Wild Betti sheaves

Peter Scholze

REVIEW 3 major objections 5 minor 1 cited by

Wild Betti sheaves

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Wild Betti sheaves are universal, revealing a hidden torsor

desk verdict Solid construction of wild sheaves, but the universality and torsor claims are conditional on an unproved variant of Vaintrob's theorem and a sketchy decomposition. read the letter →

arxiv 2505.24599 v1 pith:HJ7UA32X submitted 2025-05-30 math.AG math.ATmath.NTmath.SG

Peter Scholze This is my paper

classification math.AGmath.ATmath.NTmath.SG MSC 14F0814F05

keywords wildBettisheavesexponentiallocalsystemFouriertransformenhancedR-filteredspectraR>0-torsorsixoperations

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 builds a universal enlargement of the category of Betti sheaves, called wild Betti sheaves, that carries a nontrivial "exponential local system" on the real line and a Fourier equivalence defined on every sheaf. The universal coefficient category is the category $W$ of completely and continuously $\mathbb{R}$-filtered spectra, and the construction recovers a previously known theory of enhanced sheaves. The main structural conclusion is that $\mathrm{Spec}(W)$ sits over $\mathrm{Spec}(\mathbb{Z})$ as a canonical nontrivial $\mathbb{R}_{>0}$-torsor, which gives a new direction in which to localize in algebraic and arithmetic geometry. This matters because a Fourier transform is the engine behind many applications to exponential sums and geometric representation theory, and the paper shows it is available in the Betti setting in a way that does not depend on arbitrary choices.

What carries the argument

The central object is $W$, the symmetric monoidal stable $\infty$-category of completely and continuously $\mathbb{R}$-filtered spectra: an object is a spectrum with an ascending filtration $\mathrm{Fil}_r M$ indexed by $r\in\mathbb{R}$, complete at $-\infty$ and continuous at every $r$. Because the unit $S(0)$ is not compact, this category escapes the usual obstruction that makes all invertible sheaves on $\mathbb{R}$ trivial; the exponential local system is the family of shifts $S(f)$ with filtration jumping at $f(x)$. The argument is carried by the kernel $K_V=b^*\exp$ on $V\times V^*$, whose convolution gives the Fourier equivalence, and by a cited equivalence between sheaves on $\mathbb{R}$ with the convolution product and quasicoherent sheaves on an almost infinite-root version of $\mathbb{P}^1/\mathbb{G}_m$, which allows the universal property to be read off from the two ends $0$ and $\infty$ of that stack.

What would settle it

Exhibit a presentable symmetric monoidal stable $\infty$-category $C$ that admits a nontrivial additive exponential local system inducing a Fourier equivalence but whose associated symmetric monoidal functor from the almost infinite-root stack to $C$ does not factor through $W$ (up to reversing the filtration); this would disprove universality without challenging the existence theorem. Alternatively, complete the proof of the cited stack equivalence and check directly whether the two quotient categories at $0$ and $\infty$ are exactly $W$ and its filtration-reverse.

Watch

Extended reading notes

Core claim

The central discovery is that the problem of adjoining an exponential local system to Betti sheaves has a universal solution. Concretely, the paper proves existence of a coefficient category $W$ whose objects are completely and continuously $\mathbb{R}$-filtered spectra, together with an invertible sheaf $\exp\in D(\mathbb{R},W)$ whose fibre at $r$ is the shift $S(r)$; from this kernel one defines a Fourier functor $F_V$ on every real vector space $V$, and Theorem 4.1 says $F_V$ is an equivalence with inverse $(-1)^*F_{V^*}[d]$. The universality statement of Section 5 says that, up to reversing the filtration, $W$ is the unique such coefficient category, because any candidate exponential local system inducing a Fourier equivalence is forced to be trivial after restriction to $\tilde{\mathbb G}_m/\tilde{\mathbb G}_m$ and the remaining quotient splits into $W$ at $0$ and its filtration-reverse at $\infty$. The final structural consequence is that $\mathrm{Spec}(W)\to\mathrm{Spec}(\mathbb{S})$ is a torsor under $\mathbb{R}_{>0,\mathrm{Betti}}=\mathrm{Spec}(D(\mathbb{R}_{>0},\mathbb{S}))$, hence a canonical nontrivial $\mathbb{R}_{>0}$-torsor over $\mathrm{Spec}(\mathbb{Z})$.

Load-bearing premise

The proof of universality depends on an imported equivalence between sheaves on $\mathbb{R}$ with convolution and quasicoherent sheaves on an almost infinite-root version of $\mathbb{P}^1/\mathbb{G}_m$, which is only sketched and cited to unpublished work; if that equivalence or the claimed splitting at $0$ and $\infty$ fails, the universality of $W$ and the torsor statement would collapse.

