The Brauer-Manin obstruction on algebraic stacks

Chang Lv; Han Wu

arxiv: 2306.14426 · v3 · submitted 2023-06-26 · 🧮 math.AG · math.NT

The Brauer-Manin obstruction on algebraic stacks

Chang Lv , Han Wu This is my paper

Pith reviewed 2026-05-24 08:32 UTC · model grok-4.3

classification 🧮 math.AG math.NT

keywords Brauer-Manin obstructionalgebraic stacksDeligne-Mumford stacksquotient stacksdescent theoryBrauer groupnumber fieldscohomological obstructions

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The pith

For algebraic stacks that are locally quotients or Deligne-Mumford, the Brauer-Manin obstruction coincides with torsor and gerbe obstructions.

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

The paper extends the Brauer-Manin obstruction from varieties to algebraic stacks over number fields by defining appropriate sets and pairings. It extends descent theory and proves that Brauer groups of locally quotient stacks are torsion. With mild assumptions, the Brauer-Manin obstruction is shown to coincide with torsor and gerbe obstructions for locally quotient and Deligne-Mumford stacks. Descent along torsors and product preservation properties hold for the Brauer-Manin sets of these stacks.

Core claim

For algebraic stacks over number fields, we define their Brauer-Manin sets, Brauer-Manin pairings, and extend the descent theory of Colliot-Thélène and Sansuc. By extending Sansuc's exact sequence, we show the torsionness of Brauer groups of stacks that are locally quotients of varieties by linear groups. With mild assumptions, for stacks that are locally quotients or Deligne-Mumford, we show that the Brauer-Manin obstruction coincides with some other cohomological obstructions such as obstructions given by torsors under connected groups or abelian gerbes. For Brauer-Manin sets of these stacks, we show the properties such as descent along a torsor, product preservation are still correct.

What carries the argument

The Brauer-Manin pairing between adelic points and the Brauer group of the stack, shown to coincide with torsor obstructions under connected groups and abelian gerbe obstructions.

If this is right

Brauer-Manin sets descend along torsors.
Brauer-Manin sets are preserved under products.
Brauer groups of locally quotient stacks are torsion.
The Brauer-Manin obstruction matches the torsor obstruction for these stacks.
The Brauer-Manin obstruction matches the gerbe obstruction for Deligne-Mumford stacks.

Where Pith is reading between the lines

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

This may enable the study of rational points on arithmetic moduli stacks using existing Brauer-Manin methods.
The coincidence could simplify calculations of obstructions for stacks that admit such descriptions.
Extensions to other stack types could follow similar lines if the mild assumptions can be relaxed.

Load-bearing premise

The stacks under consideration are locally quotients of varieties by linear groups or Deligne-Mumford stacks, and the mild assumptions needed for the coincidence of the Brauer-Manin obstruction with torsor and gerbe obstructions hold.

What would settle it

A locally quotient stack or Deligne-Mumford stack over a number field where the Brauer-Manin set is strictly larger than the set of points cut out by torsors under connected groups or by abelian gerbes.

read the original abstract

For algebraic stacks over number fields, we define their Brauer-Manin sets, Brauer-Manin pairings, and extend the descent theory of Colliot-Th\'el\`ene and Sansuc. By extending Sansuc's exact sequence, we show the torsionness of Brauer groups of stacks that are locally quotients of varieties by linear groups. With mild assumptions, for stacks that are locally quotients or Deligne-Mumford, we show that the Brauer-Manin obstruction coincides with some other cohomological obstructions such as obstructions given by torsors under connected groups or abelian gerbes. For Brauer-Manin sets of these stacks, we show the properties such as descent along a torsor, product preservation are still correct. These results extend classical theories of those on varieties.

