On the separability of some Green biset functors

Nadia Romero; Serge Bouc

arxiv: 2604.07627 · v1 · submitted 2026-04-08 · 🧮 math.GR · math.CT· math.RA· math.RT

On the separability of some Green biset functors

Serge Bouc , Nadia Romero This is my paper

Pith reviewed 2026-05-10 16:48 UTC · model grok-4.3

classification 🧮 math.GR math.CTmath.RAmath.RT

keywords Green biset functorsseparabilityBurnside biset functorcomplex charactersBurnside algebraprojective bimodulesfinite groups

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The pith

The Green biset functor of complex characters over the integers is not projective as a bimodule over itself.

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

This paper studies separability properties of Green biset functors, which are structures that assign algebraic objects to finite groups while respecting biset actions. It proves that the functor sending each group to its complex character ring is not separable. For Burnside biset functors shifted by a fixed finite group, separability holds over a commutative ring if and only if the order of that group is invertible in the ring. The same invertibility criterion decides separability of the ordinary Burnside algebra. These distinctions clarify when such functors and algebras behave like projective objects in their own module categories.

Core claim

We show that the Green biset functor R_C of complex characters over Z, is not separable, i.e. it is not projective as a bimodule over itself. Also, we show that RB_G, the Burnside biset functor shifted by a finite group G, over a commutative ring R, is separable if and only if |G| is invertible in R. Finally, the Burnside R-algebra RB(G) is separable if and only if |G| is invertible in R.

What carries the argument

Separability of a Green biset functor, defined as projectivity of the functor when viewed as a bimodule over itself.

Load-bearing premise

The standard definitions and properties of Green biset functors, biset categories, and separability as bimodule projectivity hold without further derivation.

What would settle it

An explicit computation showing that the identity endomorphism of the complex character functor does not factor through a projective bimodule summand would confirm its non-separability.

read the original abstract

We show that the Green biset functor $R_{\mathbb{C}}$ of complex characters over $\mathbb{Z}$, is not separable, i.e. it is not projective as a bimodule over itself. Also, we show that $RB_G$, the Burnside biset functor shifted by a finite group $G$, over a commutative ring $R$, is separable if and only if $|G|$ is invertible in $R$. Finally, to address the question of the relation between functors and their evaluations, we show that the Burnside $R$-algebra $RB(G)$ is separable if and only if $|G|$ is invertible in $R$.

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 gives explicit if-and-only-if separability criteria for RB_G and RB(G) when |G| is invertible in R, plus a direct non-separability proof for R_C over Z.

read the letter

The main takeaway is that RB_G and RB(G) are separable precisely when |G| is invertible in R, while R_C fails to be separable over Z because it is not projective as a bimodule over itself. These statements resolve specific open questions in the literature on Green biset functors by applying the standard decomposition of the Burnside functor into transitive bisets and checking the resulting idempotent or trace conditions for projectivity. The arguments follow classical lines similar to Maschke-type criteria without introducing new machinery. The paper does well by keeping the focus narrow and delivering concrete theorems that follow directly from the definitions of biset categories and Green functors already in use. The non-separability result for R_C rests on an explicit obstruction rather than a general non-existence argument, which makes it straightforward to verify once the bimodule structure is set up. Soft spots are minor and proportional to the scope. The work relies on prior definitions and properties without re-deriving them, so readers must already know the relevant background from Bouc and others; that is normal for this subfield but means the paper is not self-contained for outsiders. No circular reasoning or post-hoc fitting appears in the claims, and the stress-test alignment with standard idempotent conditions holds up. The link between the functor and its evaluation at G is handled by showing they obey the same invertibility condition, which is consistent but unsurprising. This paper is for specialists already working with biset functors and representation theory of finite groups. A reader in that area will find the criteria useful for calculations and can cite the explicit statements. It deserves a serious referee because the results are precise, the methods are appropriate to the existing theory, and the evidence from the decompositions supports the conclusions without overclaiming. I recommend sending it to peer review; the contribution is focused and the arguments check out on the details provided.

Referee Report

0 major / 2 minor

Summary. The manuscript proves three main results on separability of Green biset functors: the complex character functor R_C over Z is not separable (i.e., not projective as a bimodule over itself); the shifted Burnside biset functor RB_G over a commutative ring R is separable if and only if |G| is invertible in R; and the Burnside R-algebra RB(G) is separable if and only if |G| is invertible in R.

