Duality between $W_n$-Cartier crystals and $\mathbb{Z}/p^n\mathbb{Z}$-perverse sheaves

Jefferson Baudin

REVIEW 3 major objections 5 minor 2 cited by

Duality between $W_n$-Cartier crystals and $\mathbb{Z}/p^n\mathbb{Z}$-perverse sheaves

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read On Noetherian F-finite schemes, W_n-Cartier crystals and constructible perverse Z/p^nZ-sheaves are equivalent up to opposite categories.

desk verdict A genuine Witt-vector extension of the author's n=1 Cartier-perverse duality, with new finiteness results, but the advertised Z/p^nZ coefficients are broader than the W_n(F_q) theorem actually proved, and the central proof leans on imports from BL19 and Bau23 that a referee should check. read the letter →

arxiv 2506.13346 v1 pith:ACVQFGFT submitted 2025-06-16 math.AG

Jefferson Baudin This is my paper · ORCID

classification math.AG MSC 14G1713A3513F35

keywords perversesheavesCartiercrystalsWittvectorsRiemann-HilbertcorrespondenceFrobeniusmodulesconstructiblet-structuresdualizingcomplexes

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 proves that a coherent algebraic object—a $W_n$-Cartier crystal, roughly a coherent module over the $n$-truncated Witt vectors equipped with a Frobenius-linear Cartier operator, considered up to nilpotence—is the same data as a constructible perverse sheaf of $\mathbb{Z}/p^n\mathbb{Z}$-modules on the \'etale site. The duality is an equivalence of derived categories, compatible with proper pushforwards, and it matches the canonical $t$-structure on the crystal side with the middle perverse $t$-structure on the sheaf side. The result extends the known $n=1$ duality to every truncation level $n$, which matters because it lets coherent techniques prove statements about perverse sheaves with $W_n(\mathbb{F}_q)$ coefficients and, conversely, topological constructibility results about Witt-vector canonical sheaves. The proof goes through a new duality between $W_n$-Frobenius modules and $W_n$-Cartier modules, built from dualizing-complex pairings against a $W_n$-unit dualizing complex.

What carries the argument

The load-bearing mechanism is a pair of Hom pairings between $W_n$-Frobenius modules and $W_n$-Cartier modules, both given by $\mathrm{Hom}_{W_n\mathcal{O}_X}(-,-)$ with the Frobenius action on one side and the Cartier operator on the other. Pairing against a $W_n$-unit dualizing complex $W_n\omega_X^\bullet$ yields two duality functors $D$, shown to be mutual inverses; through derived equivalences between coherent and quasi-coherent variants, this gives the Frobenius–Cartier duality. On the topological side, the functor $\mathrm{Sol}$ extracts the kernel of $\kappa-1$ and, by the imported Riemann–Hilbert correspondence, identifies $W_n$-Frobenius crystals with constructible $W_n(\mathbb{F}_q)$-sheaves. Composing $\mathrm{Sol}\circ D$ is what converts the coherent $t$-structure into the middle perverse $t$-structure; the inductive lemmas decompose a complex into its $p$-torsion and $W_{n-1}$ pieces to reduce to $n=1$.

What would settle it

Take an $F$-finite imperfect field $k$, set $X = \mathbb{A}^1_k$ and $n=2$, and explicitly compute $\mathrm{Sol}(D(M))$ for the $W_2$-Frobenius modules corresponding to the constant, skyscraper, and rank-one sheaves; the theorem predicts their perverse cohomology is concentrated in the predicted degree with the predicted $W_2(\mathbb{F}_q)$-ranks. A mismatch in any one of these explicit computations would disprove the $t$-structure part of the main theorem.

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Extended reading notes

Core claim

The central theorem (Theorem 5.2.7) states: let $X$ be a Noetherian, $F$-finite, semi-separated $\mathbb{F}_q$-scheme admitting a $W_n$-unit dualizing complex $W_n\omega_X^\bullet$. After shifting this dualizing complex on each connected component, the functor $\mathrm{Sol}\circ D$ sends the canonical $t$-structure on $D^b(\mathrm{Crys}^C_{W_nX})^{\mathrm{op}}$ to the perverse $t$-structure on $D^b_c(X_{\acute{e}t},W_n(\mathbb{F}_q))$, and therefore induces an equivalence $(\mathrm{Crys}^C_{W_nX})^{\mathrm{op}} \cong \mathrm{Perv}_c(X_{\acute{e}t},W_n(\mathbb{F}_q))$. Here $D$ is the new duality between $W_n$-Frobenius and $W_n$-Cartier modules, and $\mathrm{Sol}$ is the solution functor taking a Cartier module $(M,\kappa_M)$ to $\ker(\kappa_M-1)$ as an \'etale sheaf. The proof reduces to the $n=1$ case by an induction that decomposes any $W_n$ object into its mod-$p$ and $W_{n-1}$ parts, using the imported Riemann–Hilbert correspondence for $W_n$-Frobenius modules as the bridge to the topological side.

