{"id":"fafec06d-2b10-49e1-adcd-ad0a12239a84","arxiv_id":"2505.04848","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In tensor categories satisfying (GR) and (MN1-2), any quotient G/H of algebraic groups exists as an algebraic, separated scheme, and is affine, quasi-affine, or proper exactly when the underlying classical quotient G0/H0 is.","lead":"This paper proves that inside certain exotic mathematical universes, any quotient of a symmetry group by a subgroup exists as a well-behaved geometric object called a scheme. The result gives algebraists and representation theorists a working geometry in positive-characteristic tensor categories, the setting tied to modular representations of finite groups.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Circular dependency in proof of Theorem 7.5(1): Lemma 7.6 invokes [7, Lem. 7.25], which Corollary 7.8 says was written under the hypothesis that Theorem 7.5 is valid; algebraicity and separatedness of G/H are therefore not established as written.","rationale":"The reader's weakest-assumption analysis focused on the imported body-subgroup premise G0 ⊆ G from the companion paper [6]. That is a real dependency, but the present stress-test identifies a sharper and manuscript-flagged circular step: Lemma 7.6, which is needed to prove that G/H is algebraic and separated, invokes [7, Lem. 7.25], and Corollary 7.8 explicitly states that [7, Lem. 7.25] was written under the assumption that Theorem 7.5 is valid. The reader's rationale did mention that [7, Lem. 7.25] is conditional on this paper's theorem, so there is partial agreement, but the reader did not elevate this to the weakest assumption. The concern is load-bearing because the existence proof alone only yields a scheme, not an algebraic separated scheme; the finite-type and separatedness conclusions of Theorem 7.5(1) depend on Lemma 7.6. I recommend keeping the verdict CONDITIONAL: the central theorem may well be true, and the existence part of the proof appears coherent, but the algebraic/separated claim is not proven as written until the circular dependence on [7, Lem. 7.25] is removed or an independent proof of Lemma 7.6 is supplied. I do not recommend REJECT because the gap is localized and potentially repairable, and the body-subgroup issue from the reader remains a secondary imported-infrastructure concern.","tokens_in":31116,"tokens_out":7586,"duration_ms":74725,"concrete_test":"Inspect the proof of [7, Lem. 7.25] and list every line that invokes the existence of quotients G/H or G/G_x, or any statement equivalent to Theorem 7.5. If such an invocation occurs, attempt to prove Lemma 7.6 directly from the defining coequalizer property of G/H in the category of faisceaux, using only Lemmas 4.4, 4.9, and 5.1, without passing through [7, Lem. 7.25]. If the direct proof requires separatedness or algebraicity of G/H, or otherwise uses Theorem 7.5, the circularity is confirmed.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof that G/H is algebraic and separated rests on Lemma 7.6, which asserts that (G/H)[r] is a homogeneous space for G[r]. In the proof of Lemma 7.6, the orbit map G[1] → (G/H)[1] is claimed to factor through an immersion G[1]/H′ → (G/H)[1] by invoking [7, Lem. 7.25]. But Corollary 7.8 of the present paper states explicitly that [7, Lem. 7.25] 'was written under the hypothesis that Theorem 7.5 is valid.' Thus the argument for the algebraic/separated part of Theorem 7.5 uses a lemma whose proof in the companion paper assumes the very theorem being proved. The earlier part of the proof does establish existence of G/H as a scheme via the Cartesian square and Lemma 7.2, but existence alone does not imply algebraic or separated. The finite morphism G/H → (G/H)[r] in the finishing argument only transfers these properties from (G/H)[r] to G/H if (G/H)[r] is already known to be an algebraic scheme, which is exactly what Lemma 7.6 was supposed to supply. Unless [7, Lem. 7.25] has an independent proof that does not assume existence of quotients G/G_x, the text as written contains a genuine circular step in the proof of Theorem 7.5(1). The same circularity does not affect the existence part of Theorem 7.5, and it may be repairable, but the claim that G/H is algebraic and separated is not justified by the present argument.