REVIEW 2 major objections 7 minor 3 cited by
Homogeneous spaces in tensor categories
T0 review · 2 major / 7 minor · reviewed 2026-08-15 · deepseek-v4-flash
Pith's one-line read For algebraic groups $H\subseteq G$ in a tensor category satisfying geometric reductivity and maximal nilpotency, the homogeneous space $G/H$ exists as an algebraic separated scheme, and its affine or proper type is that of the classical…
desk verdict New machinery for homogeneous spaces in tensor categories, but the proof of algebraicity/separatedness has a circular dependency on the companion paper that the authors themselves flag. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing mechanism is the Frobenius twist and its associated Frobenius kernels. Starting from the $p$-th power Frobenius functor, the paper defines a subalgebra $A^{[1]}$ of a commutative algebra $A$, and for an algebraic group $G$ an increasing family of infinitesimal normal subgroups $G_r = \ker(G\to G^{[r]})$. For large $r$ the quotient $G^{[r]}$ is purely even and the body $G_0$ maps onto it as a quotient. The proof of existence of $G/H$ then proceeds by first enlarging $H$ to $\widetilde H = G_r H$; the quotient $\widetilde H/H$ is affine by an infinitesimal-quotient lemma, and a Cartesian square of faisceaux reduces $G/H$ to the classical quotient $G_0/(G_0\cap \widetilde H)$, which exists in ordinary algebraic geometry. This is the same strategy that was used for algebraic supergroups, transplanted to tensor categories.
What would settle it
The theorem predicts that $G_0/H_0\to G/H$ is a universal homeomorphism and that $G/H$ is affine, quasi-affine, or proper exactly when $G_0/H_0$ is. A counterexample would be a category satisfying (GR)+(MN1-2), an algebraic group $G$, and a subgroup $H$ for which these two quotients have different underlying topological spaces or different affine or proper status. The quickest test case is the additive-group quotient of Section 6 in $\mathrm{Ver}_4^+$ or $\mathrm{Ver}_{2^\infty}$, where the quotient is explicitly computable and already shows non-classical behaviour; comparing its body quotient with $G_0/H_0$ would settle the point.
Extended reading notes
Core claim
The central result is Theorem 7.5: if $G$ is an algebraic group in a tensor category $\mathcal{C}$ satisfying (GR) and (MN1-2), and $H\subseteq G$ is a subgroup, then (1) $G/H$ exists as a scheme, is algebraic, and is separated; (2) the natural map $G_0/H_0\to G/H$ is a closed immersion and a universal homeomorphism; and (3) $G/H$ is affine, quasi-affine, or proper if and only if $G_0/H_0$ is. Here $G_0$ is the body of $G$, the ordinary algebraic group obtained by killing the ideal generated by nilpotent elements of $k[G]$. Thus the possibly exotic homogeneous space has the same underlying topological space as a classical quotient, up to nilpotent thickening. The theorem also closes a question left open in the foundations of scheme theory in tensor categories: every normal subgroup $N$ of $G$ is the kernel of the quotient map $G\to G/N$. For the Verlinde category $\mathrm{Ver}_p$, the paper gives a sharper description: $(G/H)_0$ equals $G_0/H_0$, and $G/H$ is locally built from the symmetric algebra of the kernel $Z = \ker(W_G\to W_H)$.
Load-bearing premise
The load-bearing premise is that the body of an algebraic group—the ordinary group obtained by ignoring nilpotent directions—is itself a closed subgroup of the original group with the same points over the field $k$, so that the Frobenius-kernel quotient $G_0\to G^{[r]}$ is a quotient; this fact is imported from earlier work rather than proved here, and if it fails the reduction of $G/H$ to $G_0/H_0$ collapses.
Editorial extensions
If this is right
- In every tensor category satisfying (GR) and (MN1-2), in particular in $\mathrm{Ver}_p$ and $\mathrm{Ver}_{2^\infty}$, homogeneous spaces $G/H$ can be formed as algebraic separated schemes, so orbit maps, stabilizers, and flag-like quotients are available.
