Infinitesimal derived foliations

Bertrand To\"en; Gabriele Vezzosi

Infinitesimal derived foliations

T0 review · 1 major / 0 minor · reviewed 2026-05-24 · grok-4.3

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Pith's one-line read Infinitesimal derived foliations are defined to match classical infinitesimal cohomology and obey formal integrability.

desk verdict Toën and Vezzosi introduce infinitesimal derived foliations as an extension of their prior derived foliations work and link the new notion to classical infinitesimal cohomology plus formal integrability. read the letter →

arxiv 2305.13010 v1 submitted 2023-05-22 math.AG

Bertrand Toën , Gabriele Vezzosi This is my paper · ORCID

classification math.AG

keywords infinitesimalderivedfoliationsalgebraicgeometrycohomologyformalintegrability

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 authors introduce a new structure called an infinitesimal derived foliation inside the setting of derived algebraic geometry. They establish a direct relation between this structure and the existing notion of infinitesimal cohomology. The definition is shown to satisfy formal integrability properties, allowing local data to extend in a controlled way. Brief comparisons are offered to an earlier version of derived foliations introduced by the same authors. The work supplies a refined object for handling infinitesimal geometric data in algebraic settings.

What carries the argument

Infinitesimal derived foliation, a new object in derived algebraic geometry whose definition encodes data that aligns with infinitesimal cohomology and supports formal integrability.

What would settle it

An explicit example of an infinitesimal derived foliation, constructed in a concrete derived scheme, that fails to correspond to any class in infinitesimal cohomology would show the claimed relation does not hold.

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Extended reading notes

Core claim

We introduce a notion of infinitesimal derived foliation. We prove it is related to the classical notion of infinitesimal cohomology, and satisfies some formal integrability properties. We also provide some hints on how infinitesimal derived foliations compare to our previous notion of derived foliations.

Load-bearing premise

The notion of infinitesimal derived foliation can be defined inside derived algebraic geometry so that the stated link to infinitesimal cohomology and the formal integrability properties both hold.

Editorial extensions

If this is right

Infinitesimal derived foliations align with the classical theory of infinitesimal cohomology.
These foliations satisfy formal integrability, so local solutions extend under the stated conditions.
The new objects admit direct comparison with the authors' earlier derived foliations.

Reading between the lines

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

The definition could be used to study deformations of geometric structures where classical foliations are insufficient.
Explicit computations on simple derived schemes might reveal how the integrability condition behaves in practice.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper defines a notion of infinitesimal derived foliation and proves its relation to classical infinitesimal cohomology along with some formal integrability properties. It also gives brief comparisons to the authors' earlier derived foliations. This sits squarely inside their ongoing program in derived algebraic geometry and adds one more technical object for handling certain cohomology questions in that setting. The main concrete advance is the new definition itself together with the stated connections; if those hold up under the usual derived-category hypotheses, the work supplies a usable extension rather than a wholesale shift. The comparison hints to previous derived foliations are useful for readers already inside the program. The abstract is short, so the real test is whether the definition is set up cleanly without extra site or boundedness conditions that are not spelled out. No internal contradictions jump out from the claims as written, and the relation to infinitesimal cohomology looks like an external comparison rather than a circular fit. This is aimed at people already working in derived algebraic geometry and foliation theory. Specialists tracking the authors' sequence of papers will get the most out of it; outsiders will find it narrow. The paper deserves peer review because the claims are specific enough to check and the authors have a track record of delivering rigorous definitions in this area.

Referee Report

1 major / 0 minor

Summary. The paper introduces a notion of infinitesimal derived foliation in derived algebraic geometry. It claims to prove that this notion is related to the classical notion of infinitesimal cohomology and that it satisfies some formal integrability properties. It also provides hints comparing infinitesimal derived foliations to the authors' previous notion of derived foliations.

Significance. If the definition is rigorously well-posed in existing derived-algebraic-geometry frameworks and the claimed relations and integrability properties hold without additional unstated hypotheses, the work would introduce a new concept potentially useful for bridging classical infinitesimal cohomology with derived settings and for studying integrability questions.

major comments (1)

The central claims rest on the rigorous definition of 'infinitesimal derived foliation' and the subsequent proofs of its relation to infinitesimal cohomology and formal integrability. The provided abstract and reader's assessment indicate that no explicit derivations, definitions, or verification steps are available to check these assertions against the paper's own mathematics.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review. The major comment concerns the availability of explicit definitions and proofs for verification. We address this directly below, based on the content of the full manuscript.

read point-by-point responses

Referee: The central claims rest on the rigorous definition of 'infinitesimal derived foliation' and the subsequent proofs of its relation to infinitesimal cohomology and formal integrability. The provided abstract and reader's assessment indicate that no explicit derivations, definitions, or verification steps are available to check these assertions against the paper's own mathematics.

