{"id":"d96f6920-b144-431e-a440-592f461fdb50","arxiv_id":"2504.16602","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For H1F groups of type FP∞, the Balmer spectrum of dualisable objects is Proj(H*(G,k)) and the telescope conjecture holds, while infinite free products give Stone-Cech spectra and failures of the telescope conjecture and stratification.","lead":"This paper computes the Balmer spectrum of the stable module category for certain infinite groups and shows that a version of the telescope conjecture holds exactly for H1F groups of type FP∞ but fails for many infinite free products. The results give the first explicit tensor-triangular spectra for non-rigidly-compactly generated categories and provide counterexamples to stratification and to the telescope conjecture.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Prop 6.9's surjectivity proof rests on an unproved module-finiteness claim: H*(E,k) finite over H*(G,k) via restriction is asserted from an F-isomorphism, which does not justify it.","rationale":"The reader's weakest assumption points to Proposition 6.9's closedness claim, and our independent read lands on the same spot. The proof of surjectivity depends on realizing V as a closed subset of Proj(H*(G,k)), and the only justification given for closedness is an inference from Henn's F-isomorphism that does not logically force module-finiteness. This is a genuine gap, though likely fixable by a known finiteness result. It does not change the verdict: CONDITIONAL is appropriate. We also note a smaller slip in Proposition 6.8 where the text writes X^{⊗m} after only having a bound for X; this appears to be a typo and is not the core issue. The paper is otherwise coherent, with substantial independent support from published results, and no theatrical or ad hominem critique is warranted.","tokens_in":25567,"tokens_out":19647,"duration_ms":174760,"concrete_test":"Check the module-finiteness claim: for G an H1F group of type FP∞ and E a finite elementary abelian subgroup, prove or find a published proof that H*(E,k) is a finitely generated module over H*(G,k) via restriction. One concrete way is to use the finite free kG-resolution of k to build a finite resolution of k over kE and see whether it yields finite generation over H*(G,k); if that argument fails, test G = C_p * C_p with E = C_p by computing the restriction map H*(G,k) → H*(E,k) and determining whether H*(E,k) is finite as an H*(G,k)-module. If the example is not finite, Proposition 6.9's closedness step is false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Proposition 6.9, surjectivity of Thickid^f(G^d) → lim_E Thickid^f(E^d) is proved by forming V = ∪_E res*_{G,E}(Supp_E(N_E)) ⊆ Proj(H*(G,k)) and invoking Proposition 6.7 to realize V as the support of a dualisable object. For V to be closed, the paper needs each res*_{G,E} to be a closed map, which it justifies by asserting that 'H*(E,k) is finitely generated as a module over lim_{F∈AE(G)} H*(F,k) and hence by the F-isomorphism of Theorem 6.4 ... finitely generated as a module over H*(G,k)'. This 'hence' is not a consequence of F-isomorphism: a uniform F-isomorphism A → B does not imply that a B-module finite over B is finite over A. The finiteness of H*(E,k) over H*(G,k) via restriction is a separate, nontrivial assertion, possibly true via the finite free resolution of k over kG, but it is not proved or cited. As written, the step is unsupported, and it is load-bearing: if the finiteness failed, the union V might not be closed, Proposition 6.7 would not apply, and the surjectivity proof would collapse. The central FP∞ spectrum theorem (Theorem 1.1, Corollary 6.11) therefore rests on an unproved premise.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper determines the Balmer spectrum of the subcategory of dualisable objects in the stable module category for two classes of infinite groups: H1F groups of type FP∞ over a field of characteristic p>0, and infinite free products of finite groups of p-rank one. In the FP∞ case the spectrum is shown to be Proj(H*(G,k)), extending the finite-group theorem of Benson, Carlson, and Rickard, and the telescope conjecture is proved. In the free-product case the spectrum is the Stone-Čech compactification of N, while for more general free products the restriction map from the spectral coproduct is shown not to be surjective. The paper also gives examples where the stable category is not stratified by the spectrum of dualisable objects and where the telescope conjecture fails.","tokens_in":25812,"tokens_out":12407,"duration_ms":106811,"significance":"If the arguments are correct, these