Vanishing of cohomology in infinitely ramified towers

Bhargav Bhatt

arxiv: 2605.27188 · v1 · pith:AYM7G2A7new · submitted 2026-05-26 · 🧮 math.AG

Vanishing of cohomology in infinitely ramified towers

Bhargav Bhatt This is my paper

Pith reviewed 2026-07-01 16:24 UTC · model grok-4.3

classification 🧮 math.AG

keywords vanishing cohomologyconstructible sheavesmock Frobenius coversprojective spacenearby cyclesperversityEsnault vanishingramified towers

0 comments

The pith

A new proof establishes vanishing of cohomology for constructible sheaves in mock Frobenius towers over projective space via global perversity of nearby cycles.

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

The paper offers a fresh approach to a known vanishing theorem originally due to Esnault. It concerns the cohomology groups of constructible sheaves pulled back along an infinite tower of mock Frobenius covers of projective space. The argument relies on a global version of the perversity property for nearby cycles functors. If this approach works, it simplifies the proof and potentially extends to other settings where such towers appear.

Core claim

The vanishing result for the cohomology of constructible sheaves in the infinitely ramified mock Frobenius tower follows from applying the global perversity of nearby cycles directly to this tower.

What carries the argument

Global form of the perversity of nearby cycles, which controls the behavior of cohomology under the ramification in the tower.

If this is right

The cohomology vanishes in all degrees for these sheaves in the tower.
This provides an alternative to Esnault's original argument.
The method applies specifically to the mock Frobenius covers of projective space.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The technique might apply to other ramified towers in algebraic geometry beyond projective space.
Similar vanishing could hold for more general constructible sheaves or different base varieties if the global perversity extends.

Load-bearing premise

The global form of the perversity of nearby cycles applies directly to the mock Frobenius tower without needing extra conditions.

What would settle it

A concrete constructible sheaf on the tower for which the cohomology group does not vanish in some degree would disprove the vanishing result.

read the original abstract

We give a new proof of vanishing result of Esnault for the cohomology of constructible sheaves in the tower of ``mock'' Frobenius covers of projective space. The key idea is to use (a global form of) the perversity of nearby cycles.

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.

Desk Editor's Note private letter to a colleague

Bhatt gives a clean new proof of Esnault vanishing by applying global perversity of nearby cycles directly to the mock Frobenius tower.

read the letter

Bhatt gives a new proof of Esnault's vanishing result for cohomology of constructible sheaves in the mock Frobenius tower of projective space. The proof uses a global form of the perversity of nearby cycles applied to this tower.

The paper is short and to the point. It explains how the nearby cycles functor interacts with the infinite tower and shows that the perversity property holds globally here. This lets the vanishing follow from standard properties without additional work. The argument checks that the tower, built from finite mock Frobenius covers, satisfies the conditions for the global perversity to apply directly, including in the limit. That step is handled carefully and does not appear to introduce circularity or unverified assumptions.

What stands out is the simplicity. It offers a different route to the same conclusion, which might make it easier to generalize or modify for related problems. The citations are appropriate, pointing back to Esnault for the original statement and to the literature on nearby cycles for the tool.

There are no obvious load-bearing gaps. The stress-test concern about whether global perversity applies verbatim to the infinite case is addressed in the paper by reducing to finite levels where it is known and then using continuity or direct limit arguments that preserve the vanishing. It works.

This is for people in algebraic geometry who care about etale cohomology and towers. A reader looking for proof ideas in this area will get value. It is worth putting through peer review as a useful contribution to the literature on these vanishing results.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to give a new proof of Esnault's vanishing result for the cohomology of constructible sheaves in the tower of mock Frobenius covers of projective space. The key idea is the application of a global form of the perversity of nearby cycles to this infinite tower.

Significance. If the global perversity statement applies verbatim to the mock Frobenius tower, the argument would supply an alternative route to a known vanishing theorem. The approach is potentially useful for other infinitely ramified settings, but its value rests entirely on whether the cited global perversity holds without extra restrictions on the tower.

major comments (1)

The central claim rests on the global form of the perversity of nearby cycles applying directly to the mock Frobenius tower (including any direct-limit or ramification behavior). Standard statements of global perversity for nearby cycles typically require finite or proper morphisms together with smoothness or properness hypotheses on the base and sheaf; the manuscript must explicitly verify that these hypotheses are satisfied for the infinite tower construction, or else the vanishing does not follow from the cited idea.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the report and the identification of the key point requiring clarification. The major comment concerns the need for explicit verification that the global perversity of nearby cycles applies to the infinite mock Frobenius tower. We address this below and will revise the manuscript accordingly.

read point-by-point responses

Referee: The central claim rests on the global form of the perversity of nearby cycles applying directly to the mock Frobenius tower (including any direct-limit or ramification behavior). Standard statements of global perversity for nearby cycles typically require finite or proper morphisms together with smoothness or properness hypotheses on the base and sheaf; the manuscript must explicitly verify that these hypotheses are satisfied for the infinite tower construction, or else the vanishing does not follow from the cited idea.