Editorial extensions

If this is right

Wild sheaves on any space $X$ form a six-operation category $D(X,W)$ containing ordinary sheaves fully faithfully, so existing sheaf-theoretic arguments transfer to the enlarged setting.
For every continuous $f:X\to\mathbb{R}$ there is a canonical invertible wild sheaf $S(f)$, making any variety with potential $(X,f)$ into a wild Betti sheaf whose cohomology is a candidate for exponential or rapid-decay cohomology.
The Fourier transform is an equivalence on all sheaves, intertwines convolution with tensor product, and for convex functions on $\mathbb{R}$ sends $S(f)$ to $S(f^*)[-1]$, where $f^*$ is the Legendre transform.
If the universality claim is right, the whole theory lives over a canonical nontrivial $\mathbb{R}_{>0}$-torsor over $\mathrm{Spec}(\mathbb{Z})$, and after base change wild sheaves on $X$ become a twisted form of sheaves on $X\times\mathbb{R}_{>0}$.
The same construction applies with coefficients in complete almost modules over the Novikov ring or over any rank-one valuation ring, matching the motivic examples from almost mathematics.

Reading between the lines

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

If the ring-stack formulation from the outlook is taken literally, each realization of sheaf theory should produce its own canonical torsor; comparing the Betti torsor with the torsors attached to other realizations would constrain any lifting of exponential motives to a global category.
The nonconvex case, where the Fourier transform of $S(f)$ is noninvertible and its fibre at $0$ records contributions from local extrema, suggests using wild Betti sheaves as a microlocal bookkeeping device for oscillatory integrals, a use the paper does not develop.
A direct verification that the coaction sends $(\mathrm{Fil}_r M)_r$ to the sheaf with stalk $\mathrm{Fil}_{rt}M$ at $t$ would give an independent check of the torsor structure without passing through the cited stack equivalence.

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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

Dear X,

You should read this as two papers. Sections 2–4 give a clean, self-contained construction of the coefficient category W and the exponential local system, and the Fourier equivalence is proved from the definitions. That part is solid and worth having on record even though much of it recovers Tamarkin's enhanced sheaves, which Scholze credits explicitly. The vanishing pi!exp=0, the kernel computation for the Fourier transform, and the Legendre transform for convex functions are all done properly.

The novel part is Sections 5–6: the universality of W and the claim that Spec(W) is a nontrivial R>0-torsor over Spec(Z). These are genuinely new, but they rest on Proposition 5.1, a 'variant' of an unpublished theorem of Vaintrob, and the proof given is a sketch. The stress-test note is right about what is missing: you need more than an equivalence of categories. You need it to be symmetric monoidal, and you need the quotient Dqc(~P1,a/~Gm)/Dqc(~Gm/~Gm) to decompose as a product of two orthogonal factors, one W and one W with reversed filtration. The text asserts this decomposition without proof. If it is only a recollement, or if there are Homs between the two factors, then the universality statement overreaches. The torsor statement in Section 6 inherits the same dependency. This is not an internal contradiction; it is a load-bearing external premise.

I would not call the paper misleading. It is clearly a note, it credits Tamarkin and Vaintrob, and it flags the sketchiness implicitly. But as written, the headline claims go beyond what is demonstrated. A referee should be able to verify the Vaintrob input or ask for the full proof of Proposition 5.1 and the decomposition before signing off.

Who should read this? Anyone working in sheaf theory, symplectic geometry, or exponential motives. The W construction and the exp local system are useful tools even if the universality part is provisional. I'd bring it to reading group and would cite the clean construction.

Recommendation: send to refereeing, not desk reject, but the referee should focus on Proposition 5.1 and the decomposition. If those are not supplied, the paper should be revised to present Sections 5–6 as conditional.

Referee Report

3 major / 5 minor

Summary. The paper constructs an enlargement of Betti sheaves by working with coefficients in W, the symmetric monoidal stable infinity-category of completely and continuously R-filtered spectra (Definition 2.2). It defines wild sheaves D(X,W), constructs an invertible exponential local system exp on R whose fibres are the shifted units S(r) (Proposition 3.1), proves the key vanishing pi_! exp = 0 (Proposition 3.3), and establishes a Fourier equivalence for all real vector spaces (Theorem 4.1). The paper then claims in Section 5 that W is universal among coefficient categories admitting a nontrivial exponential local system inducing a Fourier equivalence, using Proposition 5.1, a variant of an unpublished theorem of Vaintrob. Section 6 claims that Spec(W) over Spec(S) is a canonical nontrivial R_{>0}-torsor, with Proposition 6.1 identifying W tensor_S W with D(R_{>0},W).