are the first computations of Balmer spectra for non-rigidly-compactly generated stable module categories, and they provide a meaningful extension of the telescope conjecture to a setting where compact and dualisable objects diverge. The paper is clearly written and uses a coherent lattice-theoretic framework; it builds on the author's prior classification of localising tensor ideals, and the free-product counterexamples are explicit and instructive. The main theorems are substantial and would be of considerable interest to the tensor-triangular geometry and modular representation theory communities.","major_comments":[{"comment":"The proof of surjectivity of Thickid^f(G^d) → lim_E Thickid^f(E^d) relies on the assertion that V = ∪_E res*_{G,E}(Supp_E(N_E)) is closed in Proj(H*(G,k)). The text states that 'H*(E,k) is finitely generated as a module over lim_{F∈AE(G)} H*(F,k) and hence by the F-isomorphism of Theorem 6.4 ... finitely generated as a module over H*(G,k)'. The second step is not a consequence of uniform F-isomorphism: a uniform F-isomorphism A → B does not imply that a B-module is finite over A. The finiteness of H*(E,k) over H*(G,k) via restriction is a separate, non-obvious assertion; it is needed to conclude that res*_{G,E} is a closed map, hence that V is closed, and hence that Proposition 6.7 applies. If this finiteness fails, the surjectivity proof collapses, and with it the spectrum computation of Corollary 6.11 and the injectivity part of Corollary 6.15. Please provide a proof or a precise reference (for example, a result in [12]) for this module finiteness.","section":"§6, Proposition 6.9"},{"comment":"In the proof of injectivity, the text passes from the vanishing (ξ_N^{⊗n}⊗X)↓_E = 0 for each elementary abelian E to the statement (ξ_N^{⊗m}⊗X^{⊗m})↓_E = 0 for some m, and then invokes Proposition 6.2. This transition is not justified as written: one must first tensor the zero morphism (ξ_N^{⊗n}⊗X) with X^{⊗(n-1)} to get (ξ_N^{⊗n}⊗X^{⊗n})↓_E = 0, and then tensor with suitable copies of X and ξ_N to obtain the claimed exponent m on both factors. Moreover, the application of Proposition 6.2 requires the hypothesis that a single morphism f has f↓_E = 0 for all E, which should be stated explicitly. The gap appears repairable with a short explanation, but it is load-bearing for the injectivity of the lattice map and hence for the spectrum theorem.","section":"§6, Proposition 6.8"}],"minor_comments":[{"comment":"There are typographical errors in the abstract: 'W e' should be 'We', and 'FP 8' appears where 'FP∞' is clearly intended.","section":"Abstract"},{"comment":"The tensor unit is rendered as '/BD' (likely a LaTeX artifact) in several places, for example in Proposition 2.5 and Definition 2.6; this should be replaced by a proper symbol such as 𝟙.","section":"§2, Definition 2.6 and elsewhere"},{"comment":"In the proof, the notation 'M òG_EÓG_F' in the first sentence is confusing; the subscript in the restriction should likely be E rather than F, or the sentence should be rewritten for clarity.","section":"§4, Lemma 4.7"},{"comment":"The sentence 'Since there are only finite many conjugacy classes ... we may take some m > 0 such that (ξ_N^{⊗m}⊗X^{⊗m})↓_E = 0' would benefit from an explicit explanation of how the exponent m on X is obtained, as discussed in the corresponding major comment.","section":"§6, Proposition 6.8"},{"comment":"In the proof, the phrase 'If N is projective, and Y ∈ Thick(N), then Y is projective and so ξ_N ⊗ Y = 0' should be expanded to clarify that this covers the case where M↓_{H_n} is projective, and that the conclusion needed is (ξ_M^{⊗C}⊗X)↓_{H_n} = 0.","section":"§5, Theorem 5.14"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically ambitious and largely well-organised, but the two gaps identified in the major comments are load-bearing for the FP∞ spectrum theorem. The finiteness issue in Proposition 6.9 is the more serious one; if the author can supply a proof or precise reference for the module finiteness of H*(E,k) over H*(G,k), the paper would be close to publishable. The paper relies heavily on the author's previous work [27,28], which is reasonable, but the current manuscript does not make the dependence fully transparent. I recommend major revision rather than rejection, as the identified issues appear fixable within the scope of the manuscript."