{"id":"345408e9-9a4b-43e7-89d0-fc203c6a8e3f","arxiv_id":"2505.00513","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Groups obtained by closing finite or weakly Gorenstein regular groups under the LH and Φ operations have virtually Gorenstein group algebras and satisfy Moore's conjecture.","lead":"This mathematics paper proves that two large families of groups, built by repeatedly applying Kropholler's and Talelli's group-class operations, satisfy two homological conditions: virtually Gorenstein group algebras and Moore's conjecture on projectivity. It enlarges the supply of known examples for both properties and shows the closure technique transfers both conditions faithfully.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem A is not verifiable from the text: it rests on Theorem 2.1, quoted from the authors' unpublished preprint [23], which supplies the X/Y closure properties used throughout §3–§4.","rationale":"I traced both main proof chains in full. Theorem B: Corollary 5.8 follows from Corollary 5.7 and Chouinard's theorem by the abstract closure argument; the hierarchical description is not needed for it. Theorems 5.3–5.6 are correct as far as I can tell: the base case uses Moore's conjecture for M-groups, the Hα-induction uses the cellular resolution with induced modules and Proposition 5.2, and the LH case uses flatness of N via the colimit of induced modules followed by the published Benson–Goodearl theorem. No unpublished input is used in §5. Theorem A: Corollary 4.4 is also an abstract closure argument, but Theorem 4.2's proof does need the full strength of Theorems 3.1/3.4, whose proofs repeatedly use Theorem 2.1 (properties (4), (5), (6), (4'), (5'), the closures LHX = X, etc.). I specifically checked the steps of Theorem 3.1: the induction on ordinals (part (i)), the Eklof-lemma colimit argument (part (ii)), and the use of the Φ/Φ_flat definitions to get finite projective/flat dimension from pointwise projectivity (parts (iii), (iv)) are all internally sound. The one assertion that initially looked like an error — 'if κ ≤ ℵ0, then the LHX0-group H is contained in HX0' in 3.1(ii)/3.4(ii) — is actually true: writing H as a countable union of HC-subgroups K_n and forming the tree with vertex set ⊔n H/K_n and edges (hK_n, hK_{n+1}) gives a 1-dimensional contractible H-CW-complex with isotropy conjugate to the K_n's, so H ∈ H_{sup α_n + 1}C. So the paper's own arguments hold together; what cannot be verified is Theorem 2.1 itself and, with it, the base of the X/Y universe on which Theorem A's transfer machinery depends. The reader's weakest-assumption analysis identifies exactly this point. The remedy — publishing [23] or including its proofs — is straightforward, which is why CONDITIONAL (unchanged) is the right verdict rather than REJECT.","tokens_in":22667,"tokens_out":62431,"duration_ms":595188,"concrete_test":"Make [23] (Emmanouil–Talelli, 'Total acyclicity of complexes over group algebras') public, then verify that [23, Cor. 2.3] proves X = X' ⊇ Y, [23, Thm. 3.3] proves X is closed under LH, Φ and Φ_flat, [23, Thm. 4.1] proves Y is closed under LH and Φ_inj, and [23, Lemma 1.5] proves Φ_injC ⊆ Φ_flatC, exactly in the form quoted in §2 and §4. Decisive sub-check: re-derive the Φ- and LH-closure of X from published results alone ([37, Thms. 4.4, 4.9], [29, Thm. 2.20], [20]); if this requires genuinely new arguments that appear only in [23], the dependence on the unpublished preprint is essential and Theorem A's conditional status is justified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The single load-bearing gap is the provenance of Theorem 2.1. Theorem A is proved by showing (Thm 4.2) that the class Z of Y-groups with virtually Gorenstein group algebra is closed under LH and Φ_inj, and (Prop 4.3) that the class W of weakly Gorenstein-regular groups lies in Z; the conclusion W ⊆ Z then follows by the abstract closure argument. Theorem 4.2's proof is purely an application of Corollaries 3.2 and 3.5, obtained from Theorems 3.1 and 3.4. Every step of these results invokes Theorem 2.1: restriction-stability (4)/(4'), heredity and completeness of the Gorenstein