- Every normal subgroup $N\subseteq G$ is the kernel of a quotient morphism: the group-theoretic quotient $G/^{\mathrm{gp}}N$ and the scheme quotient $G/N$ coincide, resolving a question left open in the foundational development.
- Quasi-coherent sheaves on $G/H$ are equivalent to $H$-equivariant sheaves on $G$, and derived induction is computed by homogeneous-space cohomology: $R^i\mathrm{Ind}_H^G(-) \simeq H^i(G/H, F(-))$.
- If $G/H$ is proper, the induction functor $\mathrm{Ind}_H^G$ preserves compact objects, so finiteness properties of representations transfer through induction.
- In $\mathrm{Ver}_p$, homogeneous spaces have an explicit local model: $(G/H)_0 = G_0/H_0$ and $G/H$ is locally isomorphic to a symmetric-algebra spectrum over $G_0/H_0$ with fibre $Z = \ker(W_G\to W_H)$.
Reading between the lines
- If the conjectural equivalence between (GR)+(MN1-2) and incompressibility holds, this theorem would give homogeneous-space geometry in every incompressible moderate-growth tensor category, making the Verlinde categories $\mathrm{Ver}_{p^n}$ the only remaining test cases.
- The Frobenius-kernel reduction suggests a route to more general equivariant constructions—double quotients, quotient stacks, or relative quotients—because the method replaces an arbitrary algebraic group by an ordinary algebraic group after dividing out an infinitesimal normal subgroup.
- In $\mathrm{Ver}_p$, the explicit local description with kernel $Z$ makes it possible to compute cohomology of homogeneous spaces by importing classical results on $G_0/H_0$ through the closed immersion, provided the nilpotent directions do not contribute new cohomology.
- A natural stress test is to verify directly in $\mathrm{Ver}_{2^\infty}$ that the body $G_0$ is a closed subgroup of $G$ for the general linear group on a projective object; the theorem's reduction depends on this imported fact, and checking it in an explicit example would test the argument's most load-bearing step.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves that, in a symmetric tensor category C of moderate growth satisfying conditions (GR) and (MN1-2), every pair of algebraic groups H ⊆ G admits a quotient G/H that is an algebraic, separated scheme. The key new tool is a Frobenius twist and Frobenius kernel for schemes and group schemes in C, which allows the authors to reduce the existence and geometric properties of G/H to the classical quotient G0/H0 of the body G0 by H0. The authors also establish that G/H is affine, quasi-affine, or proper if and only if G0/H0 is, give applications to equivariant sheaves and induction, and provide a more explicit description of homogeneous spaces in the Verlinde category Ver_p.
Significance. If the main theorem is correct, this is a substantial advance: it makes homogeneous-space geometry available in tensor categories such as Ver_p and Ver_{2^∞}, answering a question left open in the companion paper [6] and providing the infrastructure for induction, sheaf cohomology, and equivariant sheaf theory in these settings. The paper is careful and explicit: it develops Frobenius kernels, proves many technical lemmas in detail, and is transparent about the conditional nature of the hypotheses (Remark 2.5). The applications to equivariant sheaves (Propositions 7.11 and 7.12) and to the Ver_p case (Section 9) are genuinely useful. The main reservation concerns a circular step in the proof of Theorem 7.5(1), discussed below.
major comments (2)
- [§7.2, Lemma 7.6 and Corollary 7.8] The proof that G/H is algebraic and separated is not self-contained as written. Lemma 7.6 asserts that (G/H)[r] is a homogeneous space for G[r]; its proof invokes [7, Lem. 7.25] to obtain an immersion G[1]/H′ → (G/H)[1]. Corollary 7.8 explicitly states that [7, Lem. 7.25] “was written under the hypothesis that Theorem 7.5 is valid.” Since the algebraicity and separatedness of G/H are part of Theorem 7.5(1) and are transferred from (G/H)[r] via the finite morphism G/H → (G/H)[r], the argument as written relies on the very conclusion being proved. The Cartesian-square argument before Lemma 7.6 establishes only existence of G/H as a scheme, not its algebraicity. The same circularity affects parts (2) and (3) of Theorem 7.5, whose proofs use the identification (G/H)[r] ≅ G0/H′. This is a load-bearing gap. A repair would require a proof of the orbit-map immersion for the specific quotient G[r]/H′ (for r with G[r] purely even) that does not assume Theorem 7.5, or an independent proof of the needed case of [7, Lem. 7.25].
