Some applications of the Nygaard filtration and quasisyntomic descent in positive characteristic

Livia Grammatica

REVIEW 2 major objections 4 minor 28 references

Some applications of the Nygaard filtration and quasisyntomic descent in positive characteristic

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The Nygaard filtration gives an elementary route to Illusie's p-adic comparison theorem.

desk verdict Honest, useful re-derivation of known comparisons plus one new corollary and a nice explicit example; the load-bearing lifting input in Theorem 3.2.3 deserves a check, but the paper is worth refereeing. read the letter →

arxiv 2507.08568 v1 pith:HWBKGD5G submitted 2025-07-11 math.AG math.NT

Livia Grammatica This is my paper · ORCID

classification math.AGmath.NT MSC 14F3014F2014K15

keywords NygaardfiltrationquasisyntomicdescentcrystallinecohomologyfppfIllusiecomparisonFrobenius-smoothalgebraselementaryquasiregularsemiperfectBrauergroup

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

This paper establishes a new proof of Illusie's comparison theorem: for a smooth proper variety $X$ over an algebraically closed field $k$ of characteristic $p$, the fppf cohomology with $\mathbb{Q}_p(1)$ coefficients is isomorphic to the slope-1 Frobenius-isotypical part of crystalline cohomology, $H^i_{\mathrm{cris}}(X/W(k))[1/p]^{F=p}$. The proof goes through an exact triangle relating fppf cohomology with $\mathbb{Z}_p(1)$ to the first Nygaard filtration piece of crystalline cohomology via the map $F/p-1$, using quasisyntomic descent to reduce to elementary quasiregular semiperfect algebras. Along the way the paper gives a new proof of Ogus' comparison between infinitesimal and crystalline cohomology, derives structural results on $H^i_{\mathrm{fppf}}(X,\mathbb{Z}_p(1))$ (a direct sum of a free module and a $p$-torsion group of finite exponent), and determines the action of multiplication-by-$n$ on the fppf cohomology of abelian varieties, answering a question of Skorobogatov. This approach avoids the formalism of $\infty$-categories, replacing Bhatt-Lurie's machinery with canonical complexes built from perfections.

What carries the argument

The named machinery is the first piece of the Nygaard filtration, $F^1_N R\Gamma_{\mathrm{cris}}(X/W(k))$, defined as the cohomology of the kernel $I_{\mathrm{cris}}$ of the surjection of crystalline structure sheaves $O_{\mathrm{cris}} \to \mathbb{G}_a$; it sits in an exact triangle $F^1_N R\Gamma_{\mathrm{cris}} \to R\Gamma_{\mathrm{cris}} \to R\Gamma(X,O_X)$. A completed first Chern class $\hat{c}_1 : R\Gamma_{\mathrm{fppf}}(X,\mathbb{Z}_p(1)) \to F^1_N R\Gamma_{\mathrm{cris}}$ refines the usual Chern class, and the map $F/p-1$ (Frobenius divided by $p$ minus identity, defined on elementary quasiregular semiperfect algebras via explicit descriptions of $A_{\mathrm{cris}}$) completes the triangle. Descent along $R \to R_{\mathrm{perf}}$ for crystalline cohomology, proved using lifts of Frobenius-smooth algebras to $W_n(k)$ quoted from Berthelot-Messing, reduces everything to algebras $C = B[x_1^{p^{-\infty}}, \dots, x_n^{p^{-\infty}}]/(x_1,\dots,x_n)$, where $A_{\mathrm{cris}}(C)$ is an explicit divided-power power series ring and exactness of the triangle is checked modulo $p$.

What would settle it

Exhibit a Frobenius-smooth $\mathbb{F}_p$-algebra $R$ and a divided-power thickening $(A,I)$ over $W_n(k)$ for which the map $R \to A/I$ does not lift to a map $R_n \to A$ compatibly with Frobenius lifts; that would break Theorem 3.2.3 and the descent argument underlying Theorem 4.5. Alternatively, find an elementary quasiregular semiperfect algebra $C$ for which the mod-$p$ sequence $(1+J)^\times/p \to F^1_N A_{\mathrm{cris}}(C)/p \to A_{\mathrm{cris}}(C)/p$ has a nonzero kernel, contradicting Theorem 4.4.