cotorsion pairs over X- and Y-groups (5)/(5'), property (6), the inclusion Y ⊆ X, and the closure identities LHX = X, ΦX = X, Φ_flat X = X, LHY = Y, Φ_inj Y = Y. Theorem 2.1 is cited to [23, Cor. 2.3, Thm. 3.3, Thm. 4.1], an unpublished preprint by one of the present authors with no public identifier, and Theorem 4.2(ii) also uses [23, Lemma 1.5]. I could not detect an internal error: the countable 'LH ⊆ H' assertion in Thms 3.1(ii)/3.4(ii) is true by the standard ray/tree construction, and [23, Lemma 1.5] follows from Pontryagin duality. The load-bearing item is exactly Theorem 2.1, and Theorem A cannot be checked from the text alone. Theorem B does not depend on [23] and is checkable from published work [2], [3], [9], [15].","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies two group-class operations from the literature, Kropholler's LH and Talelli's Φ (together with its flat and injective variants), and proves that certain homological properties propagate along the resulting hierarchical closures. The first main result, Theorem A, states that if kG is weakly Gorenstein regular for all groups in a starting class W, then every group in the closure of W under LH and Φ_inj has a virtually Gorenstein group algebra. The second main result, Theorem B, states that every group in the smallest class containing the finite groups and closed under LH, Φ, and Φ_flat satisfies Moore's conjecture over k. The proof of Theorem A proceeds by characterizing Ext^1-orthogonal classes of Gorenstein modules over group algebras of groups obtained by performing LH or Φ on suitable group classes (§3), then proving that the class Z of Y-groups with virtually Gorenstein group algebra is closed under LH and Φ_inj (§4). Theorem B is proved in §5 by extending the Aljadeff–Meir argument through the operations LH, Φ, and Φ_flat, together with results of Bahlekeh–Salarian and Benson–Goodearl. The paper is carefully written and the arguments in §3 and §5 are detailed, but the proof of Theorem A depends crucially on Theorem 2.1, which is imported from an unpublished preprint [23] by one of the authors.","tokens_in":23053,"tokens_out":3363,"duration_ms":36087,"significance":"If Theorem A and Theorem B are correct, the paper establishes that very large hierarchically defined classes of groups inherit two nontrivial homological properties: virtual Gorensteinness of the group algebra and Moore's conjecture. Theorem B is a genuine extension of the previously known LHF result, and its proof is essentially checkable from published sources. The proof of Theorem A is also conceptually interesting: it shows how cotorsion-pair completeness and orthogonality can be transported along LH and Φ operations, provided the classes X, X′, and Y have the closure properties stated in Theorem 2.1. However, the significance of Theorem A is currently conditional, because Theorem 2.1—the single most load-bearing ingredient—is cited to an unpublished preprint with no public identifier and is not proved in the manuscript. The paper's other contributions, especially the orthogonality results in §3 once Theorem 2.1 is accepted, and the Moore-conjecture results in §5, are solid and well-motivated.","major_comments":[{"comment":"Theorem 2.1 is the central load-bearing result of the paper, yet it is quoted from the unpublished preprint [23] (Emmanouil and Talelli, \"Total acyclicity of complexes over group algebras\"). The theorem supplies the subgroup-closure and the LH-, Φ-, and Φ_flat-closure of X and the LH- and Φ_inj-closure of Y; these are used directly in Theorem 3.1, Theorem 3.4, Corollary 3.2, Corollary 3.5, and Theorem 4.2, and therefore in the proof of Theorem A. Since [23] has no public identifier and is co-authored by one of the present authors, the proof of Theorem A cannot be checked from the text alone. The authors should include a proof of Theorem 2.1 (or at least a detailed statement with full proof in an appendix), or replace the reference with a published, publicly verifiable source. Without this, Section 4, and hence Theorem A, is not self-contained.","section":"§2, Theorem 2.1"},{"comment":"In the proof of Theorem 