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An integral analogue of Fontaine's crystalline functor

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Pith's one-line read Integral p-adic Hodge theories unify in the Fontaine–Laffaille range

desk verdict A substantial and well-written construction of an integral D_crys that completes the comparison of crystalline, prismatic, and syntomic approaches, but with a load-bearing step in Proposition 3.3 that imports a lemma from an unreviewed preprint as a black box. read the letter →

arxiv 2504.16282 v2 pith:3J5J2YNM submitted 2025-04-22 math.NT math.AG

Naoki Imai , Hiroki Kato , Alex Youcis This is my paper · ORCID
classification math.NTmath.AG MSC 14F3014L1511F80
keywords prismaticF-crystalsFontaine–Laffaillemodulesfilteredcrystalline-deRhamcomparisonNygaardfiltrationp-divisiblegroupsDieudonnétheoryintegralp-adicHodge
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper constructs a functor $D_{\mathrm{crys}}$ from prismatic F-crystals and prismatic F-gauges on a smooth formal scheme over the Witt vectors into filtered F-crystals, and proves it behaves like an integral version of Fontaine's crystalline functor: after inverting $p$ it matches the rational functor via the étale realization. Its main theorem, for $p>2$, is a 2-commutative diagram in which prismatic F-crystals, prismatic F-gauges, crystalline local systems, and Fontaine–Laffaille modules with Hodge–Tate weights in $[0,p-2]$ are all $\mathbb{Z}_p$-linearly equivalent through $D_{\mathrm{crys}}$, the forgetful functor, and the two realization functors. It also shows that the prismatic Dieudonné crystal of a $p$-divisible group recovers the filtered crystalline Dieudonné crystal, and that the Breuil–Kisin–Kim Dieudonné functor is evaluation of the prismatic Dieudonné crystal at the Breuil–Kisin prism. The argument rests on a canonical comparison between crystalline and de Rham realizations, and on a reduction to power-series rings where the lowness of the Hodge–Tate weights forces the relevant filtered pieces to be free.

What carries the argument

The load-bearing object is the functor $D_{\mathrm{crys}}$, defined as follows: take a prismatic F-crystal $(E,\varphi_E)$, form its crystalline realization $E_{\mathrm{crys}}$ and its de Rham realization $E_{\mathrm{dR}}$, and use the canonical identification $\iota_X$ between them to transfer the Nygaard filtration of $\varphi^*E$ onto $E_{\mathrm{crys}}$; the resulting filtered F-crystal is $D_{\mathrm{crys}}(E,\varphi_E)$. Local filtered freeness of the Nygaard filtration is the condition that makes $D_{\mathrm{crys}}$ land in strongly divisible filtered F-crystals and lets it detect lffness and control exactness. In the proof of the Fontaine–Laffaille equivalence, the argument reduces to power-series rings and uses a lemma asserting that the relative Frobenius map on the associated Breuil–Kisin module has finite free cokernel, together with a finiteness result for the graded pieces over $W$.

What would settle it

Look for a prismatic F-crystal of height at most $p-2$ over $R=W[[t_1,\dots,t_d]]$ for which some graded piece $\mathrm{Gr}^i(\mathrm{Fil}^\bullet)$ of the Nygaard filtration on $\varphi^*E/(E)$ is not finite free; Proposition 3.3 asserts none exists, so one such example would falsify the main equivalence. Concretely, compute the cokernel of the relative Frobenius map $\varphi_M:\varphi^*M\to M/E^i$ for an explicit Breuil–Kisin module and check whether the quoted Lemma 52 of the cited preprint holds there.

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Extended reading notes

Core claim

The central claim is that the crystalline, prismatic, and syntomic routes to integral $p$-adic Hodge theory carry equivalent information inside the Fontaine–Laffaille range. For a smooth formal $W$-scheme $X$ and $p>2$, the paper defines $D_{\mathrm{crys}}(E,\varphi_E)$ using the crystalline realization $E_{\mathrm{crys}}$ and the de Rham realization $E_{\mathrm{dR}}$, joined by a canonical isomorphism $\iota_X$ (Theorem 1.19); the filtration is pulled back from the Nygaard filtration on $\varphi^*E$. Theorem 3.1 then states that the diagram linking locally filtered free prismatic F-crystals, prismatic F-gauges, all prismatic F-crystals in the range, crystalline local systems, and strongly divisible filtered F-crystals commutes and every arrow is a $\mathbb{Z}_p$-linear equivalence. Theorem 4.8 upgrades the earlier prismatic Dieudonné comparison to a filtered one: $D_{\mathrm{crys}}\circ M_\Delta$ is naturally the filtered crystalline Dieudonné functor $D$, and Proposition 4.16 identifies the Breuil–Kisin–Kim functor with evaluation of $M_\Delta$ on the Breuil–Kisin prism.