Load-bearing premise

The proof inherits the imported Riemann–Hilbert correspondence for $W_n$-Frobenius crystals and the $n=1$ duality result; if either of those has hidden hypotheses outside the stated framework, the induction proving the full duality collapses.

Editorial extensions

If this is right

Every theorem about constructible perverse $\mathbb{Z}/p^n\mathbb{Z}$-sheaves translates into a theorem about $W_n$-Cartier crystals, so topological proofs can settle coherent questions about $W_n$-canonical sheaves.
Vanishing theorems and finiteness results for the Witt canonical sheaf $W_n\omega_X$ can be approached through perverse sheaves with $W_n(\mathbb{F}_q)$ coefficients, extending the $n=1$ applications.
Objects of $\mathrm{Crys}^C_{W_nX}$ are both Noetherian and Artinian, and direct images of ind-coherent $W_n$-Cartier modules satisfy the corresponding finiteness and stability properties.
The equivalence is compatible with proper pushforwards and with the inclusions from level $m$ to level $n$, so perverse cohomology of direct images matches Cartier-crystal cohomology in a way that is stable under truncation.

Reading between the lines

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

If the duality holds, any perverse-sheaf theorem over $\mathbb{Z}/p^n\mathbb{Z}$ should have a purely coherent counterpart; an immediate test case is to translate known generic vanishing statements across the equivalence for all $n$.
The induction on $n$ suggests a general d\'evissage principle: many properties of perverse $W_n(\mathbb{F}_q)$-sheaves can be proven by reducing first to $\mathbb{F}_p$-sheaves and then to $W_{n-1}$-sheaves, possibly simplifying future constructibility arguments.
The duality may extend to derived $\infty$-categories of Cartier modules, where the $t$-structure comparison could be obtained formally rather than by induction; the paper notes this as a plausible but unimplemented route.
A consequence the author leaves implicit: the equivalence should induce a bijection between support conditions on perverse $W_n(\mathbb{F}_q)$-sheaves and support-theoretic strata of $W_n$-Cartier crystals, so classification results on one side can be read off from the other.

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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

The paper is what it says on the tin: a generalization of Baudin's n=1 Cartier-perverse duality to W_n-Cartier crystals and W_n(F_q)-perverse sheaves. That is real new content, not a renaming. The finiteness theorems in Section 4 (pushforwards stay ind-coherent, crystals are Noetherian and Artinian, exactness of the solution functor on ind-coherent Cartier modules) are the strongest part of the paper and are genuinely new at the W_n level. The duality Theorem 5.1.12 also appears to be proven by a sound reduction to the n=1 case, using the equivalence of derived categories established earlier. I read carefully through the reduction arguments and found no obvious circularity or algebraic error.

Two soft spots, one mostly cosmetic and one load-bearing. The cosmetic one: the abstract and introduction say the topological side is perverse Z/p^nZ-sheaves, but Theorem 5.2.7 is over W_n(F_q). For r>1 these are different coefficient rings. That should be fixed in a revision; it matters for potential applications and for anyone citing the advertised statement.

The load-bearing spot is the Riemann-Hilbert import. Theorem 3.2.3 restates [BL19, Thm 9.6.1] as an equivalence for all Noetherian F-finite F_q-schemes and all n, with constructible sheaves on the étale site. The proof given here only reduces to the affine case and invokes [BL19]. Nothing in the paper verifies that the hypotheses of [BL19, Thm 9.6.1] match this generality. If that theorem is actually proved only for finite-type schemes or only for n=1, then the equivalence Sol in Theorem 5.2.7 is not established by the text. The same holds for the n=1 base case, which is taken from an unpublished preprint [Bau23]. This is not a fatal flaw on its face, but it is exactly what a referee should check before the paper is accepted. The author should either quote the precise theorem with its hypotheses or prove the needed generality.

Who is this for? Anyone working on the positive-characteristic Riemann-Hilbert program, especially on Witt-vector coefficients, and anyone who wants to use W_n-canonical sheaves for vanishing theorems. It deserves a serious referee: the main construction is plausible, the n-level extension is significant, and the gaps I found are fixable rather than fundamental. I would send it to review, with a specific request to verify the BL19 hypotheses and to correct the coefficient ring statement in the abstract.