4.2(ii), the assertion that Φ_inj Z ⊆ Φ_flat Z is justified by the citation \"cf. [23, Lemma 1.5]\". This is another dependence on the unpublished preprint [23]. Even if a reader were willing to accept Theorem 2.1 from [23], the additional fact that every Φ_inj-group is a Φ_flat-group for the class Z is needed to apply Corollary 3.2 in the Φ_inj closure argument. The paper should either prove this inclusion directly or supply a publicly available reference for [23, Lemma 1.5]. As written, the Φ_inj closure of Z is not verifiable from the manuscript.","section":"§4, Theorem 4.2(ii)"}],"minor_comments":[{"comment":"The name \"Echmann-Shapiro\" should be \"Eckmann-Shapiro\" in the displayed isomorphisms and in the surrounding text.","section":"§1.I"},{"comment":"In the proof of properties (1')–(5') for Gorenstein injective modules, the sentence \"whereas (2) is an immediate consequence\" should read \"whereas (2') is an immediate consequence\"; the numbering of the prongs is otherwise confusing.","section":"§2, properties (1')–(5')"},{"comment":"In Proposition 5.2(i), the complex X* is right-bounded and acyclic with projective terms except possibly at the end; the contractibility argument would be clearer if the authors explicitly say that the restricted complex is a projective resolution of res_H^G M, so that Hom_{kH}(res_H^G X*, Q) is acyclic. The current wording is acceptable but slightly terse.","section":"§5, Proposition 5.2"},{"comment":"Reference [23] should include an arXiv identifier or a journal publication status; as it stands, the reader cannot locate the preprint. Similarly, [16] and [35] are arXiv preprints and could benefit from submission status information.","section":"References"},{"comment":"The continuity of the operations LH, Φ, Φ_flat, and Φ_inj is cited to [23, Lemmas 1.3 and 1.4]. Since these lemmas are also part of the unpublished preprint [23], they should either be proved in the paper or explicitly marked as standard results with a published reference.","section":"§1.IV"}],"recommendation":"major_revision","confidential_remarks":"The main issue for the editor is provenance: Theorem A rests on Theorem 2.1 and Lemma 1.5 of the unpublished preprint [23], which is co-authored by one of the present authors. The referee cannot verify the central claim of the paper from the submitted text alone. This is fixable: the authors could include a proof of Theorem 2.1 (and the needed lemma) in the paper, or make the preprint publicly available and provide a stable identifier. Given that Theorem B and the §3 orthogonality framework are checkable from published work, a major revision rather than a rejection seems appropriate. The referee is not alleging any circularity or error, only that a load-bearing part of the proof is currently unverifiable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a careful, useful paper. It proves two new statements: every group in the LH/Phi_inj closure of the weakly Gorenstein regular groups has a virtually Gorenstein group algebra, and every group in the LH/Phi/Phi_flat closure of the finite groups satisfies Moore's conjecture. Theorem B is checkable from published work. Theorem A is not fully checkable from the text, because its load-bearing Theorem 2.1 is imported from the authors' own unpublished preprint [23]. That is the one real soft spot, and it is a verification gap rather than an evident error.\n\nWhat the paper does well: Section 3 extends Kendall's orthogonality transfer to the LH, Phi, Phi_flat, and Phi_inj settings with careful transfinite induction and the standard Eklof-lemma argument. The proofs are detailed and I did not find an internal inconsistency. The use of Eaton-Shapiro and the cotorsion pair machinery is sound. Section 5 is particularly clean: the closure of the Moore classes under the operations is proved with published tools, and Theorem B follows without any recourse to [23]. I agree with the reader that the paper is a substantive application of existing operations, not a new framework, but that is fine; it produces large new classes of examples.