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. This is the first real computation of Balmer spectra for stable module categories of infinite groups in the non-rigidly-compactly generated setting, and it gives the first counterexamples there to the telescope conjecture and to stratification. Those results are genuinely new. But the main FP∞ spectrum theorem has a load-bearing gap: Proposition 6.9 needs the map res*_{G,E}: Proj H*(E,k) -> Proj H*(G,k) to be closed, and the author justifies this by asserting that H*(E,k) is finitely generated as a module over H*(G,k), claiming this follows from Henn's F-isomorphism. It doesn't. A uniform F-isomorphism between algebras does not imply module-finiteness; the stress-test note is right about that. The statement may well be true and provable from known results (Benson [12] is cited nearby), but it is not proved or cited here, and the surjectivity argument collapses without it.\n\nWhat the paper does well: the free-product computations are clean. For p-rank-one factors the spectrum is the Stone-Cech compactification of N, and Theorem 5.19 shows that in the p-rank-two case the restriction map is not an epimorphism, a useful counterpoint to the rigidly-compactly generated theory. The telescope conjecture failures in Section 5 are simple and convincing, and the lattice-theoretic setup is efficient. The paper is clearly written and honest about the limits of the generalisation.\n\nSoft spots, in proportion. The step in Proposition 6.8 flagged by the reader is real but minor: from vanishing of (ξ_N^{⊗m}⊗X)↓E you do get (ξ_N^{⊗m}⊗X^{⊗m})↓E = 0, and then Proposition 6.2 needs to be applied to the morphism ξ_N^{⊗m}⊗X^{⊗m}; the text skips that, so as written it doesn't follow, but it's a two-line fix. The bigger concern is the dependence on the two unpublished same-author preprints [27] and [28] for the stable category construction and the classification of localising ideals. This is understandable in the area, but it means a referee cannot fully verify the foundations from this paper alone.\n\nWho this is for: people working in tensor triangular geometry and modular representation theory of infinite groups. It gives them the first concrete picture of how non-rigid categories behave differently from the rigid case. Recommendation: send it to peer review. The referee should ask for a proper proof or citation of the finiteness of H*(E,k) over H*(G,k), and check that the preprints do contain the quoted classification results. If the finiteness claim holds up, this is a substantial result.","headline":"Genuinely new Balmer spectra for infinite group stable categories and telescope counterexamples, but the FP∞ spectrum theorem has a load-bearing finiteness assertion that needs a real proof or citation.","tokens_in":26376,"tokens_out":13496,"would_cite":true,"duration_ms":125695,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20C20","20J06","18G80"],"pacs":[],"model":"deepseek-v4-flash","headline":"For type-FP∞ groups the dualisable-module spectrum is the prime-ideal space of group cohomology, and for p-rank-one free products it is the Stone-Čech compactification of N; the paper also settles the telescope conjecture in both settings.","keywords":["Balmer spectrum","stable module category","telescope conjecture","H1F groups","FP-infinity groups","Stone-Čech compactification","tensor triangular geometry","dualisable objects"],"falsifier":"For $G = \\mathbb{Z}^n$, an $\\mathrm{H}_1\\mathfrak{F}$ group of type $\\mathrm{FP}_\\infty$, and $E = (C_p)^n$ its maximal elementary abelian $p$-subgroup, verify whether $H^*(E,k)$ is finitely generated as a module over $H^*(G,k)$ under restriction: Proposition 6.9 requires this for every such pair, and one failure would break the surjectivity step and with it the identification of the spectrum with $\\mathrm{Proj}(H^*(G,k))$. On the free-product side, the load-bearing claim is that for $H = C_p$ every non-projective finitely generated module $M$ has $\\xi_M = 0$ in the stable category; computing the coevaluation morphism $\\xi_M$ for one such module, for instance the augmentation ideal, either confirms or refutes the uniform nilpotence bound on which Theorem 5.14 rests.","tokens_in":25300,"feed_emoji":"♾️","tokens_out":60060,"duration_ms":453912,"temperature":0.7,"pith_summary":"This paper computes the Balmer spectrum — the space that classifies thick tensor ideals — of dualisable objects in the stable module category for two families of infinite groups, where the category is not rigidly compactly generated and few spectra were previously known. For an $\\mathrm{H}_1\\mathfrak{F}$ group of type $\\mathrm{FP}_\\infty$ over a field of characteristic $p > 0$ — a group with a finite-dimensional classifying space for proper actions and a finitely generated projective resolution of the trivial module — the spectrum is homeomorphic to $\\mathrm{Proj}(H^*(G,k))$, the