Referee Report

3 major / 5 minor

Summary. Fix a prime p, integers n, r ≥ 1, and q = p^r. The paper develops a theory of W_n-Cartier modules and crystals on Noetherian F-finite F_p-schemes, proves several finiteness and duality results for these categories, and states as its main theorem (Theorem 5.2.7) that for a Noetherian, F-finite, semi-separated F_q-scheme X equipped with a W_n-unit dualizing complex, the functor Sol∘D induces an equivalence D^b(Crys^Cr_{W_nX})^op ≅ D^b_c(X_ét, W_n(F_q)) that sends the canonical t-structure to the perverse t-structure, hence (Crys^Cr_{W_nX})^op ≅ Perv_c(X_ét, W_n(F_q)). The proof is an induction on n: Theorem 5.1.12 establishes the duality D between W_n-Cartier and W_n-Frobenius crystals, and the Riemann-Hilbert equivalence Sol between W_n-Frobenius crystals and constructible W_n(F_q)-sheaves is imported as Theorem 3.2.3 from [BL19, Theorem 9.6.1]; the base case n = 1 is the author's earlier theorem [Bau23, Theorem 5.2.7].

Significance. If the hypotheses on the imported results are verified, this is a substantial extension of the Cartier-crystal/perverse-sheaf duality of [Bau23] to truncated Witt-vector coefficients. The paper's own contributions—the finiteness theory of W_n-Cartier modules (Propositions 4.4.4 and 4.4.8), the duality Theorem 5.1.12, and the devissage induction in Lemmas 5.2.5–5.2.6—are well structured and appear correct. The main risk is that Theorem 5.2.7 depends entirely on the imported Riemann-Hilbert correspondence and on the unpublished base case [Bau23, Theorem 5.2.7], and the manuscript does not provide enough detail to confirm that the cited results apply in the stated generality. The abstract and introduction further state the result with Z/p^nZ-sheaves and F_p-schemes, while the theorem is over W_n(F_q)-sheaves and F_q-schemes; this ambiguity must be resolved.

major comments (3)

[Section 3.2, Theorem 3.2.3] The proof of Theorem 3.2.3 reduces the statement to the affine case by citing [BL19, Theorem 10.2.7], then invokes Lemma 3.2.6 and [BL19, Theorem 9.6.1], but it does not state the precise hypotheses of [BL19, Theorem 9.6.1] nor verify that they hold for every Noetherian F-finite F_q-scheme, for all n ≥ 1 and r ≥ 1. Since Theorem 5.2.7 uses this equivalence as the entire bridge from W_n-Frobenius crystals to constructible W_n(F_q)-sheaves, an unstated hypothesis in the imported theorem would invalidate the main claim. Please provide the exact statement of the cited theorem and a complete verification of the hypotheses, or restrict the main theorem to the generality for which the Riemann-Hilbert correspondence is established.
[Abstract and Introduction, §1.1] The abstract and the theorem displayed in the Introduction claim an equivalence with perverse Z/p^nZ-sheaves (and, in the Introduction, for X an F_p-scheme), whereas Theorem 5.2.7 is stated for F_q-schemes and W_n(F_q)-sheaves. Since W_n(F_q) is different from Z/p^nZ when r > 1, and an arbitrary F_p-scheme need not be an F_q-scheme for r > 1, the precise relationship between these statements is unclear. The paper should consistently state the main theorem in the full W_n(F_q)-form and explain that the Z/p^nZ version is the special case r = 1, or prove the Z/p^nZ version as advertised.
[Section 5.2, proof of Theorem 5.2.7] The induction in Theorem 5.2.7 has as its base case [Bau23, Theorem 5.2.7], which is an unpublished preprint. The proof of the n-level statement reduces entirely to this base and to Theorem 3.2.3, so the paper should either include the precise statement of [Bau23, Theorem 5.2.7] with its hypotheses, or reproduce enough of its proof to make the induction self-contained. As written, the central equivalence is not fully verifiable from the manuscript alone.

minor comments (5)

[Section 1.1, first paragraph] The sentence 'The link between W_n–Frobenius crystals and étale Z/p^nZ–sheaves was worked out in [BL19, Section 9]' should read 'W_n(F_q)-sheaves' to be consistent with Theorem 3.2.3.
[Lemma 3.1.6 and its proof] There are small typographical errors: 'essential image consist of objects' should be 'essential image consists of objects', and in the proof 'CrysF r WnX' appears where 'CrysF r WmX' or 'CrysF r WnX' is intended depending on context; please proofread this paragraph.
[Lemma 4.2.2] The displayed exact sequences use periods in place of slashes or arrows (for example 'M[p]∩p^iM. M[p]∩p^{i+1}M'); these should be typeset properly as quotients in the exact sequences.
[Proof of Proposition 4.4.16] The word 'asusme' should be 'assume'.
[References, [MW24]] The second author's name appears as 'T. Wei β'; if this is a rendering issue with a German character, it should be corrected (likely 'Weiß' or 'Weiss').