Load-bearing premise

The proof of the key Proposition 3.3 assumes on the authority of an unreviewed preprint that a relative Frobenius map on the relevant Breuil–Kisin module has finite free cokernel, so the Fontaine–Laffaille-range equivalence would collapse if that lemma were false.

Editorial extensions

If this is right

  • In Hodge–Tate weights $[0,p-2]$, the prismatic, syntomic, and crystalline categories are equivalent, so a crystalline local system admits matching integral models and one can pass freely between the three frameworks.
  • $D_{\mathrm{crys}}$ gives a canonical filtered lattice inside the rational crystalline functor, so the filtration recorded by Grothendieck–Messing theory is determined by the prismatic Dieudonné crystal, not just its underlying F-crystal.
  • The forgetful functor from prismatic F-gauges to locally filtered free prismatic F-crystals is bi-exact, which controls extension groups in both categories.
  • For smooth proper $X$, the comparison yields an isomorphism between étale cohomology of $T_{\acute{e}t}(V)_{\mathrm{alg}}$ modulo $p^n$ and $T_{\acute{e}t}(R^b f_* V/p^n)$ whenever $a+b<p-2$.
  • For every formally framed $W$-algebra $R$, the categories $Vect_{[0,1]}(R_{\mathrm{syn}})$, $Vect^\varphi_{[0,1]}(R_\Delta)$, and $Vect^\varphi_{[0,1]}(S_R,\nabla^0)$ are equivalent, identifying prismatic Dieudonné theory with Breuil–Kisin–Kim Dieudonné theory.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper, the same comparison should make it possible to transport quantitative Fontaine–Laffaille estimates, such as bounds on extension groups or filtration jumps, into the prismatic category, where such bounds are harder to see; one test would be to reprove the classical strong-divisibility results directly through $D_{\mathrm{crys}}$.
  • Beyond the paper, the construction suggests a derived or $\infty$-categorical analogue of $D_{\mathrm{crys}}$ for perfect complexes on the prismatic site, since the crystalline–de Rham comparison is naturally formulated with perfect complexes and the authors note the isomorphism should persist after inverting $p$.
  • Beyond the paper, the explicit comparison on Breuil–Kisin prisms provides a coordinate formula for $D_{\mathrm{crys}}$, so a reader could use it as a computational tool to compute filtrations attached to explicit prismatic F-crystals and $p$-adic representations.
  • Beyond the paper, bi-exactness of the forgetful functor indicates that the prismatic and syntomic sites compute the same extension groups; a concrete check is to compare $\mathrm{Ext}^1$ in both categories for small $p$-divisible groups.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper constructs an integral analogue D_crys of Fontaine's crystalline functor for prismatic F-crystals and prismatic F-gauges on a smooth formal scheme over W(k), landing in categories of filtered F-crystals. It proves a crystalline--de Rham comparison for prismatic F-crystals (Theorem 1.19), compares D_crys with the rational crystalline functor (Theorem 2.10), characterizes local filtered freeness through D_crys (Proposition 2.13), and proves bi-exactness of the forgetful functor from prismatic F-gauges to lff prismatic F-crystals (Proposition 2.17). The central result is Theorem 3.1, which asserts that in the Fontaine--Laffaille range [0,p-2] the diagram relating prismatic F-crystals, prismatic F-gauges, local systems, and filtered F-crystals is 2-commutative and every arrow is a Z_p-linear equivalence. The paper also applies D_crys to prismatic Dieudonné theory, proving a filtered identification D ≃ D_crys ∘ M_Δ and relating the prismatic Dieudonné functor to Breuil--Kisin--Kim Dieudonné functors.

