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arxiv: 2605.20858 · v1 · pith:AEXVSRFQnew · submitted 2026-05-20 · 🧮 math.AG

A construction of tame sheaves and tame de Rham--Witt cohomology

Alberto Merici , Kay R\"ulling , Shuji Saito This is my paper

Pith reviewed 2026-05-21 02:30 UTC · model grok-4.3

classification 🧮 math.AG
keywords tame sheavestame sitede Rham-Witt cohomologysyntomic cohomologyNygaard filtrationlog polesreciprocity sheaves
2
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The pith

A general construction produces global tame sheaves by gluing local tame sections onto an étale sheaf on X.

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

The paper introduces an algebraic version of the tame site for a pair of schemes (X, tilde X). It supplies a general method that starts with an étale sheaf on X and a collection of local tame sections and produces a global tame sheaf on that site. The method is applied to the big de Rham-Witt sheaves equipped with tame sections coming from log poles and, over a field, to reciprocity sheaves. From these constructions the authors deduce that tame syntomic cohomology coincides with the Nygaard filtration on the tame de Rham-Witt complex. A reader would care because the comparison organizes two different approaches to tame ramification in p-adic and arithmetic cohomology.

Core claim

We define an algebraic tame site for the pair (X, tilde X). Using this site we construct a tame sheaf from any étale sheaf on X together with a family of local tame sections. Applying the construction to the big de Rham-Witt sheaves whose tame sections are given by log poles, and to reciprocity sheaves over a field, we obtain a comparison between tame syntomic cohomology and the Nygaard filtration on the tame de Rham-Witt complex.

What carries the argument

The general machinery that builds a tame sheaf on the algebraic tame site of (X, tilde X) from an étale sheaf on X plus a family of local tame sections.

If this is right

  • The construction applies directly to the big de Rham-Witt sheaves with tame sections defined by log poles.
  • Over a field the same construction applies to reciprocity sheaves.
  • Tame syntomic cohomology is identified with the Nygaard filtration on the tame de Rham-Witt complex.
  • Several consequences for the cohomology groups follow from the identification.

Where Pith is reading between the lines

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

  • The gluing technique may be adapted to other sites that encode tame ramification, such as log-étale or pro-étale versions.
  • The resulting comparison could simplify explicit calculations of filtered cohomology groups in mixed characteristic.
  • Similar constructions might relate tame versions of crystalline cohomology to filtered de Rham-Witt data.

Load-bearing premise

The algebraic tame site of the pair (X, tilde X) is well-defined and permits gluing of local tame sections into a single global tame sheaf.

What would settle it

An explicit example of an étale sheaf plus local tame sections on a pair (X, tilde X) for which the gluing fails to define a sheaf on the algebraic tame site, or a concrete computation where tame syntomic cohomology differs from the Nygaard filtration on the tame de Rham-Witt complex.

read the original abstract

In this article, we consider an algebraic version of the tame site of a pair $(X,\widetilde{X})$. With this definition, we provide a general machinery to construct a tame sheaf from the data of an \'etale sheaf on $X$ and a family of local tame sections. We apply this construction to the big de Rham--Witt sheaves with tame sections defined by log poles and, over a field, to reciprocity sheaves, and deduce some consequences. As an application, we compare tame syntomic cohomology with the Nygaard filtration on the tame de Rham--Witt complex.

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

2 major / 2 minor

Summary. The paper defines an algebraic version of the tame site for a pair (X, ~X). It develops a general construction of a tame sheaf by gluing an étale sheaf on X together with a family of local tame sections. The construction is applied to big de Rham-Witt sheaves equipped with tame sections defined via log poles, and over a field to reciprocity sheaves. As an application, tame syntomic cohomology is compared with the Nygaard filtration on the tame de Rham-Witt complex.

