REVIEW 1 major objections 17 references
Frobenius--Witt cotangent complexes
T0 review · 1 major / 0 minor · reviewed 2026-06-30 · grok-4.3
Pith's one-line read Frobenius-Witt cotangent complexes detect regularity of noetherian local rings in derived form.
desk verdict The paper defines a new Frobenius-Witt cotangent complex and ties it to regularity of noetherian rings via perfectoid computations, but the central claim stands or falls on those specific calculations. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The Frobenius-Witt cotangent complex, a chain complex that serves as the derived analogue of the module of Frobenius-Witt differentials and simultaneously as an arithmetic lift of the usual cotangent complex.
What would settle it
Exhibit a regular noetherian local ring for which the Frobenius-Witt cotangent complex fails to be concentrated in degree zero or fails to match the expected cotangent complex of its perfection.
Extended reading notes
Core claim
The Frobenius-Witt cotangent complex of a noetherian local ring is quasi-isomorphic to the cotangent complex of its perfection or satisfies a regularity criterion if and only if the ring is regular; the proof proceeds by reducing to computations in the perfectoid case and then using the resulting identification to control the deformation theory of delta structures on the ring.
Load-bearing premise
Explicit computations of the complex on perfectoid rings correctly capture the expected vanishing behavior.
Editorial extensions
If this is right
- Regularity of a noetherian local ring can be read off from the acyclicity of its Frobenius-Witt cotangent complex.
- Delta structures on rings admit a deformation theory controlled by the cohomology of this complex.
- The same complex supplies an arithmetic version of the usual cotangent complex that is compatible with Frobenius actions.
- The criterion extends Saito's classical test to the derived category while remaining valid for rings in mixed characteristic.
Reading between the lines
- The construction may furnish new obstructions to regularity that are invisible to ordinary cotangent complexes.
- It suggests a route to lift classical regularity results to derived schemes or stacks equipped with Frobenius lifts.
- Computations in the perfectoid case could be reused to test regularity for rings arising in p-adic geometry.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper introduces the Frobenius--Witt cotangent complex as a derived variant of the module of Frobenius--Witt differentials (in the sense of T. Saito) and as an arithmetic analogue of the cotangent complex. It establishes a relationship between these complexes and the regularity of noetherian local rings, framed as a derived version of Saito's regularity criterion; the argument relies on explicit computations in the perfectoid case. The paper also develops the deformation theory of delta structures via these complexes.
Significance. If the perfectoid computations are accurate and the reduction to general noetherian local rings is valid, the result supplies a derived arithmetic regularity criterion with potential utility for deformation problems and p-adic geometry. The introduction of the new complex and its application to delta structures would constitute a substantive contribution to arithmetic algebraic geometry.
major comments (1)
- [Abstract (and the section containing the main regularity theorem)] The central claim (derived Saito regularity criterion) rests on the computations of Frobenius--Witt cotangent complexes for perfectoid rings, as stated in the abstract. These computations form a load-bearing step; the manuscript must supply explicit, verifiable details of the calculation (including any spectral sequences or base-change arguments used) so that the reduction from the perfectoid case to arbitrary noetherian local rings can be checked.
Simulated Author's Rebuttal
We thank the referee for their thorough review and for identifying the need for greater explicitness in the key computational step. We agree that the perfectoid calculations are load-bearing for the derived Saito criterion and will revise the manuscript to address this.
read point-by-point responses
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Referee: [Abstract (and the section containing the main regularity theorem)] The central claim (derived Saito regularity criterion) rests on the computations of Frobenius--Witt cotangent complexes for perfectoid rings, as stated in the abstract. These computations form a load-bearing step; the manuscript must supply explicit, verifiable details of the calculation (including any spectral sequences or base-change arguments used) so that the reduction from the perfectoid case to arbitrary noetherian local rings can be checked.