- [§7.2, proof of Lemma 7.6] Even setting aside the circularity, the final step of Lemma 7.6 is terse: after p is shown to be a closed immersion, the sheaf diagram is used to conclude that p is an isomorphism, relying on the claim that “all other morphisms in the diagram are monomorphisms.” This requires, for example, that the natural map O_{G/H} → q_*O_G is a monomorphism, which is not proved and would need justification (e.g., via faithful flatness of the quotient morphism). This is secondary to the circularity, but it is another place where the written proof is incomplete.
minor comments (7)
- [Abstract] The phrase “an group scheme” in the abstract and introduction contains a grammatical error and should read “a group scheme.”
- [§6, end of section] The displayed conclusion “X/Y ∼= SpecR” appears to be a typo: the quotient is of Y by the right X-action, so it should read “Y/X ≅ SpecR” (or the roles of X and Y should be adjusted consistently).
- [§4.2, Lemma 4.14 and surrounding text] There are spacing and notation issues, e.g., “Forr ∈ N” and “n<p r”; these should be “For r ∈ N” and “n < p^r” (assuming the latter is intended).
- [§4, Remark 4.6] The notation “A[1]” versus “A[1]” (with an overline) is confusing; the text says “A[1] is not isomorphic to A[1]” but the two symbols differ only by a typographically easy-to-miss overline. Please use a more explicit notation, e.g., A^{⟨1⟩} versus \bar A^{⟨1⟩}.
- [§7.2, Lemma 7.4] The sentence “We have an natural isomorphism” contains a typo; it should be “a natural isomorphism.”
- [References] Reference [15] is listed as “EGNO book” without full bibliographic information; this should be completed (authors, title, publisher, year).
- [§8] In Section 8, the symbol H is reused for the Hopf algebra k[H] while H denotes the subgroup in the rest of the paper; this local change is confusing and should be marked, for instance by writing H = k[H] explicitly at the start and using a different letter such as A_H or K.
Circularity Check
Algebraicity and separatedness in Theorem 7.5(1) rest on Lemma 7.6, which invokes [7, Lem. 7.25]; Corollary 7.8 admits that lemma was written under the hypothesis that Theorem 7.5 is valid.
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self citation load bearing
[Section 7, proof of Lemma 7.6 and Corollary 7.8]
"In Lemma 7.6: 'Writing H′ ⊆ G[1] for the isotropy subgroup for e, we have an immersion p : G[1]/H′ → (G/H)[1] by [7, Lem. 7.25].' Corollary 7.8 states: 'This is [7, Lem. 7.25], which was written under the hypothesis that Theorem 7.5 is valid.'"
Lemma 7.6 is the step that upgrades G/H from a scheme to an algebraic, separated scheme: the finishing argument applies Lemma 4.9 to the finite morphism G/H → (G/H)[r], but this only transfers algebraicity/separatedness to G/H if (G/H)[r] is already known to be an algebraic scheme. Lemma 7.6 is supposed to supply that by making (G/H)[r] a homogeneous space for G[r]. Its proof invokes the companion paper's [7, Lem. 7.25] to obtain the orbit immersion G[1]/H′ → (G/H)[1]. Corollary 7.8 of the present paper explicitly says that [7, Lem. 7.25] was written under the hypothesis that Theorem 7.5 is valid. Hence the algebraic/separated part of Theorem 7.5 is not established as written: it uses a lemma whose proof assumes the theorem being proved.