Significance. If the main theorem holds, the paper gives a genuinely useful bridge among the crystalline, prismatic, and syntomic approaches in the Fontaine--Laffaille range, and it substantially clarifies the integral relation between prismatic Dieudonné theory and classical crystalline Dieudonné theory. The paper is carefully organized and contains substantial proofs from definitions, especially for the crystalline--de Rham comparison (Theorem 1.19), the comparison with rational D_crys (Theorem 2.10), the lff characterization (Proposition 2.13), and the Dieudonné-theoretic applications (Theorem 4.8 and Corollary 4.18). The main reservation is that the proof of the crucial Proposition 3.3 delegates a key freeness assertion to an unpublished preprint and applies another external lemma without verifying its hypotheses; since Proposition 3.3 is what upgrades all other known equivalences to the full statement of Theorem 3.1, the central claim is conditional on that step.

major comments (2)
  1. [§3.4, proof of Proposition 3.3] The proof of Proposition 3.3 invokes 'the argument in [Hok24, Lemma 52]' to show that coker(f_i) is a finite free R-module for f_i : φ^*_{S_R}M → M/E^i. This is a load-bearing assertion: Proposition 3.3 is exactly the statement used in Proposition 3.4 to turn fully faithful or partially known arrows in diagram (3.1.1) into equivalences in the Fontaine--Laffaille range. However, [Hok24] is an unreviewed preprint (arXiv:2407.21327), and the manuscript does not reproduce the argument or state the precise hypotheses under which the lemma applies. The statement in Remark 1 that the proof is 'independent' of [Hok24] does not resolve this, because the proof as written uses that preprint as a black box. Please either include a complete proof of the cokernel freeness assertion or restate and prove the needed lemma in this paper, and verify explicitly that the setting of Proposition 3.3 satisfies all of its hypotheses.
  2. [§3.4, use of [Gao19, Lemma 3.8]] After reducing to R = WJt_1,...,t_dK, the proof applies [Gao19, Lemma 3.8] to the base-changed Breuil--Kisin module M_0 := M⊗_{S_R} S_W with its induced filtration, and uses the conclusion that Gr^i(Fil_0) is finite free over W(k) as the input for a Nakayama lifting argument. The manuscript does not verify that M_0 satisfies the hypotheses of [Gao19, Lemma 3.8]; in particular, it does not check the relevant strong divisibility or filtered-freeness conditions before applying the lemma. This matters because the freeness of Gr^i(Fil_0) is what produces the surjection R^{r_i} → Gr^i(Fil^bar) and ultimately the desired free basis of the graded pieces of Fil^bar. Please add the missing verification, or give a reference that contains it, so that the application of [Gao19, Lemma 3.8] is fully justified.
minor comments (4)
  1. [Abstract and §4 heading] The word 'Dieduonné' should be 'Dieudonné' in the abstract and in the Section 4 heading; there are also typographical slips 'crysal' in §1.3 and 'siomorphism' in §2.1.3.
  2. [Introduction, display (1)] The target of the composite functor in display (1) is written 'V ectφ,div(Xcrys)', but elsewhere, including (3.1.2), the same functor is said to take values in 'V ectFφ,div(Xcrys)'; the notation should be made consistent.
  3. [Notation and terminology] In the list of notation, the entry 'V ectan,φφ(X∆)' appears to be a typo for 'V ectφ,an(X∆)', and should be corrected.
  4. [§3.4, reduction to power-series case] The one-sentence reduction to R = WJt_1,...,t_dK should be expanded: after localizing at a maximal ideal of R/p, the residue field may be a finite extension of k rather than k itself, so the coefficient ring is a Witt ring W(κ) with κ≠k; please spell out the Cohen structure theorem step and how flat base change handles the residue-field extension.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: D_crys is built from internally proved comparison isomorphisms, and Theorem 3.1 assembles external equivalences rather than assuming its conclusion.