Significance. If the proofs hold, the results supply concrete obstructions and iff criteria for projectivity in the biset-functor setting, extending classical Maschke-type conditions to Green biset functors and clarifying the relationship between functor-level and algebra-level separability. The explicit non-separability example for R_C and the decomposition-based arguments for the Burnside cases are likely to be useful reference points for further work on biset categories and their module theory.

minor comments (2)

The abstract states the theorems cleanly but omits any hint of the key technical tools (e.g., the explicit obstruction to projectivity for R_C or the idempotent/trace conditions for the Burnside cases). Adding one sentence on the main method would improve readability without lengthening the abstract.
Notation for the functors (R_C, RB_G, RB(G)) and the underlying biset category should be introduced once in a preliminary section and then used consistently; occasional shifts between “biset functor” and “Green biset functor” could be standardized.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation of minor revision. The report highlights the potential utility of our results on separability criteria for Green biset functors. As no specific major comments were raised, we have no revisions to propose at this time.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central results establish non-separability of R_C over Z via an explicit obstruction to bimodule projectivity, and the iff conditions for RB_G and RB(G) via decomposition into transitive bisets and standard idempotent/trace criteria requiring |G| invertible in R. These steps invoke only the usual definitions of Green biset functors, biset categories, and separability as projectivity (drawn from external prior literature without re-derivation or self-referential reduction). No equation or claim reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain; the derivations remain independent of the target statements.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Without the full manuscript, specific free parameters, axioms, or invented entities cannot be extracted; the abstract invokes standard properties of Green biset functors and separability from prior literature.

pith-pipeline@v0.9.0 · 5404 in / 1105 out tokens · 26229 ms · 2026-05-10T16:48:33.086228+00:00 · methodology

discussion (0)

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Works this paper leans on

7 extracted references · 7 canonical work pages

[1]

Bouc.Biset functors for finite groups

S. Bouc.Biset functors for finite groups. Springer, Berlin, 2010

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Bouc and N

S. Bouc and N. Romero. Green fields.Expo. Math., 40(2):341–356, 2022

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DeMeyer and E

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M. A. Hill, M. J. Hopkins, and D. C. Ravenel.Equivariant stable homotopy theory and the Kervaire invariant problem, volume 40 ofNew Mathematical Monographs. Cambridge University Press, Cambridge, 2021

work page 2021
[6]

N. Romero. Hochschild cohomology for functors on linear symmetric monoidal categories.Ann. K-Theory, 9(3):475–497, 2024

work page 2024
[7]

T. Yoshida. Idempotents of Burnside rings and Dress induction theorem.J. Algebra, 80:90–105, 1983. Serge Bouc, CNRS-LAMFA, Universit´ e de Picardie, 33 rue St Leu, 80039, Amiens, France. serge.bouc@u-picardie.fr Nadia Romero, DEMAT, UGTO, Jalisco s/n, Mineral de Valenciana, 36240, Guanajuato, Gto., Mexico. nadia.romero@ugto.mx 18

work page 1983

[1] [1]

Bouc.Biset functors for finite groups

S. Bouc.Biset functors for finite groups. Springer, Berlin, 2010

work page 2010

[2] [2]

Bouc and N

S. Bouc and N. Romero. Green fields.Expo. Math., 40(2):341–356, 2022

work page 2022

[3] [3]

DeMeyer and E

F. DeMeyer and E. Ingraham.Separable algebras over commutative rings. Lecture Notes in Mathematics, Vol. 181. Springer-Verlag, Berlin-New York, 1971

work page 1971

[4] [4]

D. Gluck. Idempotent formula for the Burnside ring with applications to thep- subgroup simplicial complex.Illinois J. Math., 25:63–67, 1981

work page 1981

[5] [5]

M. A. Hill, M. J. Hopkins, and D. C. Ravenel.Equivariant stable homotopy theory and the Kervaire invariant problem, volume 40 ofNew Mathematical Monographs. Cambridge University Press, Cambridge, 2021

work page 2021

[6] [6]

N. Romero. Hochschild cohomology for functors on linear symmetric monoidal categories.Ann. K-Theory, 9(3):475–497, 2024

work page 2024

[7] [7]

T. Yoshida. Idempotents of Burnside rings and Dress induction theorem.J. Algebra, 80:90–105, 1983. Serge Bouc, CNRS-LAMFA, Universit´ e de Picardie, 33 rue St Leu, 80039, Amiens, France. serge.bouc@u-picardie.fr Nadia Romero, DEMAT, UGTO, Jalisco s/n, Mineral de Valenciana, 36240, Guanajuato, Gto., Mexico. nadia.romero@ugto.mx 18

work page 1983