Authors: We agree that the computations for perfectoid rings constitute the central technical step and that the current presentation would benefit from additional explicitness to allow verification of the reduction argument. In the revised manuscript we will expand the relevant section (and update the abstract if necessary) with a self-contained, step-by-step account of the calculation. This will include: (i) the explicit identification of the Frobenius--Witt cotangent complex in the perfectoid case, (ii) the spectral sequences used to compute its homology, and (iii) the base-change and deformation arguments that reduce the general noetherian local case to the perfectoid situation. We believe these additions will render the proof fully checkable while preserving the overall structure of the argument. revision: yes
Circularity Check
No circularity: derivation relies on independent computations in special cases
full rationale
The paper introduces the Frobenius--Witt cotangent complex as a new derived object and proves a relationship to regularity of noetherian local rings via explicit computations in the perfectoid case. No equations, self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations are visible in the provided text. The central claim is presented as following from case-by-case calculations rather than reducing to its own inputs by construction. The derivation chain is therefore self-contained against external benchmarks and receives the default non-finding.
Assumptions & free parameters
Cite this review
Pith. "Pith review of Frobenius--Witt cotangent complexes." pith.science (2026). https://pith.science/paper/CIVRD7QK
@misc{pith2026260514803,
author = {Pith},
title = {Pith review of: Frobenius--Witt cotangent complexes},
year = {2026},
howpublished = {\url{https://pith.science/paper/CIVRD7QK}},
note = {Machine review of arXiv:2605.14803}
}
read the original abstract
We introduce the notion of the Frobenius--Witt cotangent complex, which can be considered as a derived variant of the module of Frobenius--Witt differentials defined by T. Saito. This new object also can be seen as an arithmetic variant of the notion of cotangent complex. We explain the suitability of these two viewpoints through a series of propositions. Furthermore, we establish a relationship between Frobenius--Witt cotangent complexes and the regularity of noetherian local rings, which can be considered as a derived variant of Saito's regularity criterion. This proof relies heavily on computations of Frobenius--Witt cotangent complexes in the case of perfectoid rings. We also study the deformation theory of delta structures using Frobenius--Witt cotangent complexes.
Reference graph
Works this paper leans on
-
[1]
B. Bhatt. p-adic cohomology theories via stacks . A conference on the occasion of Takeshi Saito's 60th birthday (2021), https://youtu.be/v2Jfk-NTjp4?si=5qSYokoVrr_4RhJO
work page 2021
- [2]
-
[3]
The prismatization of $p$-adic formal schemes
B. Bhatt, J. Lurie. The prismatization of p-adic formal schemes . arXiv:2201.06124. (2022)
work page Pith review arXiv 2022
- [4]
- [5]
- [6]
-
[7]
K. Česnavičius, P. Scholze. Purity for flat cohomology . Ann. of Math. (2) 199 (2024), no. 1, 51--180. MR4681144
work page 2024
-
[8]
R. Ishizuka, K. Nakazato. Prismatic Kunz's theorem . J. Algebra 693 (2026), 732--769; MR5022261
work page 2026
Show all 17 references
-
[9]
S. B. Iyengar. Andr\'e-Quillen homology of commutative algebras . Interactions between homotopy theory and algebra (2007), 203--234, Contemp. Math., 436, Amer. Math. Soc., Providence, RI, ; MR2355775
2007
-
[10]
J. Lurie. Higher topos theory . Ann. of Math. Stud., 170 Princeton University Press, Princeton, NJ, 2009. xviii+925 pp. ISBN:978-0-691-14049-0 ISBN:0-691-14049-9
2009
-
[11]
J. Lurie. Higher algebra . https://people.math.harvard.edu/ lurie/papers/HA.pdf, sep 2017
2017
-
[12]
J. Lurie. Spectral algebraic geometry . https://www.math.ias.edu/ lurie/papers/SAG-rootfile.pdf, feb 2018
2018
-
[13]
J. Lurie. Kerodon . https://kerodon.net/, apr 2026
2026
-
[14]
Z. Mao. Revisiting derived crystalline cohomology . Bull. Soc. Math. France 152 (2024), no. 4, 659–784. MR4851406
2024
-
[15]
T. Saito. Frobenius--Witt differentials and regularity . Algebra Number Theory 16 (2022), no. 2, 369–391. MR4412577
2022
-
[16]
Stacks Project
Stacks Project Authors. Stacks Project . https://stacks.math.columbia.edu/
-
[17]
Takeuchi
R. Takeuchi. A criterion for log regularity via log Frobenius-Witt differentials . arXiv:2604.17394. (2026)
2026 arXiv
Reviewed June 30, 2026 · model on record in the stance chip above.
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