{"id":"997a23bf-dfb9-44fe-8e93-b3c6b99f5acf","arxiv_id":"2506.21525","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Compact derived VI-modules are classified by their support types, and the Balmer spectrum for families of bounded-rank abelian p-groups is computed explicitly.","lead":"This paper studies families of compatible representations attached to the automorphism groups of finite groups, and describes the full tensor-triangular spectrum that organizes them. It delivers a complete classification in the important case of VI-modules and reveals surprisingly rich geometry for finite abelian p-groups.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The VI-module classification and infinite Krull dimension of Theorem C hinge on the imported growth Theorem 4.16, which is not proved in this paper; if the companion paper's theorem fails, these headline results collapse.","rationale":"The reader's weak-assumption analysis is accurate: Theorem 4.16 is the single most consequential unproven input. I verified that Theorem 12.5's proof (homeomorphism via diagram with Theorem 11.13, Corollary 9.5, Theorem 5.3) does not invoke Theorem 4.16, so the bounded-rank spectra (Theorem B) are safe. However, the VI-module classification—a headline application—passes through Theorem 6.13, whose proof explicitly relies on Theorem 4.16 for both the support-monotonicity property (1) and the cofinite-support property (2); property (2) actually follows automatically from finiteness of the generating set of a compact object, but property (1) is the genuine load-bearing content and is equivalent to standardness of D(Ep)^c. The infinite Krull dimension of Theorem C also flows through Theorem F/Corollary 4.19, again from Theorem 4.16. I found no internal contradiction in the rest of the spectral machinery; the minor edge case in Theorem 6.13 where S = {0} (i.e., hsupp(X) = N\\{0}) makes the displayed equality hsupp(X) = ∩ hsupp(m_s) false because m_0 = 0, but the conclusion remains true by a separate maximality argument, so it does not create a structural risk. The only check that would settle the external dependency is a full verification of Theorem 4.16 in the companion paper, or a concrete counterexample. I therefore keep the reader's CONDITIONAL verdict unchanged.","tokens_in":59833,"tokens_out":32656,"duration_ms":319448,"concrete_test":"Independently re-derive Theorem 4.16 for the representative case U = Ep: take a nonzero compact object X, explicitly construct a torsion-free class x, and prove e_n ∈ thick⊗⟨X⟩ for some n using only the projective generators e_m and the partial order on N. Concretely, verify that for X = cof(e_1 → 1), one has e_2 ∈ thick⊗⟨X⟩ and consequently hsupp(X) = N\\{1} is cofinite; then check whether the same argument generalizes to an arbitrary X. If this independent derivation fails for any explicit X, Theorem A is refuted; if it succeeds, the external dependency is discharged and the verdict can be upgraded.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem A (classification of derived VI-modules) and the infinite Krull dimension half of Theorem C both depend on Theorem 4.16, imported from [BBP+25, Thm 7.7 & Cor 7.11]. This theorem asserts that in any multiplicative global family U, every nonzero compact X has a torsion-free element x ∈ H^*(X)(G) (nonvanishing under all epimorphism pullbacks), and that any such x forces e_G ∈ thick⊗⟨X⟩. It is used verbatim to prove Proposition 4.17 (zero ideal prime), Corollary 4.19 (strict descent of family primes, hence infinite Krull dimension), and the two support-theoretic properties (1)-(2) in Theorem 6.13 that yield standardness of D(Ep)^c and the VI-module bijection. The present paper contributes no proof, no special-case verification, and no alternative route; Theorem 6.13 explicitly says both properties 'rely on Theorem 4.16'. If the companion statement is false or has a hidden hypothesis, Theorem A, Theorem F, and the first part of Theorem C collapse. Note that Theorem H (Theorem 12.5) and Theorem B are not affected—they use only reflective filtration, Theorem 5.3, and Proposition 4.4—so the bounded-rank spectral computations do not inherit this risk.