\n\nThe soft spot: Theorem 2.1 supplies the closure properties of the classes X, X', Y, restriction stability for Gorenstein modules, and completeness of the relevant cotorsion pairs. It is cited as [23, Cor. 2.3, Thm. 3.3, Thm. 4.1], an unpublished preprint by one of the present authors with no public identifier. Since Theorem 4.2 and hence Theorem A chain directly through that theorem, the central result currently cannot be checked from the text alone. This is not a fatal flaw, but it is a genuine condition: the authors should be asked to post the preprint or provide the proofs in an appendix. The continuity lemmas for LH and Phi are also cited to [23]; those are likely standard, but they fall under the same condition.\n\nWho this is for: homological group theorists and people working on Gorenstein module classes over group algebras. The paper is worth a serious referee. If [23] is made public and the arguments check out, Theorems A and B are publishable. I would send it to peer review with that condition attached.","headline":"Genuinely new closure theorems for virtually Gorenstein algebras and Moore's conjecture, but Theorem A rests on an unpublished preprint [23] that must be made available before the main proof is independently checkable.","tokens_in":23554,"tokens_out":2074,"would_cite":true,"duration_ms":23300,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20J05","16E65","18G25"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that closing weakly Gorenstein regular groups under the operations LH and Φ_inj preserves virtual Gorensteinness of the group algebra, and closing finite groups under LH, Φ, and Φ_flat preserves Moore's conjecture.","keywords":["virtually Gorenstein algebras","Moore's conjecture","LH-hierarchy","Phi-operation","Gorenstein projective modules","Gorenstein injective modules","relative projectivity","group cohomology"],"falsifier":"Exhibit a group $G$ that is a single $\\Phi_{\\mathrm{inj}}$-step or LH-step away from a base class for which some $kG$-module violates the relevant Ext$^1$-orthogonality equality, or an $\\overline{F}$-group with a module projective on every Moore-condition finite-index restriction but not globally projective; either would refute Theorem 3.1, Theorem 4.2, or Theorem B. A cheaper check is to verify Theorem 2.1's closure claim in the companion preprint on the smallest nontrivial classes.","tokens_in":22452,"feed_emoji":"🔁","tokens_out":13196,"duration_ms":120467,"temperature":0.7,"pith_summary":"This paper proves two closure theorems for hierarchical group-class operations. Theorem A says that if you start with any group whose group algebra is weakly Gorenstein regular and repeatedly apply the operation LH and the operation $\\Phi_{\\mathrm{inj}}$, every group you reach still has a virtually Gorenstein group algebra. Theorem B says that if you start with finite groups and repeatedly apply LH, $\\Phi$, and $\\Phi_{\\mathrm{flat}}$, every group you reach satisfies Moore's conjecture, meaning a module that becomes projective after restriction to a suitable finite-index subgroup was already projective. These results matter because the two operations construct very large classes of groups, so the theorems move two homological properties from finite and LHF groups to a much wider universe.","feed_headline":"LH and Φ closures yield two homological theorems","feed_subtitle":"Closures under LH and Φ operations extend virtual Gorenstein regularity and Moore's conjecture to new classes.","key_machinery":"The machinery is the pair of group-class operations. LH starts from a class $C$ and closes under groups that admit a cellular action on a finite-dimensional contractible complex whose isotropy subgroups come from earlier stages of a transfinite induction, then adds all groups whose finitely generated subgroups lie in that hierarchy; $\\Phi$ puts $G$ into $\\Phi C$ when a $kG$-module has finite projective dimension exactly when its restrictions to $C$-subgroups have bounded projective dimension, with $\\Phi_{\\mathrm{flat}}$ and $\\Phi_{\\mathrm{inj}}$ obtained by replacing projective dimension by flat or injective dimension. The paper couples these operations with Ext$^1$-orthogonality: the technical core shows