space of homogeneous prime ideals of the group's cohomology. For an infinite free product of finite groups of $p$-rank one, the spectrum is instead the Stone-Čech compactification of $\\mathbb{N}$. The paper also settles the telescope conjecture — whether every smashing tensor ideal is generated by dualisable objects — in these settings: it holds for the $\\mathrm{FP}_\\infty$ groups and fails for the free-product examples, giving counterexamples in the infinite-group setting. Along the way it shows that the stable category of an infinite free product of cyclic groups is not stratified by the spectrum of its dualisable objects: the spectrum has $2^{2^{\\aleph_0}}$ points, and there are only $2^{\\aleph_0}$ localising tensor ideals, so no bijection can exist.","feed_headline":"Telescope conjecture is true for FP∞ and false for free products","feed_subtitle":"True for FP∞ groups, false for p-group free products; cyclic factors give the Stone-Čech compactification of N.","key_machinery":"The argument runs through the lattice-theoretic description of the Balmer spectrum via Stone duality: the spectrum of a tensor-triangulated category is the spectral space attached to the distributive lattice $\\mathrm{Thickid}_f$ of finitely generated thick tensor ideals, and restriction to finite elementary abelian $p$-subgroups assembles a map $\\mathrm{Thickid}_f(\\underline{\\mathrm{Mod}}(kG)^d) \\to \\lim_{E \\in \\mathcal{A}_E(G)} \\mathrm{Thickid}_f(\\underline{\\mathrm{Mod}}(kE)^d)$, which the paper proves to be an isomorphism in both settings. For $\\mathrm{H}_1\\mathfrak{F}$ groups of type $\\mathrm{FP}_\\infty$, the decisive input is the paper's Theorem 6.4: a uniform $F$-isomorphism $H^*(G,k) \\to \\lim_E H^*(E,k)$, finite generation of $H^*(G,k)$, and finitely many conjugacy classes of elementary abelian $p$-subgroups; this yields the homeomorphism of projective spectra and, crucially, finite generation of each $H^*(E,k)$ as a module over $H^*(G,k)$, so that unions of supports over the conjugacy classes are closed. For free products, the decisive tool is the $\\xi_M$ criterion of [4]: $\\mathrm{Thick}(M) = \\{X \\mid \\xi_M^{\\otimes n} \\otimes X = 0 \\text{ for some } n \\geq 0\\}$, where $\\xi_M$ fills the coevaluation triangle $C_M \\to k \\to M \\otimes M^*$; for a finite group of $p$-rank one a uniform bound $C$ satisfies $\\xi_M^{\\otimes C} = 0$ for every $M$, and this uniform nilpotence lets the restriction map to the product of the factor lattices be inverted. The telescope-conjecture dichotomy has the same root: for $\\mathrm{FP}_\\infty$ groups every smashing ideal is generated by dualisable objects, while in the free-product case the tensor unit is not compact, and distinct thick ideals of dualisable objects can generate the same smashing subcategory.","core_discovery":"The central assertion is Theorem 1.1. Let $k$ be a field of characteristic $p > 0$. If $G$ is an $\\mathrm{H}_1\\mathfrak{F}$ group of type $\\mathrm{FP}_\\infty$ — a group with a finite-dimensional model for the classifying space for proper actions whose trivial module has a resolution by finitely generated projectives — then the Balmer spectrum $\\mathrm{Spc}(\\underline{\\mathrm{Mod}}(kG)^d)$ of dualisable objects in the stable module category is homeomorphic to $\\mathrm{Proj}(H^*(G,k))$, the homogeneous prime spectrum of group cohomology; if instead $G$ is an infinite free product of finite groups of $p$-rank one, the same spectrum is the Stone-Čech compactification of $\\mathbb{N}$. These are the first Balmer spectra for infinite-group stable module categories that are not rigidly compactly generated. The paper further proves that the telescope conjecture holds in the $\\mathrm{FP}_\\infty$ case — every smashing localising tensor ideal is generated by dualisable objects — and fails for the free-product examples, where distinct thick tensor ideals of dualisable objects generate the same smashing subcategory. In the rank-one free-product case the stable category is not stratified by the spectrum of its dualisable objects, since $\\beta\\mathbb{N}$ has $2^{2^{\\aleph_0}}$ points but there are only $2^{\\aleph_0}$ localising tensor ideals.","pith_inferences":["If the spectrum computation is right, the same uniform-nilpotence mechanism ($\\xi_M^{\\otimes C} = 0$ for $p$-rank-one factors) should compute the spectrum of dualisable objects for other graphs of finite groups with cyclic edge groups — amalgamated free products and HNN extensions — as an appropriate compactification of the vertex or edge set, with the Stone-Čech compactification of $\\mathbb{N}$ a","The cardinality mismatch