Algebraic and analytic Brauer groups of homogeneous spaces
Pith reviewed 2026-05-22 03:27 UTC · model grok-4.3
The pith
The algebraic and analytic Brauer groups of homogeneous spaces under connected simply connected semisimple complex groups with closed connected stabilizers are computed explicitly.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
In this article, we compute both the algebraic and the analytic Brauer groups of a homogeneous space under the action of a connected, simply connected, semisimple complex algebraic group, where the stabilizer subgroup is closed and connected.
What carries the argument
The homogeneous space G/H formed by the group action, through which the algebraic and analytic Brauer groups are determined via cohomology associated to the group and stabilizer.
If this is right
- The Brauer groups reduce to explicit expressions involving the fundamental group or character group of the stabilizer.
- The algebraic and analytic versions of the Brauer group are related by a natural map whose kernel and cokernel are determined.
- The formulas apply uniformly to all such spaces, including many flag varieties and other quotients appearing in algebraic geometry.
Where Pith is reading between the lines
- The same reduction might be attempted over the real numbers by replacing the complex group with its real form and adjusting for topological differences.
- The explicit Brauer groups could be inserted into the Brauer-Manin obstruction to study existence of rational points on these homogeneous spaces.
Load-bearing premise
The base field must be the complex numbers and the acting group must be semisimple and simply connected with the stabilizer closed and connected.
What would settle it
An explicit computation of the algebraic or analytic Brauer group on a homogeneous space where the stabilizer fails to be connected, compared against the formulas given in the paper.
read the original abstract
In this article, we compute both the algebraic and the analytic Brauer groups of a homogeneous space under the action of a connected, simply connected, semisimple complex algebraic group, where the stabilizer subgroup is closed and connected.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript computes both the algebraic Brauer group Br_alg(X) and the analytic Brauer group Br_an(X) for a homogeneous space X = G/H, where G is a connected, simply connected, semisimple complex algebraic group and H is a closed connected stabilizer subgroup.
Significance. If the explicit computation holds, the result supplies concrete descriptions of these Brauer groups under the stated hypotheses on G and H. Such formulas are useful for studying obstructions to rational points, sections of fibrations, and comparisons between algebraic and topological invariants on homogeneous spaces.
minor comments (2)
- [Introduction] The introduction would benefit from an explicit statement of the main formulas for Br_alg(X) and Br_an(X) immediately after the setup is fixed, rather than deferring them entirely to later sections.
- [§2] Notation for the Hochschild-Serre spectral sequence (or whichever spectral sequence is employed for the computation) is introduced without a self-contained reminder of the relevant exact sequence or edge maps; a short paragraph recalling the standard setup would improve readability.
Simulated Author's Rebuttal
We thank the referee for the positive assessment of our manuscript and for recommending minor revision. We are pleased that the explicit computations of Br_alg(X) and Br_an(X) for homogeneous spaces X = G/H are viewed as potentially useful for studying rational points and comparisons of invariants.
Circularity Check
No significant circularity; derivation self-contained
full rationale
The paper presents a direct computation of the algebraic and analytic Brauer groups of homogeneous spaces X = G/H, with G a connected simply connected semisimple complex algebraic group and H a closed connected stabilizer. The abstract and claim summary state the result explicitly under these hypotheses without any equations, fitted parameters, or self-citations that reduce the output to the input by construction. Assumptions on the base field and group properties are listed as part of the setup rather than smuggled in. No load-bearing self-referential definitions or uniqueness theorems from prior author work are visible. The derivation chain therefore stands as independent from the provided text.
Axiom & Free-Parameter Ledger
Lean theorems connected to this paper
-
IndisputableMonolith/Foundation/RealityFromDistinction.leanreality_from_one_distinction unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
We prove that the map in Eq. (1) is an isomorphism... natural isomorphism Ext¹(π₁(H), ℤ) ≃ Br(M) (Theorem 5.3)
-
IndisputableMonolith/Cost/FunctionalEquation.leanwashburn_uniqueness_aczel unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
Haboush constructed an injective group homomorphism Eal(H, Gm) → Br(G/H)
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
- [1]
-
[2]
Knop, Friedrich and Kraft, Hanspeter and Luna, Domingo and Vust, Thierry , TITLE =. Algebraische. 1989 , ISBN =
work page 1989
-
[3]
Serre, Jean-Pierre , TITLE =. Ann. Inst. Fourier (Grenoble) , FJOURNAL =. 1955/56 , PAGES =
work page 1955
- [4]
- [5]
-
[6]
Rationality problems in algebraic geometry , SERIES =
Beauville, Arnaud , TITLE =. Rationality problems in algebraic geometry , SERIES =. 2016 , ISBN =
work page 2016
-
[7]
Artin, M. and Mumford, D. , TITLE =. Proc. London Math. Soc. (3) , FJOURNAL =. 1972 , PAGES =. doi:10.1112/plms/s3-25.1.75 , URL =
-
[8]
Dix expos\'es sur la cohomologie des sch\'emas , SERIES =
Grothendieck, Alexander , TITLE =. Dix expos\'es sur la cohomologie des sch\'emas , SERIES =. 1968 , MRCLASS =
work page 1968
- [9]
-
[10]
Kumar, Shrawan and Neeb, Karl-Hermann , TITLE =. Studies in. 2006 , ISBN =. doi:10.1007/0-8176-4478-4\_13 , URL =
- [11]
-
[12]
Fossum, R. and Iversen, B. , TITLE =. J. Pure Appl. Algebra , FJOURNAL =. 1973 , PAGES =. doi:10.1016/0022-4049(73)90014-5 , URL =
-
[13]
Schr\"oer, Stefan , TITLE =. Math. Ann. , FJOURNAL =. 2001 , NUMBER =. doi:10.1007/s002080100236 , URL =
-
[14]
Schr\"oer, Stefan , TITLE =. Topology , FJOURNAL =. 2005 , NUMBER =. doi:10.1016/j.top.2005.02.005 , URL =
-
[15]
Borel, Armand , TITLE =. 1991 , PAGES =. doi:10.1007/978-1-4612-0941-6 , URL =
- [16]
-
[17]
Colliot-Th\'el\`ene, Jean-Louis and Skorobogatov, Alexei N. , TITLE =. [2021] 2021 , PAGES =. doi:10.1007/978-3-030-74248-5 , URL =
-
[18]
Iversen, Birger , TITLE =. J. Algebra , FJOURNAL =. 1976 , NUMBER =. doi:10.1016/0021-8693(76)90100-9 , URL =
-
[19]
Iversen, Birger , TITLE =. Advances in Math. , FJOURNAL =. 1976 , NUMBER =. doi:10.1016/0001-8708(76)90170-5 , URL =
- [20]
-
[21]
Popov, V. L. , TITLE =. Uspekhi Mat. Nauk , FJOURNAL =. 2023 , NUMBER =. doi:10.4213/rm10107 , URL =
- [22]
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.