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops tensor-triangular geometry for derived categories of global representations over fields of characteristic zero, indexed by families of finite groups. It introduces homological support, group primes, family primes, profinite group primes, and reflective filtrations, and uses them to compute Balmer spectra in several important cases. The main results are: a complete tt-theoretic classification of compact derived VI-modules (Theorem A); the spectrum for abelian p-groups of p-rank at most r (Theorem B); infinite Krull dimension and infinite Cantor–Bendixson rank for the family of all finite abelian p-groups (Theorem C); an equivalence with rational global spectra (Theorem D); a surjectivity criterion for group primes in p-group families (Theorem E); primeness of family primes for multiplicative global families (Theorem F); the spectrum for elementary abelian p-groups (Theorem G); and a general homeomorphism between the profinite group prime space and the Balmer spectrum for r-submultiplicative families (Theorem H). The framework also yields standardness and classification of thick ideals in the computed cases.","tokens_in":60061,"tokens_out":32799,"duration_ms":370606,"significance":"If the main results hold, this is a substantial and original contribution to the tensor-triangular geometry of non-rigid categories. The classification of derived VI-modules is a concrete and striking application, and the construction of profinite group primes together with the use of reflective filtrations and inverse limits is a novel method likely to be influential. The proof of Theorem H is elegant and largely independent of the heavier homological input. The paper is carefully structured, with explicit references to earlier results and no fitting parameters. However, some headline results—Theorem A, the infinite Krull dimension half of Theorem C, and Theorem F—depend on a growth theorem imported from a companion paper, and one supporting lemma contains a confusing functorial identification that appears to be a typo. These points need to be resolved before the paper can be fully verified.","major_comments":[{"comment":"Theorem 4.16 is quoted from the companion paper [BBP+25, Thm 7.7 and Cor 7.11] and is load-bearing for several central results. It is used in Proposition 4.17 to prove that the zero ideal is prime, in Corollary 4.19 to prove that the family primes form a strictly descending infinite chain, and in the proof of Theorem 6.13 to establish properties (1) and (2) that yield the standardness of D(Ep)^c and the classification of derived VI-modules. The present paper provides no proof, no special-case verification, and no alternative route; Theorem 6.13 explicitly says that both properties rely on Theorem 4.16. Consequently, if the companion statement is false or has an unstated hypothesis, Theorem A, Theorem F, and the first part of Theorem C collapse. The authors should either include a proof of Theorem 4.16 (or a detailed outline) or make the dependence on the companion paper a clearly stated hypothesis, so that the reader can assess the validity of the main claims.","section":"§4, Theorem 4.16"},{"comment":"The proof of Lemma 7.15 asserts that 'q_* ≃ i!' and later that 'q_* moreover preserves coproducts'. With the conventions of Construction 2.12, q_* is the right Kan extension along q, and for a reflective inclusion q ⊣ i the correct identifications are q^* ≃ i_! and q_* ≃ i^*, not q_* ≃ i_!. Moreover, i^* is not generally fully faithful, so the argument as written is not correct. The intended argument likely uses q^*: since q^*(1) ≃ 1 and q^* preserves coproducts, the compactness claim follows from Hom_D(U)(1, −) ≃ Hom_D(V)(1, q^*(−)). This is a local and fixable issue, but the current text contains a genuine error in a proof.","section":"§7, Lemma 7.15"}],"minor_comments":[{"comment":"The claim that the category of finite groups expressible as products of simple groups is reflective is stated without proof, with the note that the details are omitted. Since this example is not used in the rest of the paper, the omission is acceptable, but it should be flagged as unproved or moved to a remark to avoid giving the impression of a fully verified assertion.","section":"Example 7.7(d)"},{"comment":"There are several minor typographical issues, including 'accommulation' in Figure 4, 'F amily' in the title, and 'choose Gn maximal among the Gis' in the proof of Proposition 5.2. These should be corrected in a final revision.","section":"Throughout"},{"comment":"It would improve readability to add a short paragraph at the beginning of Part 4 explicitly listing which of the paper's main results depend on the imported Theorem 4.16 and which do not. The reader currently has to infer this from the proofs, and the distinction matters for assessing the robustness of the different theorems.","section":"Part 4"}],"recommendation":"major_revision","confidential_remarks":"The main risk is the heavy dependence on the companion paper [BBP+25] for the growth theorem used in Theorems A, C, and F. The editors should verify that the companion paper is available, under review, or otherwise accessible, since the present paper's headline results cannot be checked independently without it. The rest of the paper is strong and the issue in Lemma 7.15 appears to be a fixable typo. I would support acceptance after the authors clarify or prove the imported theorem and correct the proof of Lemma 7.15."