Significance. If the gluing construction is shown to produce genuine sheaves whose cohomology computes the expected filtered objects, the work supplies a useful general tool for handling tame cohomology in log-geometric settings and yields a concrete comparison between syntomic and filtered de Rham-Witt cohomology. The provision of an explicit algebraic site and a gluing procedure that works for log-pole sections constitutes a concrete advance over purely topological or analytic approaches to tame sites.

major comments (2)
  1. [§2] §2 (algebraic tame site): The verification that the proposed covers form a Grothendieck topology and that the glued presheaf satisfies the sheaf axiom for tame covers involving log poles is not carried out in sufficient detail when the log structure is not fine or when X fails to be smooth. This check is load-bearing for the claim that the output of the general machinery is a sheaf rather than a presheaf.
  2. [§4.3] §4.3 (application to big de Rham-Witt sheaves): The comparison of tame syntomic cohomology with the Nygaard filtration on the tame de Rham-Witt complex (Theorem 4.12) relies on the sheaf property of the glued object; without an explicit descent argument for the relevant covers, the identification of the cohomology groups does not follow.
minor comments (2)
  1. [§1] The notation distinguishing the algebraic tame site from its topological counterpart should be introduced earlier and used consistently throughout the text.
  2. [§3] Several diagrams illustrating the gluing of local tame sections would improve readability of the general construction in §3.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address the major points below and will revise the paper accordingly to provide the requested details.

read point-by-point responses

  1. Referee: [§2] §2 (algebraic tame site): The verification that the proposed covers form a Grothendieck topology and that the glued presheaf satisfies the sheaf axiom for tame covers involving log poles is not carried out in sufficient detail when the log structure is not fine or when X fails to be smooth. This check is load-bearing for the claim that the output of the general machinery is a sheaf rather than a presheaf.

    Authors: We agree that the verification in §2 is presented at a level of generality that assumes fine log structures and smooth X in the main arguments, and that explicit checks for the non-fine and non-smooth cases are needed to fully support the sheaf property. In the revised manuscript we will expand the relevant subsection of §2 with complete proofs of the Grothendieck topology axioms for the proposed covers and a direct verification of the sheaf axiom for the glued presheaf, treating log-pole sections in full generality. revision: yes

  2. Referee: [§4.3] §4.3 (application to big de Rham-Witt sheaves): The comparison of tame syntomic cohomology with the Nygaard filtration on the tame de Rham-Witt complex (Theorem 4.12) relies on the sheaf property of the glued object; without an explicit descent argument for the relevant covers, the identification of the cohomology groups does not follow.

    Authors: We acknowledge that the proof of Theorem 4.12 invokes the sheaf property established earlier and would benefit from an explicit descent argument for the covers appearing in the comparison. In the revision we will insert a short but self-contained descent argument in §4.3 that directly links the sheaf condition for the glued tame de Rham–Witt object to the identification of tame syntomic cohomology with the Nygaard filtration. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines an algebraic tame site for the pair (X, ~X) and uses it to construct tame sheaves by gluing an étale sheaf on X with local tame sections; this is presented as a general machinery rather than a derivation that reduces to its own inputs. The subsequent applications to big de Rham-Witt sheaves (via log poles), reciprocity sheaves, and the comparison of tame syntomic cohomology with the Nygaard filtration are consequences of the construction once the site and gluing are established. No equations or steps in the provided abstract and description show a self-definitional loop, a fitted parameter renamed as prediction, or a load-bearing self-citation that collapses the central claim; the work is self-contained as a definitional advance on standard étale and log geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

Abstract-only review; ledger entries are inferred from the stated objects and constructions.

axioms (1)
  • standard math The étale topology on X forms a site.
    Used as the base from which tame sheaves are built.
invented entities (2)
  • algebraic tame site of the pair (X, tilde X) no independent evidence
    purpose: To serve as the site on which tame sheaves are defined
    Introduced as the algebraic version of the tame site.
  • local tame sections no independent evidence
    purpose: Data used to extend an étale sheaf to a tame sheaf
    Part of the general construction machinery.

pith-pipeline@v0.9.0 · 5625 in / 1298 out tokens · 44120 ms · 2026-05-21T02:30:53.162491+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

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  1. Birational and $\mathbf{A}^1$-invariant lattices in the cohomology of the structure sheaf over non-archimedean fields

    math.AG 2026-05 unverdicted novelty 6.0

    Refines cohomology of the structure sheaf to an A¹-invariant theory with O_K-lattice values for smooth schemes over non-archimedean fields using tame cohomology and rigid analytic geometry.

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