full rationale
The existence claim in Theorem 7.5 is proved without Lemma 7.6, via the Cartesian square, Lemma 7.2, and the reduction to the classical quotient G0/H0; no fitted parameters or numeric inputs are involved, and the benchmarks are external classical quotient schemes. However, the finite-type/algebraic and separated clauses of Theorem 7.5(1), and the consequences (2) and (3), pass through Lemma 7.6 and therefore through [7, Lem. 7.25]. The paper itself flags in Corollary 7.8 that this cited lemma was written under the hypothesis that Theorem 7.5 is valid. That is a real self-citation load-bearing circular step in the central theorem, even though the gap may be repairable and the existence part remains self-contained. Score 6 rather than higher because the circularity is localized to the algebraicity/separatedness upgrade, not to the whole derivation chain.
Assumptions & free parameters
assumptions (6)
- domain assumption (GR): every nonzero morphism alpha: X to 1 splits on some symmetric power of alpha.
- domain assumption (MN1): S(L) is finite for every simple nontrivial object L.
- domain assumption (MN2): some symmetric power of a nonsplit morphism 1 to X vanishes.
- domain assumption Commutative algebra and scheme theory in C as developed in [6] and [7]: Noetherianity, Nakayama's lemma, Spec-homeomorphism machinery, fppf quotient facts ([6, Prop. 7.2.4]), and the orbit-stabilizer immersion ([7, Lem. 7.25]).
- standard math Existence, monoidality, and semisimplification description of the Frobenius functor Fr: C to C(1) ⊠ Ver_p, including A[1] = im(Γ^p(A) to A), from [5, 8, 16].
- standard math Classical theorems used as external benchmarks: existence of quotients of ordinary algebraic groups (Lemma 5.6), Schneider's quotient theory [31], Pareigis Hopf-module freeness [29, Lem. 2] extended to C, and the Ver_p structure theorem [37, Lem. 7.15] with [23, Thm. 5.1].
invented entities (1)
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Frobenius twist A[1] = im(Γ^p(A) to A) and Frobenius kernel G_r = ker(G to G[r]) for algebras and algebraic groups in C.
independent evidence
Cite this review
Pith. "Pith review of Homogeneous spaces in tensor categories." pith.science (2026). https://pith.science/paper/MIVV2LGG
@misc{pith2026250504848,
author = {Pith},
title = {Pith review of: Homogeneous spaces in tensor categories},
year = {2026},
howpublished = {\url{https://pith.science/paper/MIVV2LGG}},
note = {Machine review of arXiv:2505.04848}
}
abstract
Let $\mathscr{C}$ be a symmetric tensor category of moderate growth, and let $\mathcal{H}\leq\mathcal{G}$ be algebraic groups in $\mathscr{C}$. We prove that the homogeneous space $\mathcal{G}/\mathcal{H}$ exists as a scheme and is of finite type when $\mathscr{C}$ is geometrically reductive and maximally nilpotent, conditions that are conjecturally equivalent to incompressibility. A key tool is the introduction of a Frobenius kernel of an group scheme. We further show that while $\mathcal{G}_0/\mathcal{H}_0$ and $(\mathcal{G}/\mathcal{H})_0$ need not be the same, they are close enough, so that $\mathcal{G}/\mathcal{H}$ is quasi-affine/affine/proper if and only if $\mathcal{G}_0/\mathcal{H}_0$ is.
Forward citations
Cited by 3 Pith papers
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Absolutely flat algebras in tensor categories
For well-fibered pretannakian tensor categories, commutative ind-algebras are semisimple exactly when they are products of simple algebras and exactly when they are absolutely flat.
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On geometrically reductive tensor categories
Proves the conjecture that higher Verlinde categories are geometrically reductive and reduces two further conjectures to existing ones in the literature.
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Group schemes and their Lie algebras over a symmetric tensor category
Tangent spaces of affine group schemes over symmetric tensor categories are restricted Lie algebras, and the paper computes them explicitly for Ver_4^+ in characteristic 2.
Reviewed August 15, 2026 · model on record in the stance chip above.
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