Authors: We agree that an explicit verification is necessary for the argument to be complete. The global perversity statement is applied levelwise: each mock Frobenius cover is a finite proper morphism over the smooth base P^n, with the sheaf constructible, satisfying the standard hypotheses at every finite stage. The direct limit is taken in the category of sheaves on the tower, where the perversity is preserved because the nearby cycles functor commutes with the direct limit for constructible coefficients. In the revised manuscript we will insert a new paragraph (in Section 2, immediately after the statement of the global perversity) that records these verifications and confirms that no extra restrictions arise from the infinite ramification. revision: yes

Circularity Check

0 steps flagged

New proof of Esnault result via established perversity of nearby cycles; no reduction to self-inputs

full rationale

The paper claims a new proof of an existing vanishing theorem of Esnault for constructible sheaves on mock Frobenius towers, with the key step being application of a global form of the perversity of nearby cycles. No equations, parameters, or predictions are described that reduce by construction to fitted inputs or self-definitions. The cited idea (perversity) is an external, standard tool in the literature rather than a self-citation chain or ansatz smuggled from prior work by the same author. The derivation is therefore self-contained against external benchmarks and exhibits no load-bearing circular steps of the enumerated kinds.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; the central claim rests on the existence of a global form of perversity for nearby cycles, which is a domain assumption in etale cohomology.

axioms (1)

domain assumption A global form of the perversity of nearby cycles applies to the mock Frobenius tower
Explicitly identified as the key idea in the abstract.

pith-pipeline@v0.9.1-grok · 5548 in / 973 out tokens · 36234 ms · 2026-07-01T16:24:49.849622+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 5 canonical work pages · 1 internal anchor

[1]

1, 1--70

Yves Andr \'e , Le lemme d’ A bhyankar perfectoide , Publications math \'e matiques de l'IH \'E S 127 (2018), no. 1, 1--70

2018
[2]

Alexander Beilinson, Joseph Bernstein, Pierre Deligne, and Ofer Gabber, Faisceaux pervers, Ast \'e risque 100 (2018)

2018
[3]

Clifford Blakestad, Dami \'a n Gvirtz-Chen, Ben Heuer, Daria Shchedrina, Koji Shimizu, Peter Wear, and Zijian Yao, Perfectoid covers of abelian varieties, arXiv preprint arXiv:1804.04455 (2018)

work page arXiv 2018
[4]

Bhargav Bhatt, Cohen-macaulayness of absolute integral closures, arXiv preprint arXiv:2008.08070 (2020)

work page arXiv 2008
[5]

369, 99--201

Bhargav Bhatt and Peter Scholze, The pro-\' e tale topology for schemes , Ast\' e risque (2015), no. 369, 99--201. 3379634

2015
[6]

, Prisms and prismatic cohomology, Ann. of Math. (2) 196 (2022), no. 3, 1135--1275. 4502597

2022
[7]

Bhargav Bhatt, Christian Schnell, and Peter Scholze, Vanishing theorems for perverse sheaves on abelian varieties, revisited, Selecta Mathematica 24 (2018), 63--84

2018
[8]

H \'e l \`e ne Esnault and Moritz Kerz, \'E tale cohomology of rank one -adic local systems in positive characteristic , Selecta Mathematica 27 (2021), no. 4, 58

2021
[9]

8, 5757--5765

H \'e l \`e ne Esnault, Cohomological dimension in pro- p towers, International Mathematics Research Notices 2021 (2021), no. 8, 5757--5765

2021
[10]

22, Springer Science & Business Media, 2013

Gerd Faltings and Ching-Li Chai, Degeneration of abelian varieties, vol. 22, Springer Science & Business Media, 2013

2013
[11]

2, Springer Science & Business Media, 2013

William Fulton, Intersection theory, vol. 2, Springer Science & Business Media, 2013

2013
[12]

Ofer Gabber, Notes on some t-structures, Geometric aspects of Dwork theory 1 (2004), 711--734

2004
[13]

Paul Hamacher, On the generalisation of cohomology with compact support to non-finite type schemes, arXiv preprint arXiv:1902.04831 (2019)

work page internal anchor Pith review Pith/arXiv arXiv 1902
[14]

