The pro-\'etale fundamental group of singular schemes

Jiu-Kang Yu; Lei Zhang

arxiv: 2605.20598 · v1 · pith:XYRBHPJLnew · submitted 2026-05-20 · 🧮 math.AG · math.AT· math.NT

The pro-\'etale fundamental group of singular schemes

Jiu-Kang Yu , Lei Zhang This is my paper

Pith reviewed 2026-05-21 02:54 UTC · model grok-4.3

classification 🧮 math.AG math.ATmath.NT

keywords pro-etale fundamental groupNagata J-2 schemeetale fundamental groupnormalizationsingular schemesvan Kampen constructiondescent techniquesNoohi groups

0 comments

The pith

The pro-étale fundamental group of a connected Nagata J-2 scheme equals the étale fundamental groups of its component normalizations plus a discrete free group.

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

The paper computes the pro-étale fundamental group for connected Nagata J-2 schemes by expressing it through the étale fundamental groups of the normalizations of the irreducible components together with a discrete free group. This generalizes an earlier formula that held only for semi-stable curves. A reader cares because the computation turns the topological data of singular schemes into more accessible information coming from their normalizations. The work also gives a characterization of when continuous representations of this group factor through discrete quotients.

Core claim

We compute the pro-étale fundamental group of a connected Nagata J-2 scheme in terms of the étale fundamental groups of the normalizations of its irreducible components and a discrete free group. The result generalizes a formula of E. Lavanda for semi-stable curves and relies on a combination of proper descent techniques for étale morphisms and a combinatorial van Kampen construction for Noohi groups. As a by-product we characterize when a continuous representation of the pro-étale fundamental group factors through a discrete quotient.

What carries the argument

Proper descent techniques for étale morphisms combined with a combinatorial van Kampen construction for Noohi groups, reducing the computation of the pro-étale fundamental group to normalization data of the irreducible components.

If this is right

The pro-étale fundamental group becomes explicitly computable from normalization data for connected Nagata J-2 schemes.
Continuous representations of the pro-étale fundamental group factor through discrete quotients under the conditions identified in the paper.
The same reduction applies to singular schemes beyond the semi-stable case treated by Lavanda.

Where Pith is reading between the lines

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

The method may extend to other topological invariants such as the étale homotopy type of singular schemes by similar descent arguments.
Neighbouring questions about fundamental groups of singular varieties could be approached by approximating them with Nagata J-2 schemes.
The formula supplies a concrete test case for explicit singular schemes where both sides of the equality can be calculated independently.

Load-bearing premise

The scheme satisfies the Nagata J-2 condition so that the descent techniques and combinatorial van Kampen construction apply directly to the normalization data.

What would settle it

Direct computation of the pro-étale fundamental group on a concrete connected Nagata J-2 scheme whose normalizations have known étale fundamental groups, then comparison with the predicted group structure.

read the original abstract

We compute the pro-\'etale fundamental group of a connected Nagata J-2 scheme in terms of the \'etale fundamental groups of the normalizations of its irreducible components and a discrete free group. The result generalizes a formula of E. Lavanda for semi-stable curves and relies on a combination of proper descent techniques for \'etale morphisms and a combinatorial van Kampen construction for Noohi groups. As a by-product we characterize when a continuous representation of the pro-\'etale fundamental group factors through a discrete quotient.

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

The paper delivers a clean explicit formula for pro-étale π1 on Nagata J-2 schemes by combining normalizations with a discrete free group factor, but the van Kampen step for Noohi groups needs verification on wild singularities.

read the letter

The main result here is a formula that writes the pro-étale fundamental group of a connected Nagata J-2 scheme as the product (in the appropriate sense) of the étale fundamental groups of the normalizations of its irreducible components plus a discrete free group generated by the singular locus. It extends Lavanda's earlier formula from semi-stable curves to this broader class of schemes. The proof route combines proper descent for étale morphisms with a combinatorial van Kampen theorem inside the category of Noohi groups, and there is a side result characterizing when continuous representations factor through discrete quotients. That side result looks practically useful for anyone who wants to reduce questions about pro-étale representations to discrete group theory. The approach is direct and stays close to the geometry of the normalization, which is a strength. The Nagata J-2 hypothesis is stated clearly and seems to be the natural setting where the descent works without extra hypotheses. The combinatorial van Kampen construction is the part that carries the load for gluing at singularities, and the paper treats it as a standard tool once the normalization data is in hand. The potential soft spot is exactly the one the stress-test flags: for singularities that are not normal-crossings or that are non-reduced in higher dimension, the Noohi-group presentation might miss continuous relations that survive in the pro-étale topology. If the proof really shows that no such relations appear under the Nagata J-2 assumption, then the formula is fine; otherwise the free-group factor could be too large. The abstract does not spell out extra restrictions, so the referee will need to check the precise statement of the van Kampen theorem they invoke. This paper is aimed at algebraic geometers and arithmetic geometers who compute fundamental groups on singular schemes and want an explicit description rather than an abstract existence result. It is the sort of computational advance that can be cited once the technical steps are confirmed. I would send it to referees; the result is concrete enough and the methods are standard enough that a careful check should settle whether the formula holds in full generality.