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline is real if the companion paper holds up: this paper computes Balmer spectra for global representations of several infinite families—cyclic, elementary abelian, bounded-rank abelian p-groups—and gives a complete tt-theoretic classification of compact derived VI-modules. The main new idea is to get leverage in the non-rigid setting via reflective filtrations and profinite group primes, which is a real step beyond the essentially finite cases of Xu and Wang.\n\nWhat the paper does well: the structure is careful, the proofs are explicit with cross-references, and the authors are transparent about what depends on what. Theorem H and Theorem B—the spectrum for bounded-rank abelian p-groups—are proved from reflective filtrations and colimit machinery, and they do not depend on the imported growth theorem. Those computations look solid and are themselves a substantial contribution.\n\nThe soft spot is localized but real. Theorem 4.16, quoted verbatim from the authors' companion paper [BBP+25, Thm 7.7 & Cor 7.11], asserts that in a multiplicative global family every nonzero compact object has a torsion-free element, and any such element generates the whole thick⊗-ideal. This is used to prove standardness of D(E_p)^c and hence the VI-module classification (Theorem A), as well as Theorem F and the infinite Krull dimension half of Theorem C. The authors say explicitly that Theorem 6.13 relies on Theorem 4.16, so there is no hidden assumption—but this paper gives no proof and no special-case check. If that companion theorem has a gap, those headline results collapse. The bounded-rank results and Theorem H are unaffected, so the risk is contained to the parts of the paper that advertise the most dramatic classification.\n\nOn citation pattern and novelty: the self-citations point to the foundational companion paper and to [PS22], which is legitimate given the dependency. The free parameters are honest—no fitting, no made-up entities that drive the conclusions. The conjectural description for all finite abelian p-groups is clearly labelled as future work.\n\nThis paper deserves a serious referee. The referee should read the companion paper's Theorem 7.7 and Corollary 7.11 carefully, and the authors should be asked to either include a proof of Theorem 4.16 in this paper or make the dependency even more prominent in the introduction. I would send it to peer review rather than desk reject.","headline":"A carefully built non-rigid tt-geometry computation whose headline VI-module classification rests on a growth theorem imported from the authors' companion paper; the bounded-rank spectral computations stand on their own.","tokens_in":60663,"tokens_out":1627,"would_cite":true,"duration_ms":24663,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18F99","18G80","20C99","55P91","18A25"],"pacs":[],"model":"deepseek-v4-flash","headline":"For r-submultiplicative families of finite groups, the Balmer spectrum of global representations is homeomorphic to the space of profinite groups built from the family, classifying derived VI-modules by cofinite support sets.","keywords":["global representations","tensor-triangular geometry","Balmer spectrum","VI-modules","profinite group primes","reflective filtrations","representation stability","non-rigid tt-geometry"],"falsifier":"Compute, for each integer $s \\ge 0$, the thick tensor ideal generated by $m_s = \\operatorname{cof}(e_{(\\mathbb{Z}/p)^{\\oplus s}} \\to \\mathbf{1})$ in $\\mathsf{D}(\\mathcal{E}_p)^c$ and its Balmer support. The theorem predicts the support is exactly the complement of the single prime $p_{(\\mathbb{Z}/p)^{\\oplus s}}$ and that every prime ideal is generated by a single object; exhibiting a nonzero compact object whose homological support is finite rather than cofinite, or a thick ideal that is not radical, would falsify the classification of derived VI-modules.","tokens_in":59569,"feed_emoji":"🗺️","tokens_out":12770,"duration_ms":122831,"temperature":0.7,"pith_summary":"The paper begins the tensor-triangular geometry of global representations: compatible systems of representations of the outer automorphism groups of the finite groups in a family $\\mathscr{U}$, organized into a derived category $\\mathsf{D}(\\mathscr{U};k)$. Its central theorem (Theorem H) identifies the Balmer spectrum of compact objects with a profinite space of primes attached to profinite groups: for any $r$-submultiplicative family — all groups generated by at most $r$ elements and closed under wide subgroups of products — the spectrum is homeomorphic to $\\pi_0(\\widehat{\\mathscr{U}})$, the space of finitely generated profinite groups