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":31337,"tokens_out":9426,"duration_ms":88404,"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":[{"comment":"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].","section":"§7.2, Lemma 7.6 and Corollary 7.8"},{"comment":"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.","section":"§7.2, proof of Lemma 7.6"}],"minor_comments":[{"comment":"The phrase “an group scheme” in the abstract and introduction contains a grammatical error and should read “a group scheme.”","section":"Abstract"},{"comment":"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).","section":"§6, end of section"},{"comment":"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).","section":"§4.2, Lemma 4.14 and surrounding text"},{"comment":"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⟩}.","section":"§4, Remark 4.6"},{"comment":"The sentence “We have an natural isomorphism” contains a typo; it should be “a natural isomorphism.”","section":"§7.2, Lemma 7.4"},{"comment":"Reference [15] is listed as “EGNO book” without full bibliographic information; this should be completed (authors, title, publisher, year).","section":"References"},{"comment":"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.","section":"§8"}],"recommendation":"major_revision","confidential_remarks":"The paper is commendably transparent about its dependencies: Corollary 7.8 explicitly flags that [7, Lem. 7.25] was written assuming Theorem 7.5. That transparency does not remove the circularity, which affects the central claim of algebraicity and separatedness. The result is likely correct and the gap appears repairable—for example, one could prove the needed orbit-map immersion directly in the case where G[r] is purely even, using classical quotient theory—so I recommend major revision rather than rejection. The paper is a strong contribution to the program on tensor categories in positive characteristic, and I hope the authors can close this gap."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proves that for algebraic groups H ⊆ G in a tensor category satisfying (GR)+(MN1-2), the homogeneous space G/H exists as a scheme and is separated, with affine/quasi-affine/proper detected on the body G0/H0. That is a genuinely new result, and the machinery—Frobenius twists and kernels for group schemes in arbitrary tensor categories, reducing to classical quotients of algebraic groups over k—looks like the right tool. The existence part of Theorem 7.5 is proved in the text via faisceaux and stands on its own.\n\nBut there is a real problem in the proof of the algebraicity/separatedness claim. Lemma 7.6 asserts (G/H)[r] is a homogeneous space for G[r], and its proof invokes [7, Lem. 7.25] for an immersion G[1]/H′ → (G/H)[1]. Corollary 7.8 of this paper says that [7, Lem. 7.25] was written under the hypothesis that Theorem 7.5 is valid. So the proof of Theorem 7.5(1) uses a lemma whose proof in the companion assumes the very theorem. The stress-test note is right, and the reader's report underweights it. The existence of G/H as a scheme is unaffected; the circularity only affects the algebraic/separated part. But that part is load-bearing for the applications (properness, induction, cohomology). The fix may be straightforward—if [7, Lem. 7.25] can be proven directly from the existence of G/H without assuming the full theorem—but as written the argument is incomplete.\n\nAlso worth noting: the body-subgroup G0 ⊆ G is imported from [6], and the hypotheses are conjectural for Ver_{p^∞} with odd p. Those are honest caveats, not flaws.\n\nBottom line: this is a serious paper with new ideas and a coherent architecture, but the gap in Theorem 7.5(1) should be fixed before publication. I would send it to a serious referee, with a request that they check whether the circularity can be removed. I would bring it to reading group—the Frobenius-kernel construction is worth discussing even if the proof is currently incomplete.","headline":"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.","tokens_in":32090,"tokens_out":2998,"would_cite":true,"duration_ms":27699,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14L15","14M17","18M05","14L30"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["tensor categories","algebraic groups","homogeneous spaces","quotient schemes","Frobenius kernels","Verlinde categories","moderate growth","geometric reductivity"],"falsifier":"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.","tokens_in":30711,"feed_emoji":"🧮","tokens_out":13646,"duration_ms":118938,"temperature":0.7,"pith_summary":"The paper proves that homogeneous spaces exist in a broad class of tensor categories: whenever a symmetric tensor category satisfies the two axioms of geometric reductivity and maximal nilpotency, every subgroup $H$ of an algebraic group $G$ admits a quotient $G/H$ that is a scheme, algebraic, and separated. This matters because these axioms are conjecturally equivalent to incompressibility in characteristic $p$, and they are known to hold in the Verlinde categories $\\mathrm{Ver}_p$ and $\\mathrm{Ver}_{2^\\infty}$, which are the conjectural