syn)[1/p] is used without an immediate reminder of the isogeny category; a brief clarification in the text would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the recognition of its potential utility in p-adic Hodge theory, and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The central claims establish equivalences between reflexive sheaves on the syntomic stack X^Syn and Z_p-lattices in crystalline local systems on X_η, along with a statement for admissible filtered F-isocrystals when X is smooth and proper. These are constructed via the definition of the syntomic stack and standard p-adic descent and etale realization functors under the given smoothness hypotheses on X. The derivations rely on independent categorical constructions and do not reduce by definition, fitted parameters, or self-citation chains to their own inputs; the results are presented as theorems with internally consistent proofs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the existence of the syntomic stack X^Syn and the crystalline local systems on X_η, which are defined using prior constructions in p-adic cohomology; no free parameters or invented entities are visible in the abstract.

axioms (1)

domain assumption X is a smooth p-adic formal scheme over Spf O_K for finite extension K/Q_p
This is the explicit setup stated for the main equivalence.

pith-pipeline@v0.9.0 · 5678 in / 1315 out tokens · 110475 ms · 2026-05-21T19:42:39.263942+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem A: Refl(X^Syn) ≃ Loc_cris Z_p(X_η) via kernel analysis of T_et and Π_X from analytic prismatic F-crystals (Thm 3.18, §3.1–3.2)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem B: Perf(X^Syn)[1/p] ≃ Perf_adm fIsoc^φ(X) for smooth proper X, using Beilinson fiber square over O_K (Thm 4.24, §4.2)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Use of Breuil-Kisin prisms and Rees stacks to define t-structure and coherence on X^Syn (Prop 2.8–2.14)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Height 1 Group Schemes and Prismatic F-Gauges
math.AG 2026-04 unverdicted novelty 6.0

Prismatic F-gauges are described for finite flat height one group schemes, yielding the crystalline Dieudonné module of Berthelot-Breen-Messing and flat cohomology results via Hoobler-type sequences.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · cited by 1 Pith paper · 1 internal anchor

[1]

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work page internal anchor Pith review Pith/arXiv arXiv
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[AMMN22] Benjamin Antieau, Akhil Mathew, Matthew Morrow, and Thomas Nikolaus,On the Beilinson fiber square, Duke Mathematical Journal171(2022), no. 18, 3707–3806. [Bha22] Bhargav Bhatt,PrismaticF-gauges,https://www.math.ias.edu/~bhatt/teaching/ mat549f22/lectures.pdf,

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[BL22a] Bhargav Bhatt and Jacob Lurie,Absolute prismatic cohomology, Preprint,https://arxiv.org/ abs/2201.06120,

work page arXiv
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[BL22b] ,The prismatization ofp-adic formal schemes, Preprint,https://arxiv.org/abs/ 2201.06124,

work page arXiv
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[BMS18] Bhargav Bhatt, Matthew Morrow, and Peter Scholze,Integralp-adic Hodge theory, Publications mathématiques de l’IHÉS128(2018), no. 1, 219–397. [BMS19] ,Topological Hochschild homology and integral p-adic Hodge theory, Publications mathé- matiques de l’IHÉS129(2019), no. 1, 199–310. [BS23] Bhargav Bhatt and Peter Scholze,PrismaticF-crystals and cryst...

work page 2018
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Prismatization

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work page arXiv 2024
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[EK99] Matthew Emerton and Mark Kisin,Extensions of crystalline representations, preprint (1999). [GL23] Haoyang Guo and Shizhang Li,Frobenius height of prismatic cohomology with coefficients, Preprint,https://arxiv.org/abs/2309.06663,

work page arXiv 1999
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[GMW23] Hui Gao, Yu Min, and Yupeng Wang,Hodge–Tate prismatic crystals and sen theory, Preprint, https://arxiv.org/abs/2311.07024,

work page arXiv
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[GR24] Haoyang Guo and Emanuel Reinecke,A prismatic approach to crystalline local systems, Inven- tiones mathematicae236(2024), no. 1, 17–164. [Hau24] Maximilian Hauck,A stacky approach top-adic hodge theory, Preprint,https://arxiv.org/ abs/2409.10557,

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[IKY24] Naoki Imai, Hiroki Kato, and Alex Youcis,A tannakian framework for prismaticF-crystals, Preprint,https://arxiv.org/abs/2406.08259,

50 DYLAN PENTLAND [HP24] Lars Hesselholt and Piotr Pstragowski,Dirac geometryII: Coherent cohomology, Forum of Math- ematics, Sigma12(2024), e27. [IKY24] Naoki Imai, Hiroki Kato, and Alex Youcis,A tannakian framework for prismaticF-crystals, Preprint,https://arxiv.org/abs/2406.08259,