behind the non-stratification result points to a systematic limitation: for non-rigid categories, the spectrum of dualisable objects is too coarse to stratify the category, and a support theory defined on all compact objects — which here must exist without compacts being tensor-closed — would be needed to recover a bijection with localising tensor ideals.","The paper's dichotomy separates two hypotheses that often travel together: the tensor unit being compact and full cohomological finiteness of type $\\mathrm{FP}_\\infty$; a natural test is whether the telescope conjecture continues to hold for $\\mathrm{H}_1\\mathfrak{F}$ groups whose tensor unit is compact but whose cohomology is not finitely generated, which would show the $\\mathrm{FP}_\\infty$ hypot"],"forward_implications":["For an $\\mathrm{H}_1\\mathfrak{F}$ group of type $\\mathrm{FP}_\\infty$, the Balmer spectrum of dualisable objects is $\\mathrm{Proj}(H^*(G,k))$ and the telescope conjecture holds, so every smashing localising tensor ideal is generated by dualisable objects and distinct thick ideals of dualisable objects generate distinct smashing ideals.","For an infinite free product of finite $p$-groups, the telescope conjecture fails: $\\mathrm{Thick}^b(\\{k\\uparrow^G_{H_n}\\})$ and $\\mathrm{Thick}^b(k)$ are distinct thick tensor ideals of dualisable objects, yet both generate the whole stable category as a localising tensor ideal.","For an infinite free product of cyclic groups, the stable category is not stratified by the Balmer spectrum of dualisable objects: the spectrum $\\beta\\mathbb{N}$ has $2^{2^{\\aleph_0}}$ points and there are $2^{\\aleph_0}$ localising tensor ideals.","More generally, a countable product of tensor-triangular fields has spectrum $\\beta\\mathbb{N}$ and is not stratified by the spectrum of its dualisable objects.","When the free factors have $p$-rank at least two, the restriction functors to the factors are jointly conservative yet do not induce a jointly surjective map on the Balmer spectrum, unlike what happens for rigidly compactly generated categories."],"supporting_citations":[{"why":"Constructs the stable module category for LH𝔽 groups and supplies the compact generation statement (Proposition 3.3) and compactness of the tensor unit (Lemma 6.1) on which the whole paper rests.","marker":"[28]"},{"why":"Constructs the stable category for infinite groups of type Φ and gives the gluing procedure used to build modules with prescribed restrictions to the free factors and finite subgroups (Propositions 5.10, 6.9, Theorem 5.19).","marker":"[33]"},{"why":"Classifies the localising tensor ideals of the stable category for LH𝔽 groups and establishes the detection property reducing arguments to elementary abelian subgroups; also supplies the count of localising ideals used in the non-stratification results.","marker":"[27]"},{"why":"Supplies the uniform F-isomorphism between the group cohomology ring and the limit over elementary abelian subgroups, together with finite generation and finiteness of conjugacy classes — the backbone of the FP∞ spectrum theorem.","marker":"[24]"},{"why":"Gives the finite-group spectrum theorem that the FP∞ result extends, and the non-trivial thick tensor ideals for p-rank-two factors used in Theorem 5.19.","marker":"[14]"},{"why":"Provides the criterion identifying a thick tensor ideal generated by M with the objects killed by some tensor power of the coevaluation morphism, used throughout to compare and detect thick tensor ideals.","marker":"[4]"},{"why":"Introduces tensor-triangular fields and b-faithful objects, used to prove the vanishing of the coevaluation morphism over cyclic p-groups and to generalise the free-product spectrum to arbitrary products of fields (Theorem 5.24).","marker":"[6]"},{"why":"Supplies the coevaluation-morphism and tensor-generation machinery behind the uniform nilpotence bound for p-rank-one factors (Proposition 5.12) and the unbounded generation of Theorem 5.18.","marker":"[20]"},{"why":"Proves the telescope conjecture and the support formulas for finite groups, the finite-group input for the smashing-ideal characterisation (Theorem 4.11) and for the support calculus of Section 3.","marker":"[16]"},{"why":"Shows the stable category of an infinite free product splits as a product over the factors, the step that turns factor-wise vanishing of the coevaluation morphism into vanishing in the whole category in Theorem 5.14.","marker":"[22]"}],"fun_headline_variants":["Balmer spectrum for infinite groups: FP∞ true, free products false","First Balmer spectra for infinite stable module categories","Telescope conjecture