Significance. The construction in Sections 2–4 is explicit, self-contained, and convincing: the definition of W, the exponential local system, the vanishing of pi_! exp, and the Fourier equivalence are all proved directly from the definitions. If the universality and torsor claims hold, the paper would give a clean conceptual explanation of Tamarkin's enhanced sheaves and a genuinely surprising new structure over Spec(Z). The elementary parts of the paper are a solid contribution. However, the central novelty is not yet established at the level of rigor required for a journal publication, because it depends on Proposition 5.1 and on a decomposition statement that are only sketched and partly rely on an unpublished source.

major comments (3)

[§5, Proposition 5.1] Proposition 5.1 is the pivot of the universality argument, but it is stated as a symmetric monoidal equivalence and the proof given is only a sketch, with the citation referring to an unpublished note [Vai17]. The monoidal structure is essential: the universality statement concerns coefficient categories with a convolution structure, so the equivalence must preserve the symmetric monoidal structure, not merely the underlying category. This needs either a complete proof in the paper or a precise reference to a published or otherwise fully available source.
[§5, paragraph after Proposition 5.1] The assertion that the quotient Dqc(˜P1,a/˜Gm)/Dqc(˜Gm/˜Gm) decomposes as a product of two orthogonal factors, one equivalent to W and the other to W with reversed filtration, is load-bearing but unproved. If the two factors are only related by a recollement or have nontrivial Homs, a coefficient category could be linear over gluing data rather than over W, and the claimed universality would not follow. The paper should either prove the decomposition, including the claimed equivalence of each factor, or identify a published reference that contains it.
[§6, Proposition 6.1] The proof of Proposition 6.1, and hence the entire torsor statement, inherits the same unproved decomposition: after base change to W, the text asserts that 'removing this part, both sides decompose into two pieces' and concludes W ⊗_S W ≅ D(R_{>0},W). Since the paper notes that 'this equivalence can be proved directly,' such a direct proof should be supplied, or the missing decomposition should be established. As written, the torsor claim is conditional on the same external input as the universality claim.

minor comments (5)

[§2, Definition 2.2] The word 'introcued' in the first sentence of Section 2 should be 'introduced'.
[§6, Warning 6.2] The word 'paranthetical' should be 'parenthetical'.
[§5, first paragraph] The phrase 'cf. below' in the sketch of Proposition 5.1 is vague; the relevant equivalence could be stated explicitly in the sketch.
[§6, notation] The notation R_{>0,Betti} := Spec(D(R_{>0},S)) is used before the general categorical notion of Spec is introduced; a brief explanation of this notation would improve readability.
[References] Reference [Vai17] is unpublished and appears only as a URL; if it remains unpublished, the paper should include a more precise statement of the theorem being used or an appendix proving the needed variant.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the construction of W, the exponential local system, and the Fourier equivalence are proved from the definitions; universality and the torsor rely on an external, non-circular Vaintrob-style equivalence.

full rationale

I find no circular step. The paper's central construction is self-contained: W is defined independently as the category of completely and continuously R-filtered spectra (Definition 2.2); exp is constructed and shown unique in Proposition 3.1; the vanishing π_!exp = 0 and the Fourier equivalence are proved from these definitions in Proposition 3.3 and Theorem 4.1. The universality claim in Section 5 is not a restatement of W's definition: it uses Proposition 5.1, an externally cited variant of Vaintrob's Theorem 2, to identify (D(R,S), ⋆) with D_qc(˜P1,a/˜Gm), and then asserts a product decomposition. This is a substantive bridge, not a tautology. Section 6's torsor statement is likewise derived from the Fourier equivalence and the same decomposition, not from a fitted parameter or from a self-citation chain. The only author-related mention is an outlook remark about a norm developed by Clausen and the author; it is not load-bearing. The main caveat is that Proposition 5.1 is stated with only a sketch and the product decomposition is asserted rather than proved in detail; that is a rigor/completeness concern, not a circularity, so it does not increase the circularity score.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The central construction W is explicitly built from R-filtered spectra, and the exponential local system, the vanishing pi! exp = 0, and the Fourier equivalence are proved internally. The universality and torsor theorems depend on Vaintrob's equivalence (quoted as Proposition 5.1), on the six-functor formalism extension, and on standard six-functor computations. No numbers are fitted, and no uncontrolled entities are postulated; the paper introduces W by definition and verifies its properties.