that membership in the right orthogonal of Gorenstein projective modules, or the left orthogonal of Gorenstein injective modules, is inherited under restriction from base-class subgroups to subgroups produced by any of the operations (Theorems 3.1 and 3.4). The orthogonal classes are then intersected over base subgroups to prove the virtual Gorenstein equality (Theorem 4.2). For Moore's conjecture, the operative object is the finite-index subgroup $H$ satisfying Moore's condition; Proposition 5.2 and Theorem 5.3 show projectivity, flatness, or injectivity of a restricted module climbs the LH, $\\Phi$, and $\\Phi_{\\mathrm{flat}}$ hierarchy.","core_discovery":"On the paper's own terms, the central claim is that virtual Gorensteinness of group algebras and Moore's conjecture both propagate through the closure process. If $G$ lies in $\\overline{W}$, the smallest class containing all groups with $kG$ weakly Gorenstein regular and closed under LH and $\\Phi_{\\mathrm{inj}}$, then $kG$ is virtually Gorenstein: $GProj(kG)^\\perp = {}^\\perp GInj(kG)$, i.e. the classes Ext$^1$-orthogonal to Gorenstein projective and to Gorenstein injective modules coincide. If $G$ lies in $\\overline{F}$, the smallest class containing all finite groups and closed under LH, $\\Phi$ and $\\Phi_{\\mathrm{flat}}$, then $G$ satisfies Moore's conjecture over $k$. The mechanism is an Ext$^1$-orthogonality transfer: a module whose restrictions to base-class subgroups lie in the orthogonal complement of Gorenstein modules has the same property after restriction to any subgroup built by LH, $\\Phi$, $\\Phi_{\\mathrm{flat}}$ or $\\Phi_{\\mathrm{inj}}$ (Theorems 3.1 and 3.4), and intersecting these conditions over base subgroups yields the desired equality for the whole group (Theorem 4.2). For Moore's conjecture, the proof propagates projectivity of restricted modules up the same hierarchy using dimension shifting and the adjunction between induction and restriction (Theorem 5.3).","pith_inferences":["The same Ext$^1$-orthogonality transfer is a general template: any homological property that can be stated as an equality between a module class and its Ext-orthogonal should survive the LH/Φ closure once the base class satisfies the closure axioms, so the technique may generalize to other relative homological conditions.","The closures $\\overline{F}$ and $\\overline{W}$ are likely to contain groups not of type FP$_\\infty$ and not in LHF; if so, Moore's conjecture and virtual Gorensteinness are being established for genuinely new examples, though the paper itself does not exhibit such groups.","A direct way to test the scope of the results is to look for a group in $\\overline{W}$ that is not in the closure of finite groups under the same operations; if such groups exist, the two theorems cover independent families of examples."],"forward_implications":["Every group in the closure $\\overline{W}$ has a group algebra whose Gorenstein projective and Gorenstein injective orthogonal classes coincide; the paper records that this makes the associated relative derived category triangulated equivalent to homotopy categories of Gorenstein projective and Gorenstein injective modules.","Every group in the closure $\\overline{F}$ satisfies Moore's conjecture over any commutative ring $k$, so projective-on-a-suitable-finite-index-subgroup implies projective for all $kG$-modules over such groups.","The flat and injective versions of Moore's conjecture also propagate: LH with $\\Phi_{\\mathrm{flat}}$ preserves the flat version, and LH with $\\Phi_{\\mathrm{inj}}$ preserves the injective version.","Because the closures are built by transfinite iteration, the results apply not only to LHF-groups but to every group reached by iterating the operations, including groups whose structure is far from finite."],"supporting_citations":[{"why":"Supplies Theorem 2.1, the closure of the classes X, X' and Y under LH, Φ, Φ_flat and Φ_inj, on which the orthogonality transfers and Theorems 4.2 and 5.3 depend.","marker":"[23]"},{"why":"Proves that weakly Gorenstein regular rings are virtually Gorenstein, giving the inclusion W ⊆ Z used in Proposition 