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arxiv: 2502.11055 · v2 · submitted 2025-02-16 · 🧮 math.AG · math.NT

Bounding ramification with coherent sheaves

Haoyu Hu , Jean-Baptiste Teyssier This is my paper

Pith reviewed 2026-05-23 03:01 UTC · model grok-4.3

classification 🧮 math.AG math.NT
keywords coherent sheavesétale sheaveslogarithmic conductorsramificationfinite pushforwardsalgebraic geometryétale cohomology
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The pith

Given a coherent sheaf E on X, a category of étale sheaf complexes has logarithmic conductors bounded by E and is compatible with finite pushforwards.

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

The paper constructs a category of complexes of étale sheaves on a scheme X of finite type over a perfect field whose logarithmic conductors are bounded by a given coherent sheaf E. It examines how this category interacts with finite push-forward functors. The construction links coherent sheaf data directly to bounds on ramification for étale complexes. A sympathetic reader would care because the bounds supply a mechanism to control ramification behavior under morphisms using only coherent input.

Core claim

We introduce the category of complexes of étale sheaves on X with logarithmic conductors bounded by E and study its compatibilities with finite push-forward.

What carries the argument

The category of complexes of étale sheaves whose logarithmic conductors are bounded by a coherent sheaf E, which organizes the ramification bounds and their pushforward behavior.

Load-bearing premise

Logarithmic conductors for étale sheaf complexes can be defined so that an arbitrary coherent sheaf E bounds them and the resulting collection forms a category whose pushforward compatibilities can be studied.

What would settle it

An explicit finite pushforward of a complex whose conductor is bounded by E that produces a complex whose conductor exceeds the image of E would falsify the claimed compatibilities.

read the original abstract

Given a coherent sheaf E on a scheme of finite type X over a perfect field, we introduce a category of complexes of \'etale sheaves on X with logarithmic conductors bounded by E and study its compatibilities with finite push-forward.

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

0 major / 2 minor

Summary. The manuscript defines, for a coherent sheaf E on a scheme X of finite type over a perfect field, a category whose objects are complexes of étale sheaves on X whose logarithmic conductors are bounded by E. It then studies the compatibilities of this category under finite push-forward functors.

Significance. If the definition is rigorous and the stated compatibilities hold, the construction supplies a coherent-sheaf mechanism for controlling ramification in the étale setting. This is a potentially useful addition to the toolkit of ramification theory and could facilitate comparisons between coherent and étale data on schemes over perfect fields. The paper introduces a new category together with its functorial properties; these are the primary contributions.

minor comments (2)
  1. The abstract states the definition and the push-forward study but does not indicate where in the text the precise definition of 'logarithmic conductor bounded by E' is given or how the category is shown to be abelian (or triangulated). Adding an explicit reference to the relevant section or proposition would improve readability.
  2. Notation for the new category (e.g., a symbol such as D^b_{E}(X) or similar) is not mentioned in the abstract; introducing it early would help readers track the object throughout the paper.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, noting its potential as a useful addition to ramification theory, and for recommending minor revision. No specific major comments or concerns are listed in the report.

Circularity Check

0 steps flagged

No significant circularity; definitional construction of a category

full rationale

The paper introduces a category of complexes of étale sheaves whose logarithmic conductors are bounded by a given coherent sheaf E on a scheme of finite type over a perfect field, then studies finite-pushforward compatibilities. This is presented as a direct definition and construction rather than any derivation, prediction, or first-principles result that reduces to its own inputs. No equations, fitted parameters, self-citation chains, or uniqueness theorems are invoked in the abstract or description that would create circularity. The weakest assumption is precisely the definitional step undertaken by the authors. Per the hard rules, no circularity can be claimed without quoting specific reductions from the paper's equations or self-citations, none of which appear here. This is the common case of a self-contained definitional paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central contribution rests on the existence of a well-defined notion of logarithmic conductor for étale complexes that can be bounded by an arbitrary coherent sheaf; this is taken as given in the setup.

axioms (1)
  • domain assumption X is a scheme of finite type over a perfect field.
    Explicitly stated as the ambient setting in the abstract.
invented entities (1)
  • Category of complexes of étale sheaves with logarithmic conductors bounded by E no independent evidence
    purpose: To organize étale complexes whose ramification is controlled by the coherent sheaf E
    Newly introduced in the paper as the main object of study.

pith-pipeline@v0.9.0 · 5545 in / 1344 out tokens · 38116 ms · 2026-05-23T03:01:29.332263+00:00 · methodology

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Works this paper leans on

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