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arxiv: 2305.13010 · v1 · submitted 2023-05-22 · 🧮 math.AG

Infinitesimal derived foliations

Bertrand To\"en , Gabriele Vezzosi This is my paper

Pith reviewed 2026-05-24 09:24 UTC · model grok-4.3

classification 🧮 math.AG
keywords infinitesimal derived foliationsderived algebraic geometryinfinitesimal cohomologyformal integrabilityderived foliations
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The pith

Infinitesimal derived foliations are defined to match classical infinitesimal cohomology and obey formal integrability.

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

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.

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.

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.

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.

Where Pith is reading between the lines

These are 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.

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.

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.

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

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)
  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

  1. 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

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.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 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 no independent evidence
    purpose: New mathematical structure capturing foliation-like behavior in the derived setting
    Explicitly introduced in the abstract as the central new object.

pith-pipeline@v0.9.0 · 5552 in / 1160 out tokens · 46651 ms · 2026-05-24T09:24:20.120845+00:00 · methodology

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Reference graph

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