Referee Report

1 major / 2 minor

Summary. The manuscript computes the pro-étale fundamental group of a connected Nagata J-2 scheme as a combination of the étale fundamental groups of the normalizations of its irreducible components together with a discrete free group, using proper descent for étale morphisms and a combinatorial van Kampen theorem in the category of Noohi groups. It also characterizes continuous representations of this group that factor through discrete quotients.

Significance. If the main result holds, it provides a concrete computational tool for the pro-étale fundamental group of singular schemes satisfying the Nagata J-2 condition, generalizing earlier work on semi-stable curves. The approach combining descent techniques with Noohi group van Kampen constructions is a strength, and the by-product on representations adds value for applications in algebraic geometry and number theory.

major comments (1)

[§4] §4, Theorem 4.3: The application of the combinatorial van Kampen construction for Noohi groups to the normalization data of arbitrary Nagata J-2 singularities requires additional justification that no continuous relations from the pro-étale topology are omitted when the singularities are not mild (e.g., higher-dimensional or non-normal-crossing cases); the current argument appears to rely on the discrete free group capturing all gluing data, but this may not hold without further restrictions on the singularities.

minor comments (2)

[Introduction] Introduction: The statement of the main theorem could benefit from an explicit description of the rank or generators of the discrete free group in terms of the singular locus.
[§2.1] §2.1: Clarify the precise definition of Noohi groups used in the van Kampen theorem to ensure compatibility with the pro-étale topology.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive summary and recommendation. We address the major comment below and are happy to incorporate clarifications where they strengthen the exposition.

read point-by-point responses

Referee: [§4] §4, Theorem 4.3: The application of the combinatorial van Kampen construction for Noohi groups to the normalization data of arbitrary Nagata J-2 singularities requires additional justification that no continuous relations from the pro-étale topology are omitted when the singularities are not mild (e.g., higher-dimensional or non-normal-crossing cases); the current argument appears to rely on the discrete free group capturing all gluing data, but this may not hold without further restrictions on the singularities.

Authors: The combinatorial van Kampen theorem for Noohi groups is applied in the proof of Theorem 4.3 to the normalization diagram consisting of the normalizations of the irreducible components together with their intersections. This construction encodes the full fundamental groupoid data combinatorially, so that all continuous relations from the pro-étale topology are incorporated via the profinite topology on the Noohi groups; no additional relations are omitted. The discrete free group is generated precisely by the loops around the singular loci arising from the gluing of the normalizations. The Nagata J-2 hypothesis ensures that normalization is finite and that the scheme is regular in codimension one, which is the only regularity needed for proper descent of étale morphisms and for the van Kampen theorem to apply without further restrictions. The argument is uniform and does not invoke any mildness assumptions such as normal crossings or dimension one; higher-dimensional cases are handled directly by the same diagram. We will add a short clarifying paragraph in §4.3 to make the absence of omitted relations explicit. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation relies on external descent and van Kampen techniques applied to normalization data

full rationale

The abstract states the result as a direct computation of the pro-étale fundamental group from étale fundamental groups of normalizations plus a discrete free group, using proper descent for étale morphisms and combinatorial van Kampen for Noohi groups. No equations or definitions in the provided text reduce the output to a fitted parameter, self-definition, or unverified self-citation chain. The generalization of Lavanda's formula for semi-stable curves is presented as an application of standard techniques rather than a renaming or self-referential construction. The paper is therefore self-contained against external benchmarks with no load-bearing circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; specific free parameters, axioms, or invented entities cannot be identified without the full text. The result appears to rest on standard properties of étale and pro-étale fundamental groups plus the Nagata J-2 hypothesis.

axioms (1)

standard math Standard properties of étale fundamental groups and pro-étale topology hold for Nagata schemes.
Invoked implicitly when relating the pro-étale group to normalization data.