Authors: The full manuscript contains an explicit definition of infinitesimal derived foliation (developed in the derived algebraic geometry setting using appropriate simplicial or dg-objects). The relation to classical infinitesimal cohomology is proven via a comparison theorem that identifies the cohomology of the foliation with the classical infinitesimal cohomology under the natural forgetful functor. Formal integrability is established by showing that the obstruction classes vanish in the appropriate derived deformation complex. These constructions and proofs are carried out in detail in the body of the paper (following the introduction and preliminary sections on derived foliations), using standard references for the ambient framework. The abstract is intentionally concise; the complete text supplies the required derivations and verifications. revision: no

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; new definition compared to prior work only as hints

full rationale

The paper introduces a fresh definition of infinitesimal derived foliation and claims external relations to classical infinitesimal cohomology plus formal integrability. The only self-reference is a brief comparison to the authors' earlier derived-foliation notion, presented explicitly as 'hints' rather than a load-bearing premise. No equation or theorem reduces a claimed result to a fitted parameter or to a self-citation chain; the central claims rest on the new definition inside existing derived-algebraic-geometry frameworks. This yields a minor self-citation score but leaves the derivation self-contained.

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

Only the abstract is available, so the ledger records the introduction of a new mathematical notion as the primary addition; no free parameters, background axioms, or invented physical entities are identifiable from the given text.

invented entities (1)

infinitesimal derived foliation
purpose: New mathematical structure capturing foliation-like behavior in the derived setting
Explicitly introduced in the abstract as the central new object.

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Cite this review

Pith. "Pith review of Infinitesimal derived foliations." pith.science (2026). https://pith.science/paper/2305.13010

@misc{pith2026230513010,
  author       = {Pith},
  title        = {Pith review of: Infinitesimal derived foliations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2305.13010}},
  note         = {Machine review of arXiv:2305.13010}
}

read the original abstract

We introduce a notion of \emph{infinitesimal derived foliation}. We prove it is related to the classical notion of infinitesimal cohomology, and satisfies some formal integrability properties. We also provide some hints on how infinitesimal derived foliations compare to our previous notion of derived foliations.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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Pith tools

Reviewed May 24, 2026 · model on record in the stance chip above.

[1] [1]

Pd operads and explicit partition lie algebras

Lukas Brantner, Ricardo Campos, and Joost Nuiten. Pd operads and explicit partition lie algebras. Preprint arXiv:2104.03870 , 2021

work page arXiv 2021

[2] [2]

Axiomatic homotopy theory for operads

Clemens Berger and Ieke Moerdijk. Axiomatic homotopy theory for operads. Comment. Math. Helv. , 78(4):805--831, 2003

work page 2003

[3] [3]

Shifted P oisson structures and deformation quantization

Damien Calaque, Tony Pantev, Bertrand To\" e n, Michel Vaqui\' e , and Gabriele Vezzosi. Shifted P oisson structures and deformation quantization. J. Topol. , 10(2):483--584, 2017

work page 2017

[4] [4]

Foliations and inseparable morphisms

Torsten Ekedahl. Foliations and inseparable morphisms. In Algebraic geometry, B owdoin, 1985 ( B runswick, M aine, 1985) , volume 46 of Proc. Sympos. Pure Math. , pages 139--149. Amer. Math. Soc., Providence, RI, 1987

work page 1985

[5] [5]

private communication

Jiaqi Fu. private communication

work page

[6] [6]

Deformations of a morphism along a foliation and applications

Yoichi Miyaoka. Deformations of a morphism along a foliation and applications. In Algebraic geometry, B owdoin, 1985 ( B runswick, M aine, 1985) , volume 46 of Proc. Sympos. Pure Math. , pages 245--268. Amer. Math. Soc., Providence, RI, 1987

work page 1985

[7] [7]

A note on linear stacks, 2021

Ludovic Monier. A note on linear stacks, 2021

work page 2021

[8] [8]

A universal H ochschild- K ostant- R osenberg theorem

Tasos Moulinos, Marco Robalo, and Bertrand To\" e n. A universal H ochschild- K ostant- R osenberg theorem. Geom. Topol. , 26(2):777--874, 2022

work page 2022