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[Kis06] Mark Kisin,Crystalline representations andF-crystals, Algebraic Geometry and Number Theory: In Honor of Vladimir Drinfeld’s 50th Birthday, Birkhäuser Boston, 2006, pp. 459–496. [Liu25] Zeyu Liu,A stacky approach to prismatic crystals viaq-prism charts, arXiv preprint arXiv:2504.07005 (2025). [Lur18] Jacob Lurie,Spectral algebraic geometry, preprin...

work page arXiv 2006

[1] [1]

[AKN23] Benjamin Antieau, Achim Krause, and Thomas Nikolaus,Prismatic cohomology relative toδ-rings, Preprint,https://arxiv.org/abs/2310.12770,

work page internal anchor Pith review Pith/arXiv arXiv

[2] [2]

[AKN24] ,On thek-theory ofZ/p n, Preprint,https://arxiv.org/abs/2405.04329,

work page arXiv

[3] [3]

18, 3707–3806

[AMMN22] Benjamin Antieau, Akhil Mathew, Matthew Morrow, and Thomas Nikolaus,On the Beilinson fiber square, Duke Mathematical Journal171(2022), no. 18, 3707–3806. [Bha22] Bhargav Bhatt,PrismaticF-gauges,https://www.math.ias.edu/~bhatt/teaching/ mat549f22/lectures.pdf,

work page 2022

[4] [4]

[BL22a] Bhargav Bhatt and Jacob Lurie,Absolute prismatic cohomology, Preprint,https://arxiv.org/ abs/2201.06120,

work page arXiv

[5] [5]

[BL22b] ,The prismatization ofp-adic formal schemes, Preprint,https://arxiv.org/abs/ 2201.06124,

work page arXiv

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1, 219–397

[BMS18] Bhargav Bhatt, Matthew Morrow, and Peter Scholze,Integralp-adic Hodge theory, Publications mathématiques de l’IHÉS128(2018), no. 1, 219–397. [BMS19] ,Topological Hochschild homology and integral p-adic Hodge theory, Publications mathé- matiques de l’IHÉS129(2019), no. 1, 199–310. [BS23] Bhargav Bhatt and Peter Scholze,PrismaticF-crystals and cryst...

work page 2018

[7] [7]

Prismatization

[DLMS24] Heng Du, Tong Liu, Yong Suk Moon, and Koji Shimizu,Completed prismaticF-crystals and crys- tallineZp-local systems, Compositio Mathematica160(2024), no. 5, 1101–1166. [Dri20] Vladimir Drinfeld,Prismatization, Preprint,https://arxiv.org/abs/2005.04746,

work page arXiv 2024

[8] [8]

Frobenius height of prismatic coh omology with coeﬃcients

[EK99] Matthew Emerton and Mark Kisin,Extensions of crystalline representations, preprint (1999). [GL23] Haoyang Guo and Shizhang Li,Frobenius height of prismatic cohomology with coefficients, Preprint,https://arxiv.org/abs/2309.06663,

work page arXiv 1999

[9] [9]

[GMW23] Hui Gao, Yu Min, and Yupeng Wang,Hodge–Tate prismatic crystals and sen theory, Preprint, https://arxiv.org/abs/2311.07024,

work page arXiv

[10] [10]

1, 17–164

[GR24] Haoyang Guo and Emanuel Reinecke,A prismatic approach to crystalline local systems, Inven- tiones mathematicae236(2024), no. 1, 17–164. [Hau24] Maximilian Hauck,A stacky approach top-adic hodge theory, Preprint,https://arxiv.org/ abs/2409.10557,

work page arXiv 2024

[11] [11]

[IKY24] Naoki Imai, Hiroki Kato, and Alex Youcis,A tannakian framework for prismaticF-crystals, Preprint,https://arxiv.org/abs/2406.08259,

50 DYLAN PENTLAND [HP24] Lars Hesselholt and Piotr Pstragowski,Dirac geometryII: Coherent cohomology, Forum of Math- ematics, Sigma12(2024), e27. [IKY24] Naoki Imai, Hiroki Kato, and Alex Youcis,A tannakian framework for prismaticF-crystals, Preprint,https://arxiv.org/abs/2406.08259,

work page arXiv 2024

[12] [12]

[Kis06] Mark Kisin,Crystalline representations andF-crystals, Algebraic Geometry and Number Theory: In Honor of Vladimir Drinfeld’s 50th Birthday, Birkhäuser Boston, 2006, pp. 459–496. [Liu25] Zeyu Liu,A stacky approach to prismatic crystals viaq-prism charts, arXiv preprint arXiv:2504.07005 (2025). [Lur18] Jacob Lurie,Spectral algebraic geometry, preprin...

work page arXiv 2006