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity: the W_n duality theorem is not defined in terms of its target, and its two main imports are strictly weaker prior results.

full rationale

The central theorem (Theorem 5.2.7) is an induction on n. Its base case is the n=1 theorem from the author's earlier paper [Bau23, Theorem 5.2.7], and its topological input is the W_n Frobenius-to-étale equivalence restated from [BL19, Theorem 9.6.1] as Theorem 3.2.3. Neither of these inputs contains the W_n Cartier-to-perverse statement being proved for general n; they are prior, strictly smaller results. The proof of Theorem 5.2.7 explicitly says: 'We will show the result by induction on n≥ 1. This is true for n = 1 by [Bau23, Theorem 5.2.7], so now assume that n≥ 2.' The induction step uses the paper's Lemmas 5.2.5 and 5.2.6, the duality Theorem 5.1.12, and the imported Riemann–Hilbert correspondence; it does not re-define the perverse t-structure in terms of Cartier crystals or vice versa. The repeated reductions to [Bau23] for formally identical proofs are self-citations, but they are not circular because they cite a different, lower-n theorem rather than assuming the present conclusion. The reliance on an unpublished arXiv preprint for the base case and on [BL19] for the W_n Frobenius equivalence is a verification risk, not a circularity; if those hypotheses were mismatched the theorem would fail, but that is a correctness concern. No equation in the paper makes the target equivalent to its input by construction.

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

The paper introduces no numerical free parameters and no new physical entities. The ledger instead records the imported theorems that carry the main weight: the W_n Frobenius Riemann-Hilbert correspondence, the n=1 duality base case, Gabber's t-structure, and the existence of W_n-unit dualizing complexes.

assumptions (5)

standard math Riemann-Hilbert for W_n-Frobenius crystals: Sol_n is an equivalence between IndCrys^Fr_{W_nX} and Sh(X_et,W_n(F_q)) and between Crys^Fr_{W_nX} and Sh_c(X_et,W_n(F_q)).
Imported from [BL19, Theorem 9.6.1], restated as Theorem 3.2.3, and used to identify the topological side in the main theorem.
domain assumption Base case n=1 duality between Cartier crystals and perverse F_q-sheaves.
Taken from [Bau23, Theorem 5.2.7]; the induction in Theorem 5.2.7 starts here and many W_n proofs are declared identical to it.
standard math Gabber's middle perverse t-structure exists on D(X_et,W_n(F_q)) and restricts to constructible complexes.
Used in Definition 5.2.1 and Proposition 5.2.2 to define the perverse truncations and the heart Perv_c.
standard math Schemes of finite type over a Noetherian F-finite F_q-algebra admit W_n-unit dualizing complexes.
Corollary 5.1.16 derives this from [KTT+24, Theorem 9.1] and [Sta25, Tag 0AU5], and it is needed to apply Theorem 5.2.7.
standard math Finiteness theorem: objects of the category of Cartier crystals are Noetherian and Artinian.
Imported from [BB11, Main Theorem] and used in Proposition 4.4.8 to prove the W_n analogue.

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Pith. "Pith review of Duality between $W_n$-Cartier crystals and $\mathbb{Z}/p^n\mathbb{Z}$-perverse sheaves." pith.science (2026). https://pith.science/paper/ACVQFGFT

@misc{pith2026250613346,
  author       = {Pith},
  title        = {Pith review of: Duality between $W_n$-Cartier crystals and $\mathbbZ/p^n\mathbbZ$-perverse sheaves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ACVQFGFT}},
  note         = {Machine review of arXiv:2506.13346}
}

abstract

We generalize the results in [Bau23] to obtain a duality between $W_n$-Cartier crystals and perverse $\mathbb{Z}/p^n\mathbb{Z}$-sheaves.

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Forward citations

Cited by 2 Pith papers

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Proper birational maps from smooth varieties have vanishing higher rational Witt top-forms; in low dimensions the integral Witt sheaves are annihilated by a fixed p-power, yielding Q_p-rationality of F-rational singularities.
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Smooth proper weakly ordinary varieties of maximal Albanese dimension satisfy chi(X, omega_X) >= 0, with chi = 0 for non-general-type examples and the Albanese image then fibered by ordinary abelian varieties.

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Works this paper leans on

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