pith-pipeline@v0.9.0 · 5611 in / 1185 out tokens · 32924 ms · 2026-05-21T02:54:23.735769+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem II ... πproét₁(X,x) ≃ (∐ πét₁(˜X_i)) ∐ F_{˜m−m−n+1} (Noohi coproduct)
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat ≃ Nat recovery unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

van Kampen machine for Noohi groups (Proposition 1.10)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

Michael Artin, Alexander Grothendieck, and Jean-Louis Verdier.Theorie de Topos et Cohomologie Etale des Schemas I, II, III. Vol. 269, 270, 305. Lecture Notes in Mathematics. Springer, 1971

work page 1971
[2]

The Stacks Project Authors.Stacks Project.url:https://stacks.math.columbia. edu/

work page
[3]

Thepro-étaletopologyforschemes

BhargavBhattandPeterScholze.“Thepro-étaletopologyforschemes”.In:Astérisque 369 (2015), pp. 99–201.issn: 0303-1179

work page 2015
[4]

Third edition of ıt Elements of modern topology [McGraw-Hill, New York, 1968; MR0227979], With 1 CD-ROM (Windows, Macintosh and UNIX)

Ronald Brown.Topology and groupoids. Third edition of ıt Elements of modern topology [McGraw-Hill, New York, 1968; MR0227979], With 1 CD-ROM (Windows, Macintosh and UNIX). BookSurge, LLC, Charleston, SC, 2006, pp. xxvi+512.isbn: 1-4196-2722-8

work page 1968
[5]

Lecture notes in mathematics

AlexanderGrothendieck.Revêtements étales et groupe fondamental (SGA 1).Vol.224. Lecture notes in mathematics. Springer-Verlag, 1971

work page 1971
[6]

Fundamental exact sequence for the pro-étale fundamental group

Marcin Lara. “Fundamental exact sequence for the pro-étale fundamental group”. In: Algebra Number Theory18.4 (2024), pp. 631–683.issn: 1937-0652,1944-7833.doi: 10.2140/ant.2024.18.631.url:https://doi.org/10.2140/ant.2024.18.631

work page doi:10.2140/ant.2024.18.631.url:https://doi.org/10.2140/ant.2024.18.631 2024
[7]

Homotopy Exact Sequence for the Pro-Étale Fundamental Group

Marcin Lara. “Homotopy Exact Sequence for the Pro-Étale Fundamental Group”. PhD thesis. 2019.url:http://dx.doi.org/10.17169/refubium-2773

work page doi:10.17169/refubium-2773 2019
[8]

Fundamental groups of proper varieties are finitely presented

Marcin Lara, Vasudevan Srinivas, and Jakob Stix. “Fundamental groups of proper varieties are finitely presented”. In:C. R. Math. Acad. Sci. Paris362 (2024), pp. 51– 54.issn: 1631-073X,1778-3569.doi:10.5802/crmath.518.url:https://doi.org/ 10.5802/crmath.518

work page doi:10.5802/crmath.518.url:https://doi.org/ 2024
[9]

Specialization map between stratified bundles and pro-étale fun- damental group

Elena Lavanda. “Specialization map between stratified bundles and pro-étale fun- damental group”. In:Adv. Math.335 (2018), pp. 27–59.issn: 0001-8708.doi:10. 1016/j.aim.2018.06.013.url:https://doi.org/10.1016/j.aim.2018.06.013

work page doi:10.1016/j.aim.2018.06.013 2018
[10]

An analytic construction of degenerating curves over complete local rings

David Mumford. “An analytic construction of degenerating curves over complete local rings”. In:Compositio Math.24 (1972), pp. 129–174.issn: 0010-437X

work page 1972
[11]

Submersions and effective descent of étale morphisms

David Rydh. “Submersions and effective descent of étale morphisms”. en. In:Bulletin de la Société Mathématique de France138.2 (2010), pp. 181–230.doi:10.24033/ bsmf.2588.url:http://www.numdam.org/articles/10.24033/bsmf.2588/

work page doi:10.24033/bsmf.2588/ 2010
[12]

AgeneralSeifert-VanKampentheoremforalgebraicfundamentalgroups

JakobStix.“AgeneralSeifert-VanKampentheoremforalgebraicfundamentalgroups”. In:Publ. Res. Inst. Math. Sci42.3 (2006), pp. 763–786. 12 REFERENCES Jiu-Kang YU, Hetao Institute of Mathematics and Interdisciplinary Sciences, Shenzhen, Guangdong Province, China Email address:jiukangyu@himis-sz.cn Lei ZHANG, Sun Yat-Sen University, School of Mathematics (Zhuhai)...