Watch

Extended reading notes

Core claim

Theorem 4.5 asserts that for $X$ smooth over a perfect field $k$, the sequence $R\Gamma_{\mathrm{fppf}}(X,\mathbb{Z}_p(1)) \to F^1_N R\Gamma_{\mathrm{cris}}(X/W(k)) \xrightarrow{F/p-1} R\Gamma_{\mathrm{cris}}(X/W(k))$ is an exact triangle in $D(\mathbb{Z}_p)$. From this triangle the paper recovers Illusie's comparison: when $k$ is algebraically closed and $X$ is smooth and proper, $H^i_{\mathrm{fppf}}(X,\mathbb{Q}_p(1)) \cong H^i_{\mathrm{cris}}(X/W(k))[1/p]^{F=p}$. The triangle is proved by descent: first to affine $X$, then along $X_{\mathrm{perf}} \to X$ to elementary quasiregular semiperfect algebras, where crystalline cohomology is $A_{\mathrm{cris}}(C)$, the Nygaard filtration is explicit, and exactness becomes a direct computation with power series.

Load-bearing premise

The descent theorem for crystalline cohomology along $R \to R_{\mathrm{perf}}$ assumes that every Frobenius-smooth $\mathbb{F}_p$-algebra $R$ has compatible lifts $(R_n, F_n)$ to $W_n(k)$ and that every divided-power thickening lifts as well, a deformation-theoretic input quoted from Berthelot-Messing; if that input fails, the descent step that assembles the exact triangle for general smooth $X$ collapses.

Editorial extensions

If this is right

Illusie's comparison theorem $H^i_{\mathrm{fppf}}(X,\mathbb{Q}_p(1)) \cong H^i_{\mathrm{cris}}(X/W(k))[1/p]^{F=p}$ follows as a direct corollary of the exact triangle, with properness used only to make the maps $F/p-1$ surjective after inverting $p$.
For every $i$, $H^i_{\mathrm{fppf}}(X,\mathbb{Z}_p(1))$ is the direct sum of a free $\mathbb{Z}_p$-module of rank $\operatorname{rank} H^i_{\mathrm{cris}}(X)^{F=p}$ and a $p$-torsion group of finite $p$-exponent; for $i=1,2$ the groups are finite-type $\mathbb{Z}_p$-modules.
For straight varieties (torsion-free crystalline cohomology and a degenerating Hodge-de Rham spectral sequence, e.g. abelian varieties, K3 surfaces, complete intersections), fppf cohomology is completely determined by the $F$-crystal $H^i_{\mathrm{cris}}$, and $F^1_N H^i = F^{-1}(pH^i)$.
On an abelian variety $A$, multiplication-by-$n$ acts as $n^i$ on both $H^{i+1}_{\mathrm{fppf}}(A,\mathbb{Z}_p(1))_{\mathrm{tors}}$ and $H^i_{\mathrm{fppf}}(A,\mathbb{Z}_p(1))/\mathrm{tors}$, answering Skorobogatov's question.
The same descent formalism gives a new proof of Ogus' theorem $R\Gamma_{\mathrm{inf}}(X/W(k)) \simeq R\lim_F R\Gamma_{\mathrm{cris}}(X/W(k))$.