splits for infinite group categories","Stone-Čech compactification emerges in group cohomology spectra","Non-stratified stable categories for some infinite groups"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The FP∞ spectrum theorem rests on the assertion, made in a single sentence inside the proof of Proposition 6.9, that for every elementary abelian $p$-subgroup $E$, the cohomology $H^*(E,k)$ is finitely generated as a module over $H^*(G,k)$ via restriction, because only this finite generation makes the support unions over the finitely many conjugacy classes closed subsets of $\\mathrm{Proj}(H^*(G,k))$ and lets the restriction map on lattices surject; if that premise failed, the spectrum could be strictly larger than $\\mathrm{Proj}(H^*(G,k))$.","fun_headline_variants_meta":{"raw":{"variants":["Balmer spectrum for infinite groups: FP∞ true, free products false","First Balmer spectra for infinite stable module categories","Telescope conjecture splits for infinite group categories","Stone-Čech compactification emerges in group cohomology spectra","Non-stratified stable categories for some infinite groups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001411,"raw_usage":{"total_tokens":5686,"prompt_tokens":919,"completion_tokens":4767,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":535,"completion_tokens_details":{"reasoning_tokens":4686}},"tokens_in":535,"tokens_out":4767,"duration_ms":31042,"temperature":1.0,"reasoning_tokens":4686,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:59:59.130905+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $G = \\mathbb{Z}^n$, an $\\mathrm{H}_1\\mathfrak{F}$ group of type $\\mathrm{FP}_\\infty$, and $E = (C_p)^n$ its maximal elementary abelian $p$-subgroup, verify whether $H^*(E,k)$ is finitely generated as a module over $H^*(G,k)$ under restriction: Proposition 6.9 requires this for every such pair, and one failure would break the surjectivity step and with it the identification of the spectrum with $\\mathrm{Proj}(H^*(G,k))$. On the free-product side, the load-bearing claim is that for $H = C_p$ every non-projective finitely generated module $M$ has $\\xi_M = 0$ in the stable category; computing the coevaluation morphism $\\xi_M$ for one such module, for instance the augmentation ideal, either confirms or refutes the uniform nilpotence bound on which Theorem 5.14 rests.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Constructs the stable category for infinite groups of type Φ and gives the gluing procedure used to build modules with prescribed restrictions to the free factors and finite subgroups (Propositions 5.10, 6.9, Theorem 5.19)."},{"cited_title":"(Co)stratification of stable categories for infinite groups","cited_arxiv_id":"2410.23941","evidence_quote":"Classifies the localising tensor ideals of the stable category for LH𝔽 groups and establishes the detection property reducing arguments to elementary abelian subgroups; also supplies the count of localising ideals used in the non-stratification results."},{"cited_title":"63 of Proc","cited_arxiv_id":null,"evidence_quote":"Supplies the uniform F-isomorphism between the group cohomology ring and the limit over elementary abelian subgroups, together with finite generation and finiteness of conjugacy classes — the backbone of the FP∞ spectrum theorem."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the finite-group spectrum theorem that the FP∞ result extends, and the non-trivial thick tensor ideals for p-rank-two factors used in Theorem 5.19."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the criterion identifying a thick tensor ideal generated by M with the objects killed by some tensor power of the coevaluation morphism, used throughout to compare and detect thick tensor ideals."},{"cited_title":"(N.S.) 25 (2019), no","cited_arxiv_id":null,"evidence_quote":"Introduces tensor-triangular fields and b-faithful objects, used to prove the vanishing of the coevaluation morphism over cyclic p-groups and to generalise the free-product spectrum to arbitrary products of fields (Theorem 5.24)."},{"cited_title":"Benson and Jon F","cited_arxiv_id":null,"evidence_quote":"Supplies the coevaluation-morphism and tensor-generation machinery behind the uniform nilpotence bound for p-rank-one factors (Proposition 5.12) and the unbounded generation of Theorem 5.18."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows the stable category of an infinite free product splits as a product over the factors, the step that turns factor-wise vanishing of the coevaluation morphism into vanishing in the whole category in Theorem 5.14."}],"review_version":1}