built from finite quotients in $\\mathscr{U}$. From this it derives a complete classification of compact derived VI-modules by cofinite support sets (Theorem A), explicit spectra for abelian $p$-groups of $p$-rank at most $r$ with Cantor-Bendixson rank exactly $r$ (Theorem B), and a demonstration that the spectrum for all finite abelian $p$-groups has infinite Krull dimension and infinite Cantor-Bendixson rank (Theorem C). Because $\\mathsf{D}(\\mathscr{U})$ is almost never rigidly compactly generated, the standard surjectivity results of tt-geometry fail, so the paper contributes new methods for non-rigid tt-geometry: group primes, family primes, profinite group primes, and reflective filtrations.","feed_headline":"Global representation spectra classified by profinite groups","feed_subtitle":"Profinite groups parametrize the prime ideals; compact VI-modules reduce to cofinite support sets.","key_machinery":"The load-bearing construction is the profinite group prime: to each finitely generated profinite group $G$ in the profinite extension $\\widehat{\\mathscr{U}}$ of a family $\\mathscr{U}$, the paper assigns the thick ideal $p_G = \\{X \\in \\mathsf{D}(\\mathscr{U})^c \\mid \\operatorname{colim}_{N \\in \\mathcal{N}(G;\\mathscr{U})} H^*(X(G/N)) = 0\\}$, which is prime by construction, and shows that the map $G \\mapsto p_G$ is injective whenever $\\mathscr{U}$ admits a profinite reflective filtration. Surjectivity and the homeomorphism are proven by writing $\\widehat{\\mathscr{U}}$ as an inverse limit of essentially finite families $\\mathscr{U}[n]$ through the reflections $q_n$, invoking a continuity theorem for filtered colimits of tensor-triangulated categories to compute $\\operatorname{Spc}(\\mathsf{D}(\\mathscr{U})^c)$ as the inverse limit of the discrete spectra $\\operatorname{Spc}(\\mathsf{D}(\\mathscr{U}[n])^c)$, equipped with the profinite topology. The second engine is a growth theorem imported from the companion paper: for a multiplicative global family and any nonzero compact object $X$, the homology $H^*(X)$ contains a torsion-free element that forces $e_G$ into $\\operatorname{thick}_{\\otimes}\\langle X\\rangle$; this yields primeness of the zero ideal, cofinite homological support in $\\mathsf{D}(\\mathcal{E}_p)^c$, and standardness of all the computed categories.","core_discovery":"The central claim is Theorem 12.5 (Theorem H): if $\\mathscr{U}$ is an $r$-submultiplicative family of finite groups, then the profinite group prime map sends each finitely generated profinite group $G$ built from quotients in $\\mathscr{U}$ to the prime ideal $p_G = \\{X \\in \\mathsf{D}(\\mathscr{U})^c \\mid \\operatorname{colim}_{N \\in \\mathcal{N}(G;\\mathscr{U})} H^*(X(G/N)) = 0\\}$, and this map is a homeomorphism from $\\pi_0(\\widehat{\\mathscr{U}})$ onto the Balmer spectrum $\\operatorname{Spc}(\\mathsf{D}(\\mathscr{U})^c)$. Equivalently, the homological support extended to profinite groups is the universal support datum: it bijects thick ideals of $\\mathsf{D}(\\mathscr{U})^c$ with open subsets of $\\pi_0(\\widehat{\\mathscr{U}})$, finitely generated thick ideals with clopen subsets, and prime ideals with complements of points, and every thick ideal is radical. The special case $\\mathscr{U} = \\mathcal{E}_p$, the elementary abelian $p$-groups, yields Theorem A: compact derived VI-modules are classified up to tt-equivalence by the cofinite subsets of $\\mathbb{N}$ together with the empty set, via the type map $M \\mapsto \\{n \\mid M(\\mathbb{F}_p^{\\oplus n}) \\neq 0\\}$.","pith_inferences":["The profinite-compactification template — replace the discrete indexing set of group primes by an inverse limit of essentially finite pieces — suggests a general strategy for computing Balmer spectra in other non-rigid tensor-triangulated categories, such as functor categories of FI-modules or twisted commutative algebras.","Theorem E isolates a sharp failure of rigidity: jointly conservative evaluation functors are never jointly surjective on spectra for infinite families of $p$-groups, so any non-rigid analogue of the standard surjectivity criterion must be phrased in terms of profinite completions rather than the original indexing set.","The infinite Krull dimension of the abelian $p$-group spectrum is generated by the $p$-exponent filtration of family primes; tracking the same filtration transfinitely could yield a full description of $\\operatorname{Spc}(\\mathsf{D}(\\mathcal{A}(p))^c)$, a problem the paper states it plans to return to.","Via the equivalence with rational global spectra, the computed tt-geometry furnishes a classification of thick subcategories of rational global spectra for submultiplicative families, a consequence the paper leaves implicit."],"forward_implications":["Compact derived VI-modules