Reading between the lines

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

The paper's replacement of $\infty$-categories by canonical complexes built from perfections suggests the same descent package could be adapted to other $p$-adic cohomology functors that also have explicit descriptions on semiperfect algebras.
For straight varieties the Nygaard filtration is determined by the $F$-crystal only under the Mazur-Ogus Newton-above-Hodge hypothesis; the paper's examples suggest that the exact triangle itself may still determine fppf cohomology from the $F$-crystal when that hypothesis fails.
The computation $H^3_{\mathrm{fppf}}(E\times E,\mathbb{Z}_p(1)) \cong k$ for a supersingular elliptic curve $E$ shows that torsion fppf cohomology can carry positive-dimensional information (the unipotent group $\mathbb{G}_a$), which may have consequences for Brauer-group computations on supersingular surfaces.
Corollary 5.2.4 is stated for abelian varieties; a testable extension is whether the same $n^i$ action holds on the fppf cohomology of any straight variety carrying a multiplication-by-$n$ endomorphism.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: this paper is a well-written expository account of Bhatt–Lurie's approach to Illusie's comparison theorem, and it is honest about what is new. The genuinely new items are Corollary 5.2.4 (multiplication-by-n acts as n^i on fppf cohomology of abelian varieties, answering Skorobogatov's question) and the explicit computation H^3_fppf(E×E) = k for a supersingular elliptic curve E, plus a derived-category formulation of Ogus' theorem. The paper's claim to originality is modest and correct: Sections 1–4 contain only the new proof of Ogus' theorem, Section 4 is a rewriting of part of Bhatt–Lurie, and most of Section 5 is new proofs of known results.

What the paper does well: the descent framework is kept light, the explicit algebra for elementary quasiregular semiperfect rings is spelled out in real detail (at least in one variable), and the examples at the end show the machinery actually computes something. The writing is clear and the authors are appropriately careful about credits.

Soft spots: the multi-variable case of Theorem 4.4 is dismissed with 'adapts as is', which is probably fine but should be checked. More importantly, Theorem 3.2.3 — the coperfection descent theorem that the whole proof of Theorem 4.5 depends on — quotes [BeM07, Cor 1.2.7, Prop 1.2.6] for the existence of compatible lifts (R_n, F_n) and lifts to PD thickenings for Frobenius-smooth algebras. The class includes rings like B[[x_1,...,x_n]] with B perfect, which is outside the smooth finite-type setting I associate with Dieudonné theory. The stress-test note is right: if that reference does not literally cover Frobenius-smooth algebras, the descent statement collapses and the argument for arbitrary smooth X cannot be assembled. I cannot check the original right now, but the authors should be asked to state the hypotheses of the cited results and explain why they apply. It is a quotable external input rather than an internal inconsistency, so it does not sink the paper, but it is the weakest link.

Bottom line: for a reader who wants an elementary path to Illusie's comparison and the Nygaard exact triangle without infinity-categories, this is useful. For a specialist, the new corollary and the supersingular example are worth having. I would cite it and I would send it to a referee who can verify the Berthelot–Messing citation. Recommendation: accept, with that verification.

Referee Report

2 major / 4 minor

Summary. The paper develops an elementary, ∞-category-free treatment of quasisyntomic descent and the Nygaard filtration in positive characteristic. Its central theorem (Theorem 4.5) states an exact triangle RΓ_fppf(X, Z_p(1)) → F^1_N RΓ_cris(X/W(k)) → RΓ_cris(X/W(k)) for smooth X over a perfect field k, from which Illusie's comparison H^i_fppf(X, Q_p(1)) ≅ H^i_cris(X/W(k))[1/p]^{F=p} is deduced. The proof reduces by cohomological descent to elementary quasiregular semiperfect algebras, where all three terms are computed explicitly. The paper also gives a descent proof of Ogus' comparison between infinitesimal cohomology and the unit-root part of crystalline cohomology, derives structural results for fppf cohomology (a free part plus a p-group of finite p-exponent), computes the multiplication-by-n action on fppf cohomology of abelian varieties, and works out two explicit examples.

Significance. If correct, the paper provides a valuable, more accessible route to Bhatt–Lurie's comparison theorem and its consequences, avoiding ∞-categories. The explicit Acris computations for eqrsp algebras are clear and useful, as are the worked examples (ordinary abelian varieties and E×E) and the new proof of the [n]-action on fppf cohomology. The proof of Ogus' theorem via descent is elegant. The genuinely new results are modest, however, and much of the paper is expository or a re-proof of known theorems; its main value is pedagogical and organizational rather than groundbreaking.

major comments (2)