4.3.","marker":"[19]"},{"why":"Provides the total acyclicity and cotorsion-pair results for PGF, GFlat and GInj modules that justify the module-class properties in Section 2.","marker":"[37]"},{"why":"Gives the Ext^1-orthogonality description for LH-groups that Theorem 3.1 extends to Φ and Φ_flat.","marker":"[31]"},{"why":"Proves Moore's conjecture for all LHF-groups, the base result and proof template for Theorem 5.3.","marker":"[2]"},{"why":"Establishes the flat and injective versions of Moore's conjecture, used in Proposition 5.1 and Theorem 5.5.","marker":"[3]"},{"why":"Shows finite groups satisfy Moore's conjecture, providing the base class F for Theorem B.","marker":"[15]"},{"why":"Defines the LH hierarchy and the cellular-action construction used throughout.","marker":"[32]"},{"why":"Defines groups of type Φ, the operation whose Φ, Φ_flat and Φ_inj variants are used in Theorems A and B.","marker":"[38]"}],"fun_headline_variants":["LH and Φ closures propagate Gorenstein and Moore properties","Two homological conditions survive LH and Φ enlargement","Virtual Gorenstein and Moore's conjecture extend via LH and Φ closures","LH and Φ closures make Gorenstein and Moore properties hereditary"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument's load-bearing premise is Theorem 2.1, imported from the companion preprint [23]: the classes $X$, $X'$ and $Y$ of groups where the Gorenstein module classes are perfectly behaved are closed under LH, $\\Phi$, $\\Phi_{\\mathrm{flat}}$ and $\\Phi_{\\mathrm{inj}}$, and if that closure fails the orthogonality transfers in Sections 3 and 4 and hence Theorems A and B no longer follow.","fun_headline_variants_meta":{"raw":{"variants":["LH and Φ closures propagate Gorenstein and Moore properties","Two homological conditions survive LH and Φ enlargement","Virtual Gorenstein and Moore's conjecture extend via LH and Φ closures","LH and Φ closures make Gorenstein and Moore properties hereditary"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000571,"raw_usage":{"total_tokens":2699,"prompt_tokens":944,"completion_tokens":1755,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":560,"completion_tokens_details":{"reasoning_tokens":1695}},"tokens_in":560,"tokens_out":1755,"duration_ms":13273,"temperature":1.0,"reasoning_tokens":1695,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:40:25.164844+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a group $G$ that is a single $\\Phi_{\\mathrm{inj}}$-step or LH-step away from a base class for which some $kG$-module violates the relevant Ext$^1$-orthogonality equality, or an $\\overline{F}$-group with a module projective on every Moore-condition finite-index restriction but not globally projective; either would refute Theorem 3.1, Theorem 4.2, or Theorem B. A cheaper check is to verify Theorem 2.1's closure claim in the companion preprint on the smallest nontrivial classes.","supporting_citations":[{"cited_title":"(preprint)","cited_arxiv_id":null,"evidence_quote":"Supplies Theorem 2.1, the closure of the classes X, X' and Y under LH, Φ, Φ_flat and Φ_inj, on which the orthogonality transfers and Theorems 4.2 and 5.3 depend."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves that weakly Gorenstein regular rings are virtually Gorenstein, giving the inclusion W ⊆ Z used in Proposition 4.3."},{"cited_title":"Selecta Math","cited_arxiv_id":null,"evidence_quote":"Provides the total acyclicity and cotorsion-pair results for PGF, GFlat and GInj modules that justify the module-class properties in Section 2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves Moore's conjecture for all LHF-groups, the base result and proof template for Theorem 5.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the flat and injective versions of Moore's conjecture, used in Proposition 5.1 and Theorem 5.5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows finite groups satisfy Moore's conjecture, providing the base class F for Theorem B."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines groups of type Φ, the operation whose Φ, Φ_flat and Φ_inj variants are used in Theorems A and B."}],"review_version":1}