full rationale

I walked the derivation chain and found no step where a claimed prediction or equivalence is, by the paper's own definitions, identical to its input. The functor D_crys is defined via the crystalline realization E_crys, the de Rham realization D_dR, and a canonical comparison isomorphism proved from definitions in Theorem 1.19; the filtration is pulled back from the Nygaard filtration, not fitted to a target. The central Theorem 3.1 is explicitly reduced in Proposition 3.4 to Proposition 3.2 (commutativity, proved via Tsuji's functors and external comparison results) and Proposition 3.3 (lffness of effective F-crystals of height at most p-2). Proposition 3.3's proof does import [Hok24, Lemma 52] for finite freeness of a cokernel and [Gao19, Lemma 3.8] for freeness of graded pieces; these are external inputs, and a failure there would be a correctness risk, not a circularity. The paper also cites the authors' prior work [IKY25] for the equivalence R_X, but the introduction attributes the same result to [GL23] as well, and the cited result is a separate prior theorem, not an assumption of the present theorem. The main comparison and the Fontaine-Laffaille equivalence therefore have independent content: no arrow in the main diagram is defined as the conclusion it is claimed to prove.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new empirical constants or invented objects; D_crys and the lff condition are mathematical constructions defined within the paper. The load-bearing content is carried by standard background results and two external lemmas from preprints, which are the main points of fragility.

assumptions (5)
  • domain assumption Smoothness and base assumptions: X is a smooth formal scheme over W(k), p is odd, and the categories Vect^φ(X∆), Vect(Xsyn), Vect^φ(Xcrys) are defined as in the cited literature.
    These assumptions are stated in the introduction and used throughout, defining the scope of the results.
  • domain assumption External lemmas in Proposition 3.3: [Hok24, Lemma 52] asserting finite freeness of a certain cokernel, and [Gao19, Lemma 3.8] asserting freeness of graded pieces of a Breuil-Kisin module filtration.
    These are cited as black boxes in the reduction to the power-series case; if either is false, the proof of Proposition 3.3 breaks.
  • standard math Theory of prismatic F-gauges from [Bha23] unpublished course notes, including the equivalence RX between F-gauges and lff prismatic F-crystals.
    The paper relies on [Bha23, Remark 6.3.4] and [IKY25, Proposition 1.28] for the key equivalence RX, which is central to the statement of Theorem A.
  • standard math External categorical equivalences: T_ét is fully faithful and an equivalence by [GR24] and [DLMS24]; Tcrys is fully faithful by [Fal89, Theorem 2.6].
    These results are cited as established in the literature and are used directly in the proof of Theorem 3.1.
  • standard math Standard constructions in crystalline cohomology and prismatic cohomology, including the equivalence (−)crys and p-adic faithfully flat descent.
    These are standard background results, used without proof, with references to [BS23], [IKY24], and standard texts.

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Pith. "Pith review of An integral analogue of Fontaine's crystalline functor." pith.science (2026). https://pith.science/paper/3J5J2YNM

@misc{pith2026250416282,
  author       = {Pith},
  title        = {Pith review of: An integral analogue of Fontaine's crystalline functor},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3J5J2YNM}},
  note         = {Machine review of arXiv:2504.16282}
}
abstract

For a smooth formal scheme $\mathfrak{X}$ over the Witt vectors $W$ of a perfect field $k$, we construct a functor $\mathbb{D}_\mathrm{crys}$ from the category of prismatic $F$-crystals $(\mathcal{E},\varphi_\mathcal{E})$ (or prismatic $F$-gauges) on $\mathfrak{X}$ to the category of filtered $F$-crystals on $\mathfrak{X}$. We show that $\mathbb{D}_\mathrm{crys}(\mathcal{E},\varphi_\mathcal{E})$ enjoys strong properties when $(\mathcal{E},\varphi_\mathcal{E})$ is what we call locally filtered free (lff). Most significantly, we show that $\mathbb{D}_\mathrm{crys}$ actually induces an equivalence between the category of prismatic $F$-gauges on $\mathfrak{X}$ with Hodge--Tate weights in $[0,p-2]$ and the category of Fontaine--Laffaille modules on $\mathfrak{X}$. Finally, we use our functor $\mathbb{D}_\mathrm{crys}$ to enhance the study of prismatic Dieduonn\'e theory of $p$-divisible groups (as initiated by Ansch\"{u}tz--Le Bras) allowing one to recover the filtered crystalline Dieudonn\'e crystal from the prismatic Dieudonn\'e crystal. This in turn allows us to clarify the relationship between prismatic Dieudonn\'e theory and the work of Kim on classifying $p$-divisible groups using Breuil--Kisin modules.

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