[§3.2, Theorem 3.2.3(1)] The proof of coperfection descent delegates the two crucial deformation-theoretic inputs to [BeM07, Cor. 1.2.7 and Prop. 1.2.6] without stating their precise hypotheses. This is load-bearing: the lifting lemma is what upgrades étale descent to descent along R → R_perf for every Frobenius-smooth F_p-algebra, and Theorem 4.5 uses it for arbitrary smooth X. Since the class of Frobenius-smooth algebras includes non-finite-type rings such as B[[x_1,…,x_n]] with B perfect, the reader cannot verify from the text that the cited Dieudonné-theoretic statements apply verbatim. Please state the cited results, justify their applicability to the full class of Frobenius-smooth algebras, or replace the quotation with a direct proof.
[§4, Theorem 4.4(2)] The exactness of (4.3), which is the explicit algebraic heart of the comparison theorem, is written out only for the one-variable algebra C = B[x^{p^{-∞}}]/(x). The passage to the general eqrsp case is dismissed with the sentence 'the proof adapts as is, it is only more tedious to keep track of all indices'. Since Theorem 4.5 reduces all smooth schemes to exactly these multivariable algebras, the omitted verification is load-bearing. Please provide the multivariable computation in full or give a formal reduction (e.g. an explicit isomorphism or induction) that turns the asserted adaptation into a complete proof.

minor comments (4)

[§3.3] The proof headings appear to be interchanged: the paragraph labeled 'Proof of Theorem 3.3.1' in fact proves the unit-root description that constitutes Theorem 3.3.2, while Theorem 3.3.1 is Grothendieck's characteristic-zero statement quoted earlier. Please relabel the proofs.
[§5.2, Corollary 5.2.4] In the statement, 'H^i_fppf(X, Z_p(1))/tors' should presumably be 'H^i_fppf(A, Z_p(1))/tors'. The proof would also benefit from a one-line diagram chase noting that the map H^i_fppf(A) → F^1_N H^i_cris(A) is injective only after quotient by torsion, so that the action on the free quotient is indeed determined by the action on F^1_N H^i_cris(A).
[§2.4, Proposition 2.4.9] In the proof, the algebra 'k[y^{p^{-∞}}]/(y)' should be written 'F_p[y^{p^{-∞}}]/(y)' (or with a new symbol) for consistency, since k is the base perfect field and the algebra under consideration is an F_p-algebra.
[§5.2, Proposition 5.2.5(3)] The rank formula g·binom(g, i-1) for H^i_fppf(A, Z_p(1)) needs a stated range for i, because for i = 0 the binomial coefficient is undefined.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 4.5 is an independently re-derived Bhatt–Lurie triangle, supported by external citations and explicit eqrsp computations.

full rationale

The paper makes no empirical predictions and fits no parameters, so the fitted-input and self-definitional patterns do not arise. Its central result, Theorem 4.5, is presented as a re-derivation of [BhL22, Theorem 7.3.5]; the proof reduces by Zariski and coperfection descent to the explicit eqrsp computation of Theorem 4.4, whose exactness is verified directly from the explicit descriptions of C^flat, A_cris(C), and F^1_N A_cris(C) in Section 2.4. The only load-bearing external input is the lifting/deformation machinery quoted from Berthelot–Messing [BeM07, Corollary 1.2.7, Proposition 1.2.6] in the proof of Theorem 3.2.3. That citation is genuinely external and not by the present author, and it is not equivalent to the target comparison theorem. The paper also benchmarks its results against Illusie's theorem, Ogus' theorem, and other established statements, none of which reduce by construction to the paper's own definitions. The reliance on [BeM07] is a potential correctness risk if the cited results do not literally cover all Frobenius-smooth algebras needed, but that is a question of validity of external support, not circularity. There are no self-citations that are load-bearing, no ansatz smuggled in through the author's prior work, and no known results renamed as derivations. Accordingly, the honest finding is no circularity.