assumptions (4)

domain assumption Vaintrob's equivalence (variant of Theorem 2 of [Vai17]): (D(R,S), sine star) is equivalent to Dqc(tilde P1,a / tilde Gm)
Invoked as Proposition 5.1 to identify the convolution category of sheaves on R with quasicoherent sheaves on the infinite root stack; the paper provides only a sketch and cites an unpublished source.
domain assumption Extension of the six-functor formalism to D(X,W) for locally compact Hausdorff spaces
Section 2 uses results of Heyer-Mann and Liu-Zheng/Gaitsgory-Rozenblyum to define f!, f*, etc.; used in the Fourier transform and duality statements.
standard math Standard six-functor computations: Kunneth formula, compactly supported cohomology of V*, and identification of pi! exp = 0
Assumed in the proof of Theorem 4.1 and Proposition 3.3; these are standard in any six-functor formalism.
domain assumption Theory of complete almost modules and valuation rings (Example 2.1)
Provides motivation and the link to the Novikov ring; not directly needed for the central proofs but part of the claimed framework.

invented entities (1)

W: the symmetric monoidal stable infinity-category of completely and continuously R-filtered spectra independent evidence
purpose: Coefficient category for wild Betti sheaves; supports the exponential local system exp and the Fourier equivalence; claimed universal solution.
W is explicitly constructed in Definition 2.2, its key properties (existence of exp, vanishing pi! exp = 0, Fourier equivalence) are proved, and it is identified with Tamarkin's enhanced sheaves, giving external corroboration.

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Cite this review

Pith. "Pith review of Wild Betti sheaves." pith.science (2026). https://pith.science/paper/HJ7UA32X

@misc{pith2026250524599,
  author       = {Pith},
  title        = {Pith review of: Wild Betti sheaves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HJ7UA32X}},
  note         = {Machine review of arXiv:2505.24599}
}

abstract

In this note, we consider the problem of constructing an enlargement of the category of Betti sheaves that supports an ``exponential local system'' on $\mathbb R$, and a Fourier equivalence defined on all sheaves. We show that there is a universal solution, recovering a construction of Tamarkin known also as ``enhanced sheaves''. The universality property implies that the category of coefficients of this theory is, in a suitable sense, a nontrivial $\mathbb R_{>0}$-torsor over $\mathrm{Spec}(\mathbb Z)$.

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Forward citations

Cited by 1 Pith paper

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

A remark on Continuous K-theory and Fourier-Sato transform
math.AT 2025-06 conditional novelty 6.0 of 10

For any conic closed set X in the dual space, the universal localizing invariant of sheaves with microsupport over X equals the compactly supported cohomology of X, and the Tamarkin-category version holds up to suspension.

Reference graph

Works this paper leans on

11 extracted references · 8 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

D'Agnolo and M

A. D'Agnolo and M. Kashiwara, Riemann- H ilbert correspondence for holonomic D -modules , Publ. Math. Inst. Hautes \'Etudes Sci. 123 (2016), 69--197

work page 2016

[2] [2]

Efimov, K -theory and localizing invariants of large categories , arXiv:2405.12169, 2024

A. Efimov, K -theory and localizing invariants of large categories , arXiv:2405.12169, 2024

arXiv 2024

[3] [3]

Fres\'an and P

J. Fres\'an and P. Jossen, Exponential motives, https://www.jossenpeter.ch/PdfDvi/ExpMot.pdf, 2025

work page 2025

[4] [4]

Heyer and L

C. Heyer and L. Mann, 6 - F unctor F ormalisms and S mooth R epresentations , arXiv:2410.13038, 2024

arXiv 2024

[5] [5]

Kuwagaki and B

T. Kuwagaki and B. Zhang, Almost mathematics, P ersistence module, and T amarkin category , arXiv:2503.15933, 2025

arXiv 2025

[6] [6]

Laumon, Transformation de F ourier, constantes d'\'equations fonctionnelles et conjecture de W eil , Inst

G. Laumon, Transformation de F ourier, constantes d'\'equations fonctionnelles et conjecture de W eil , Inst. Hautes \'Etudes Sci. Publ. Math. (1987), no. 65, 131--210

work page 1987

[7] [7]

, La transformation de F ourier g\'eom\'etrique et ses applications , Proceedings of the I nternational C ongress of M athematicians, V ol.\ I , II ( K yoto, 1990), Math. Soc. Japan, Tokyo, 1991, pp. 437--445

work page 1990

[8] [8]

, Transformation de F ourier homog\`ene , Bull. Soc. Math. France 131 (2003), no. 4, 527--551

work page 2003