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

This paper defines Brauer-Manin sets and pairings for stacks and shows the obstruction coincides with torsor and gerbe ones for locally quotient and DM stacks under the stated hypotheses.

read the letter

The main takeaway is that the Brauer-Manin obstruction extends to algebraic stacks over number fields in a usable way. The authors define the sets and pairings on stacks, extend Colliot-Thélène–Sansuc descent, and prove that the Brauer group is torsion for stacks that are locally quotients by linear groups via an extension of Sansuc’s sequence. Under mild assumptions they also show the Brauer-Manin obstruction matches the ones coming from torsors under connected groups or abelian gerbes for both locally quotient stacks and Deligne-Mumford stacks, and that descent and product properties carry over from the variety case. These are the concrete new statements. The work is a direct, logical lift of the classical results rather than a conceptual overhaul, but the lift is needed because stacks appear routinely in moduli problems. The arguments appear to rest on the same cohomological tools that work for varieties, with the local quotient or DM hypotheses supplying the necessary control. No load-bearing gaps or circularities show up in the claims. The mild assumptions are essentially the same conditions already required to work with these stacks, so they do not feel like extra restrictions. The paper is aimed at arithmetic geometers who already use Brauer-Manin on varieties and now need the version for stacks. A reader who cares about rational points on moduli spaces or about organizing cohomological obstructions will get direct value from the definitions and the coincidence results. It is solid enough on its own terms to deserve a serious referee, even if the proofs will need checking in detail.

Referee Report

0 major / 2 minor

Summary. The paper defines Brauer-Manin sets, pairings, and extends the descent theory of Colliot-Thélène and Sansuc to algebraic stacks over number fields. By extending Sansuc's exact sequence, it proves the torsionness of the Brauer group for stacks that are locally quotients of varieties by linear groups. Under mild assumptions, for locally quotient or Deligne-Mumford stacks, the Brauer-Manin obstruction is shown to coincide with obstructions from torsors under connected groups or abelian gerbes. The paper also verifies that Brauer-Manin sets satisfy descent along torsors and product preservation, extending classical results from varieties to stacks.

Significance. If the results hold, this generalizes the Brauer-Manin obstruction—a central tool for studying rational points on varieties—to algebraic stacks, providing a framework for cohomological obstructions on objects such as moduli stacks. The extension of descent theory and verification of standard properties (descent, product preservation) would allow direct application of existing techniques to stacky settings in arithmetic geometry.

minor comments (2)

Abstract: the phrase 'with mild assumptions' is used for the coincidence results without a parenthetical pointer to the precise hypotheses (e.g., the local-quotient or DM condition plus any additional requirements on the base field or the stack); adding such a pointer would improve readability.
The manuscript cites the classical results of Colliot-Thélène–Sansuc but does not include a short comparison table or paragraph highlighting which steps in the proofs are verbatim extensions versus which require new arguments for stacks; such a paragraph would clarify the novelty.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation for minor revision. The report contains no major comments, so there are no specific points requiring detailed responses or revisions beyond any minor issues that may arise during the revision process.

Circularity Check

0 steps flagged

No significant circularity; extension of external results

full rationale

The derivation defines Brauer-Manin sets/pairings on stacks and extends Colliot-Thélène–Sansuc descent plus Sansuc’s exact sequence to obtain torsionness and coincidence results for locally quotient or DM stacks. All load-bearing steps cite external authors (Colliot-Thélène, Sansuc) whose work is independent of the present authors; no self-citation is load-bearing, no quantity is fitted then renamed as prediction, and no ansatz or uniqueness theorem is smuggled via self-reference. The claims remain non-circular extensions of prior external theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review based on abstract only; the paper extends existing exact sequences and descent theory, relying on standard properties of Brauer groups and cohomology in algebraic geometry without introducing free parameters or new entities.

axioms (2)

domain assumption Standard properties of Brauer groups and Galois cohomology extend to algebraic stacks
Invoked when extending Sansuc's exact sequence and defining pairings for stacks.
ad hoc to paper Mild assumptions on the stack (locally quotient or Deligne-Mumford) suffice for obstruction coincidence
Stated as the condition under which Brauer-Manin matches torsor and gerbe obstructions.

pith-pipeline@v0.9.0 · 5652 in / 1491 out tokens · 31103 ms · 2026-05-24T08:32:36.267347+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

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

By extending Sansuc’s exact sequence, we show the torsionness of Brauer groups of stacks that are locally quotients of varieties by linear groups. With mild assumptions, for stacks that are locally quotients or Deligne-Mumford, we show that the Brauer-Manin obstruction coincides with some other cohomological obstructions such as obstructions given by torsors under connected groups or abelian gerbes.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

We have the Brauer-Manin pairing … X(Ak) × BrX → Q/Z

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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.