building blocks of modular representation theory. The main theorem reduces the geometry of $G/H$ to the classical quotient $G_0/H_0$ of the underlying ordinary groups, so that affine, quasi-affine, and properness questions can be answered by classical algebraic group theory. Along the way the paper introduces Frobenius kernels of group schemes in tensor categories, proves that every normal subgroup is a kernel, and connects homogeneous-space cohomology to derived induction.","feed_headline":"Homogeneous spaces exist as schemes in tensor categories","feed_subtitle":"Under two nilpotence axioms, G/H is algebraic and separated, with affine or proper type set by the classical quotient.","key_machinery":"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.","core_discovery":"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 $Z = \\ker(W_G\\to W_H)$.","pith_inferences":["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."],"forward_implications":["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 $Z = \\ker(W_G\\to W_H)$."],"supporting_citations":[{"why":"Supplies the (GR) and (MN1-2) axioms, Noetherianity of finitely generated algebras, and the subgroup theory used by the Frobenius-kernel reduction.","marker":"[6]"},{"why":"Supplies the scheme theory in tensor categories used to formulate $G/H$ as a scheme and to transfer finiteness properties through $X_0$ and $X_{(0)}$.","marker":"[7]"},{"why":"Supplies the faisceau/fppf quotient formalism and the affineness and separatedness criteria used in the quotient construction.","marker":"[12]"},{"why":"Supplies the strategy of quotienting by a large Frobenius kernel to reduce homogeneous spaces to classical and infinitesimal cases.","marker":"[26]"},{"why":"Supplies the local construction of $G/H$ used in the explicit $\\mathrm{Ver}_p$ description and the gluing argument.","marker":"[25]"},{"why":"Supplies the Hopf-module equivalence and relative injectivity and coflatness criteria that underlie the affine quotient criterion.","marker":"[31]"},{"why":"Supplies the identification of derived induction with sheaf cohomology on the homogeneous space.","marker":"[21]"},{"why":"Supplies the incompressible categories $\\mathrm{Ver}_{p^n}$ and the verification that $\\mathrm{Ver}_{2^\\infty}$ satisfies the axioms.","marker":"[9]"}],"fun_headline_variants":["Homogeneous spaces in tensor categories are schemes under nilpotence","G/H exists as algebraic scheme in nilpotent tensor categories","Classical quotient's topology decides affineness of G/H","Tensor category homogeneous spaces: existence and affineness settled","Nilpotent axioms yield scheme structure for homogeneous spaces"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Homogeneous spaces in tensor categories are schemes under nilpotence","G/H exists as algebraic scheme in nilpotent tensor categories","Classical quotient's topology decides affineness of G/H","Tensor category homogeneous spaces: existence and affineness settled","Nilpotent axioms yield scheme structure for homogeneous spaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000778,"raw_usage":{"total_tokens":3450,"prompt_tokens":965,"completion_tokens":2485,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":581,"completion_tokens_details":{"reasoning_tokens":2402}},"tokens_in":581,"tokens_out":2485,"duration_ms":18170,"temperature":1.0,"reasoning_tokens":2402,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:23:42.960129+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Demazure and P","cited_arxiv_id":null,"evidence_quote":"Supplies the faisceau/fppf quotient formalism and the affineness and separatedness criteria used in the quotient construction."},{"cited_title":"Masuoka and A","cited_arxiv_id":null,"evidence_quote":"Supplies the strategy of quotienting by a large Frobenius kernel to reduce homogeneous spaces to classical and infinitesimal cases."},{"cited_title":"Masuoka and Y","cited_arxiv_id":null,"evidence_quote":"Supplies the local construction of $G/H$ used in the explicit $\\mathrm{Ver}_p$ description and the gluing argument."},{"cited_title":"Schneider, Principal homogeneous spaces for arb itrary Hopf algebras, Israel J","cited_arxiv_id":null,"evidence_quote":"Supplies the Hopf-module equivalence and relative injectivity and coflatness criteria that underlie the affine quotient criterion."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the identification of derived induction with sheaf cohomology on the homogeneous space."},{"cited_title":"Coulembier, P","cited_arxiv_id":null,"evidence_quote":"Supplies the incompressible categories $\\mathrm{Ver}_{p^n}$ and the verification that $\\mathrm{Ver}_{2^\\infty}$ satisfies the axioms."}],"review_version":1}