over a characteristic-zero field are completely classified up to tt-equivalence by their type: a cofinite subset of the natural numbers, or the empty set.","For the family of abelian $p$-groups of $p$-rank at most $r$, the Balmer spectrum is the explicit profinite space $bS_{\\le r} = \\{v \\in (\\mathbb{N}^+)^r \\mid \\infty \\ge v_1 \\ge \\cdots \\ge v_r \\ge 0\\}$, with Cantor-Bendixson rank exactly $r$.","Every thick ideal in the compact derived category of an $r$-submultiplicative family is radical, so homological support gives a complete classification of thick ideals by open subsets of $\\pi_0(\\widehat{\\mathscr{U}})$.","The spectrum for all finite abelian $p$-groups has infinite Krull dimension and infinite Cantor-Bendixson rank, witnessed by strict descending chains of family primes and by embeddings of the bounded-rank spectra.","For the family of $r$-generated finite abelian groups, the spectrum decomposes as a product over all primes $p$ of the spectra for the corresponding $p$-parts."],"supporting_citations":[{"why":"Companion paper supplying Theorem 4.16, the growth theorem on torsion-free homology elements that underwrites primeness of the zero ideal, cofinite support, and standardness.","marker":"[BBP+25]"},{"why":"Foundation for the abelian category of global representations, its projective generators $e_G$, tensor structure, and change-of-family adjunctions used throughout.","marker":"[PS22]"},{"why":"Establishes the Balmer spectrum, the universal support formalism, and the radical-ideal criterion ($a \\in \\operatorname{thick}_{\\otimes}\\langle a \\otimes a\\rangle$) that defines standardness.","marker":"[Bal05]"},{"why":"Continuity theorem for Balmer spectra under filtered colimits of tt-categories, which turns the reflective filtration into the inverse-limit description of the spectrum.","marker":"[Gal18]"},{"why":"Provides rational global spectra and geometric fixed points, whose comparison with $\\mathsf{D}(\\mathscr{U};\\mathbb{Q})$ is sharpened in Theorem D.","marker":"[Sch18]"},{"why":"Prior rational global homotopy theory input (compact generators and geometric-fixed-point homology) used in the proof of Theorem D.","marker":"[Wim17]"},{"why":"The rigid-case surjectivity criterion whose failure in the non-rigid setting is demonstrated by Theorem E.","marker":"[BCHS24]"},{"why":"Neeman-Thomason localization theorem used to pass Verdier quotients and recollements to compact objects and compute spectra of quotients.","marker":"[Nee92]"},{"why":"Background on VI-modules and representation stability; the category of VI-modules is realized as $\\mathsf{D}(\\mathcal{E}_p)$.","marker":"[PS17]"}],"fun_headline_variants":["Profinite groups classify global representation spectra","Profinite groups parametrize Balmer spectrum of global reps","Global rep tt-spectrum homeomorphic to profinite groups","Cofinite supports classify compact VI-modules tt-theoretically"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper depends on a growth theorem proven in its companion paper: for a multiplicative global family, the homology of any nonzero compact object contains an element that stays nonzero under pullback along any epimorphism and forces a generator into the object's thick tensor ideal; if that theorem were false, the standardness of $\\mathsf{D}(\\mathcal{E}_p)^c$, the cofinite-support result, and the primeness of family primes would all collapse.","fun_headline_variants_meta":{"raw":{"variants":["Profinite groups classify global representation spectra","Profinite groups parametrize Balmer spectrum of global reps","Global rep tt-spectrum homeomorphic to profinite groups","Cofinite supports classify compact VI-modules tt-theoretically"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000982,"raw_usage":{"total_tokens":4219,"prompt_tokens":1047,"completion_tokens":3172,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":663,"completion_tokens_details":{"reasoning_tokens":3088}},"tokens_in":663,"tokens_out":3172,"duration_ms":27990,"temperature":1.0,"reasoning_tokens":3088,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:24:07.113768+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for each integer $s \\ge 0$, the thick tensor ideal generated by $m_s = \\operatorname{cof}(e_{(\\mathbb{Z}/p)^{\\oplus s}} \\to \\mathbf{1})$ in $\\mathsf{D}(\\mathcal{E}_p)^c$ and its Balmer support. The theorem predicts the support is exactly the complement of the single prime $p_{(\\mathbb{Z}/p)^{\\oplus s}}$ and that every prime ideal is generated by a single object; exhibiting a nonzero compact object whose homological support is finite rather than cofinite, or a thick ideal that is not radical, would falsify the classification of derived VI-modules.","supporting_citations":[],"review_version":1}