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

The central claims rest on standard p-adic cohomology facts and on several deep theorems taken from the literature; no free parameters or invented entities appear. The most load-bearing external inputs are the deformation theory for Frobenius-smooth algebras, the Mazur-Ogus theorem, and the Dieudonné-Manin classification.

assumptions (5)

domain assumption Frobenius-smooth algebras admit compatible Witt-vector lifts (R_n, F_n) and lifts of thickening maps
Invoked in the proof of Theorem 3.2.3, Section 3.2, to prove descent along R to R_perf; quoted from [BeM07] without proof.
domain assumption Mazur-Ogus Newton-above-Hodge theorem
Used in Proposition 5.2.2 to identify the Nygaard filtration with F^{-1}(p H^i_cris) for straight varieties.
domain assumption Dieudonné-Manin classification of F-crystals over W(k)
Used in Lemma 5.1.3 and Proposition 5.1.13 to prove finite p-exponent and surjectivity statements.
standard math Standard derived p-completeness facts from the Stacks Project
Section 2.2, used throughout via Lemmas 2.2.2, 2.2.3 and 2.2.5.
standard math Cartier isomorphism and de Rham-Witt facts
Used in Section 5.1, in particular Lemmas 5.1.8 and 5.1.12, citing [Ill79], [Oda69] and related literature.

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Cite this review

Pith. "Pith review of Some applications of the Nygaard filtration and quasisyntomic descent in positive characteristic." pith.science (2026). https://pith.science/paper/HWBKGD5G

@misc{pith2026250708568,
  author       = {Pith},
  title        = {Pith review of: Some applications of the Nygaard filtration and quasisyntomic descent in positive characteristic},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HWBKGD5G}},
  note         = {Machine review of arXiv:2507.08568}
}

abstract

This article gives an expository account of quasisyntomic descent and the Nygaard filtration in positive characteristic, complemented by several new applications to $p$-adic cohomology theories. The guiding result is a new approach to Illusie's comparison between fppf cohomology with $\mathbb{Z}_p(1)$ coefficients and the slope $1$ part of crystalline cohomology. We follow work of Bhatt-Lurie, but give a more elementary presentation which does not rely on the formalism of $\infty$-categories. We then revisit Ogus' comparison theorem between infinitesimal cohomology and \'etale cohomology, and give new proofs of several results on fppf cohomology that were previously obtained with the de Rham-Witt complex. We also determine the action of multiplication-by-$n$ on the fppf cohomology of an abelian variety, answering a question of A. Skorobogatov to the author. This is an expanded version of the author's master thesis.

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Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Supersingular K 3 Surfaces

Michael Artin. Supersingular K 3 Surfaces. In Annales scientifiques de l'École Normale Supérieure , série 4, volume 7, pages 543-567, 1974

work page 1974

[2] [2]

Cohomologie cristalline des sch\' e mas de caractéristique p>0

Pierre Berthelot. Cohomologie cristalline des sch\' e mas de caractéristique p>0 . Lecture Notes in Mathematics, volume 407, Springer, 1974

work page 1974

[3] [3]

Th \'e orie de Dieudonn \'e cristalline III

Pierre Berthelot and William Messing. Th \'e orie de Dieudonn \'e cristalline III. In The Grothendieck Festschrift I , volume 86 of Progress in Mathematics , pages 173--247. Birkh \"a user, Boston, 2007

work page 2007

[4] [4]

Notes on crystalline cohomology

Pierre Berthelot and Arthur Ogus. Notes on crystalline cohomology. Mathematical Notes, volume 21, Princeton University Press, 1978

work page 1978

[5] [5]

Absolute Prismatic Cohomology

Bhargav Bhatt and Jacob Lurie. Absolute Prismatic Cohomology. 2022. https://arxiv.org/abs/2201.06120

arXiv 2022

[6] [6]

The Brauer–Grothendieck Group

Jean-Louis Colliot-Thélène, Alexei Skorobogatov. The Brauer–Grothendieck Group. Springer, 2021

work page 2021

[7] [7]

Boundedness of the p-primary torsion of the Brauer group of an abelian variety

Marco D'Addezio. Boundedness of the p-primary torsion of the Brauer group of an abelian variety. In Compositio Mathematica , volume 160, pages 463-480, 2024

work page 2024

[8] [8]

Differentiably simple rings and ring extensions defined by $p$-basis

Celia del Buey de Andrés, Diego Sulca, Orlando Villamayor. Differentiably simple rings and ring extensions defined by p -basis 2022. https://arxiv.org/abs/2211.09125

work page Pith review arXiv 2022