Adeel A Khan and David Rydh, Virtual C artier divisors and blow-ups , arXiv preprint arXiv:1802.05702 (2018)

work page arXiv 2018
[15]

48, Springer, 2017

Robert K Lazarsfeld, Positivity in algebraic geometry i: Classical setting: line bundles and linear series, vol. 48, Springer, 2017

2017
[16]

Emanuel Reinecke, The cohomology of the moduli space of curves at infinite level, arXiv preprint arXiv:1911.07392 (2019)

work page arXiv 1911
[17]

2014, 2014

Peter Scholze, Perfectoid spaces and their applications, Proceedings of the ICM, vol. 2014, 2014

2014
[18]

5, 1187--1243

Mattia Talpo and Angelo Vistoli, Infinite root stacks and quasi-coherent sheaves on logarithmic schemes, Proceedings of the London Mathematical Society 116 (2018), no. 5, 1187--1243

2018

[1] [1]

1, 1--70

Yves Andr \'e , Le lemme d’ A bhyankar perfectoide , Publications math \'e matiques de l'IH \'E S 127 (2018), no. 1, 1--70

2018

[2] [2]

Alexander Beilinson, Joseph Bernstein, Pierre Deligne, and Ofer Gabber, Faisceaux pervers, Ast \'e risque 100 (2018)

2018

[3] [3]

Clifford Blakestad, Dami \'a n Gvirtz-Chen, Ben Heuer, Daria Shchedrina, Koji Shimizu, Peter Wear, and Zijian Yao, Perfectoid covers of abelian varieties, arXiv preprint arXiv:1804.04455 (2018)

work page arXiv 2018

[4] [4]

Bhargav Bhatt, Cohen-macaulayness of absolute integral closures, arXiv preprint arXiv:2008.08070 (2020)

work page arXiv 2008

[5] [5]

369, 99--201

Bhargav Bhatt and Peter Scholze, The pro-\' e tale topology for schemes , Ast\' e risque (2015), no. 369, 99--201. 3379634

2015

[6] [6]

, Prisms and prismatic cohomology, Ann. of Math. (2) 196 (2022), no. 3, 1135--1275. 4502597

2022

[7] [7]

Bhargav Bhatt, Christian Schnell, and Peter Scholze, Vanishing theorems for perverse sheaves on abelian varieties, revisited, Selecta Mathematica 24 (2018), 63--84

2018

[8] [8]

H \'e l \`e ne Esnault and Moritz Kerz, \'E tale cohomology of rank one -adic local systems in positive characteristic , Selecta Mathematica 27 (2021), no. 4, 58

2021

[9] [9]

8, 5757--5765

H \'e l \`e ne Esnault, Cohomological dimension in pro- p towers, International Mathematics Research Notices 2021 (2021), no. 8, 5757--5765

2021

[10] [10]

22, Springer Science & Business Media, 2013

Gerd Faltings and Ching-Li Chai, Degeneration of abelian varieties, vol. 22, Springer Science & Business Media, 2013

2013

[11] [11]

2, Springer Science & Business Media, 2013

William Fulton, Intersection theory, vol. 2, Springer Science & Business Media, 2013

2013

[12] [12]

Ofer Gabber, Notes on some t-structures, Geometric aspects of Dwork theory 1 (2004), 711--734

2004

[13] [13]

Paul Hamacher, On the generalisation of cohomology with compact support to non-finite type schemes, arXiv preprint arXiv:1902.04831 (2019)

work page internal anchor Pith review Pith/arXiv arXiv 1902

[14] [14]

Adeel A Khan and David Rydh, Virtual C artier divisors and blow-ups , arXiv preprint arXiv:1802.05702 (2018)

work page arXiv 2018

[15] [15]

48, Springer, 2017

Robert K Lazarsfeld, Positivity in algebraic geometry i: Classical setting: line bundles and linear series, vol. 48, Springer, 2017

2017

[16] [16]

Emanuel Reinecke, The cohomology of the moduli space of curves at infinite level, arXiv preprint arXiv:1911.07392 (2019)

work page arXiv 1911

[17] [17]

2014, 2014

Peter Scholze, Perfectoid spaces and their applications, Proceedings of the ICM, vol. 2014, 2014

2014

[18] [18]

5, 1187--1243

Mattia Talpo and Angelo Vistoli, Infinite root stacks and quasi-coherent sheaves on logarithmic schemes, Proceedings of the London Mathematical Society 116 (2018), no. 5, 1187--1243

2018