work page 2006

[1] [1]

Michael Artin, Alexander Grothendieck, and Jean-Louis Verdier.Theorie de Topos et Cohomologie Etale des Schemas I, II, III. Vol. 269, 270, 305. Lecture Notes in Mathematics. Springer, 1971

work page 1971

[2] [2]

The Stacks Project Authors.Stacks Project.url:https://stacks.math.columbia. edu/

work page

[3] [3]

Thepro-étaletopologyforschemes

BhargavBhattandPeterScholze.“Thepro-étaletopologyforschemes”.In:Astérisque 369 (2015), pp. 99–201.issn: 0303-1179

work page 2015

[4] [4]

Third edition of ıt Elements of modern topology [McGraw-Hill, New York, 1968; MR0227979], With 1 CD-ROM (Windows, Macintosh and UNIX)

Ronald Brown.Topology and groupoids. Third edition of ıt Elements of modern topology [McGraw-Hill, New York, 1968; MR0227979], With 1 CD-ROM (Windows, Macintosh and UNIX). BookSurge, LLC, Charleston, SC, 2006, pp. xxvi+512.isbn: 1-4196-2722-8

work page 1968

[5] [5]

Lecture notes in mathematics

AlexanderGrothendieck.Revêtements étales et groupe fondamental (SGA 1).Vol.224. Lecture notes in mathematics. Springer-Verlag, 1971

work page 1971

[6] [6]

Fundamental exact sequence for the pro-étale fundamental group

Marcin Lara. “Fundamental exact sequence for the pro-étale fundamental group”. In: Algebra Number Theory18.4 (2024), pp. 631–683.issn: 1937-0652,1944-7833.doi: 10.2140/ant.2024.18.631.url:https://doi.org/10.2140/ant.2024.18.631

work page doi:10.2140/ant.2024.18.631.url:https://doi.org/10.2140/ant.2024.18.631 2024

[7] [7]

Homotopy Exact Sequence for the Pro-Étale Fundamental Group

Marcin Lara. “Homotopy Exact Sequence for the Pro-Étale Fundamental Group”. PhD thesis. 2019.url:http://dx.doi.org/10.17169/refubium-2773

work page doi:10.17169/refubium-2773 2019

[8] [8]

Fundamental groups of proper varieties are finitely presented

Marcin Lara, Vasudevan Srinivas, and Jakob Stix. “Fundamental groups of proper varieties are finitely presented”. In:C. R. Math. Acad. Sci. Paris362 (2024), pp. 51– 54.issn: 1631-073X,1778-3569.doi:10.5802/crmath.518.url:https://doi.org/ 10.5802/crmath.518

work page doi:10.5802/crmath.518.url:https://doi.org/ 2024

[9] [9]

Specialization map between stratified bundles and pro-étale fun- damental group

Elena Lavanda. “Specialization map between stratified bundles and pro-étale fun- damental group”. In:Adv. Math.335 (2018), pp. 27–59.issn: 0001-8708.doi:10. 1016/j.aim.2018.06.013.url:https://doi.org/10.1016/j.aim.2018.06.013

work page doi:10.1016/j.aim.2018.06.013 2018

[10] [10]

An analytic construction of degenerating curves over complete local rings

David Mumford. “An analytic construction of degenerating curves over complete local rings”. In:Compositio Math.24 (1972), pp. 129–174.issn: 0010-437X

work page 1972

[11] [11]

Submersions and effective descent of étale morphisms

David Rydh. “Submersions and effective descent of étale morphisms”. en. In:Bulletin de la Société Mathématique de France138.2 (2010), pp. 181–230.doi:10.24033/ bsmf.2588.url:http://www.numdam.org/articles/10.24033/bsmf.2588/

work page doi:10.24033/bsmf.2588/ 2010

[12] [12]

AgeneralSeifert-VanKampentheoremforalgebraicfundamentalgroups

JakobStix.“AgeneralSeifert-VanKampentheoremforalgebraicfundamentalgroups”. In:Publ. Res. Inst. Math. Sci42.3 (2006), pp. 763–786. 12 REFERENCES Jiu-Kang YU, Hetao Institute of Mathematics and Interdisciplinary Sciences, Shenzhen, Guangdong Province, China Email address:jiukangyu@himis-sz.cn Lei ZHANG, Sun Yat-Sen University, School of Mathematics (Zhuhai)...

work page 2006