{"id":"a43dd867-18c2-40a7-855c-f3e3b81820bb","arxiv_id":"2608.08141","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Finite group equivariantizations of abelian model categories inherit Hovey triples, and the resulting homotopy categories are triangle-equivalent up to retracts to equivariantizations of the original homotopy categories.","lead":"The paper proves a general transfer theorem: when a finite group acts on an abelian category equipped with a Hovey triple, the model structure lifts to the category of equivariant objects, and the homotopy theories agree up to retracts. For representation theory, this gives a systematic way to pass between homological algebra over a ring and over its skew group ring.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Prop. 2.11's proof uses an invalid double-orthogonal identity; the transfer theorem's completeness argument needs a one-line repair before the central claim is fully established.","rationale":"I read the paper in good faith and its central construction—transferring Hovey triples along the Frobenius pair Ind⊣U—is plausible and largely carefully executed. The reader's identified concern about Lemma 3.1 is legitimate but it is an explicit hypothesis: the split counit is guaranteed by invertibility of |G|, and the proof does rely on that condition. My independent stress-test found a different, more local soft spot: in Proposition 2.11, the proof of the (⇐) direction uses the equality ^⊥((^⊥F(D))^⊥)=^⊥F(D), which is not generally true. This matters because that proposition is the engine behind Theorem 2.16, and hence behind Theorem A(1). The good news is that the flawed step is not essential: the needed inclusion follows directly from F(S)⊆^⊥F(D), and the rest of the proof appears to work. Thus this is a proof gap, not a counterexample to the claims. The paper would benefit from a corrected version of this paragraph, but the verdict remains conditional rather than reject. I agree partially with the reader: the transfer theorem is indeed the load-bearing core, but the specific weakest point I found is the invalid orthogonal identity rather than the invertibility assumption itself.","tokens_in":45681,"tokens_out":32319,"duration_ms":311656,"concrete_test":"Re-derive the (⇐) direction of Proposition 2.11, replacing the displayed chain by the direct argument: condition (a) yields F(S)⊆^⊥F(D), hence F(D)⊆F(S)^⊥; for X∈^⊥(F(S)^⊥), this implies Ext^1_B(X,F(D))=0 for all D∈D, so X∈^⊥F(D). Confirm that the rest of the proof, including the use of [38, Corollary 2.15(3)] to obtain completeness of the cotorsion pair cogenerated by F(S), goes through unchanged. If the repaired proof is valid, Theorem 2.16 and Theorem A stand.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 2, Proposition 2.11, proof of (⇐) contains the displayed chain ^⊥H^{-1}(D)=^⊥(F(S)^⊥)⊆^⊥((^⊥F(D))^⊥)=^⊥F(D)=H^{-1}(C). The equality ^⊥((^⊥F(D))^⊥)=^⊥F(D) is not a valid identity for arbitrary classes in an abelian category: the double left orthogonal can be strictly larger than the original left orthogonal. This step is used to establish the inclusion ^⊥H^{-1}(D)⊆H^{-1}(C), which is needed to identify pH^{-1}(C), H^{-1}(D)q as the cotorsion pair cogenerated by F(S) and hence complete. Since Proposition 2.11 is the technical foundation for Theorem 2.16 and therefore for Theorem A(1), this is a genuine gap in the written proof of the central transfer theorem. The gap is repairable: condition (a) gives F(S)⊆^⊥F(D), so for every D∈D one has F(D)∈F(S)^⊥; consequently, for X∈^⊥(F(S)^⊥), Ext^1_B(X,F(D))=0, and X∈^⊥F(D). Thus the needed inclusion follows without the invalid equality. The reverse inclusion is proved later in the same paragraph using condition (b) and is valid. The theorem may well be true, but as written the proof is incomplete at this step.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a transfer theorem for abelian model structures along Frobenius functors (Theorem 2.16 and 2.18) and applies it to the equivariant category A^G of a Grothendieck category A equipped with a finite group action with |G| invertible. Its Theorem A states that if two of the three classes of a cofibrantly generated hereditary Hovey triple are G-invariant, then the equivariant classes form such a triple, and the induced functor Ho(A^G) -> Ho(A)^G is a triangle equivalence up to retracts, with a genuine equivalence under an idempotent completeness condition. Theorem B gives conditions under which a Quillen adjunction between two such categories induces a commutative square relating L(F^G) and L(F)^G, with both horizontal functors triangle equivalences. Section 5 illustrates the results for PGF Hovey triples over skew group rings, Frobenius bimodules, and stable equivalences of adjoint type.","tokens_in":45948,"tokens_out":21366,"duration_ms":205458,"significance":"If the main theorems are correct, the paper provides a general and useful framework for equivariantizing abelian model structures, complementing the transfer results of Hovey, Gillespie, Sun, and Chen. The paper is explicit about which results are imported and provides detailed proofs for the new lifting lemmas; the PGF corollaries give concrete, checkable applications. The principal caveats are the essential hypothesis that |G| is invertible and the reliance on Sun's theorem for the triangulated comparison; these are stated clearly. The identified gap in Proposition 2.11 is localized and repairable, so the overall contribution remains valuable.","major_comments":[{"comment":"The proof of the implication (⇐) contains the displayed chain ^⊥H^{-1}(D)=^⊥(F(S)^⊥)⊆^⊥((^⊥F(D))^⊥)=^⊥F(D)=H^{-1}(C). The middle equality ^⊥((^⊥F(D))^⊥)=^⊥F(D) is not valid for arbitrary classes in an abelian category, since the double left orthogonal can be strictly larger. This step is used to prove the inclusion ^⊥H^{-1}(D)⊆H^{-1}(C), which is needed to identify (^⊥H^{-1}(D), H^{-1}(D)) with the cotorsion pair cogenerated by F(S). The gap is repairable directly from condition (a): since F(S)⊆^⊥F(D), every F(D) with D∈D lies in F(S)^⊥=H^{-1}(D); hence for X∈^⊥(F(S)^⊥) one has Ext^1_B(X,F(D))=0 for all D, so X∈^⊥F(D)=H^{-1}(C). I recommend replacing the invalid equality by this direct argument and checking the surrounding identifications in Theorem 2.16 and Theorem A(1), which rest on this proposition.","section":"Section 2, Proposition 2.11"}],"minor_comments":[{"comment":"The title and running header contain the typo 'EQUIV ARIANT'; the word should be 'EQUIVARIANT'.","section":"General"},{"comment":"The notation λ_X in the statement of Proposition 2.11 is introduced via the short exact sequence for objects of B, but the variable name is reused; writing the sequence explicitly for each object appearing in conditions (b) and (c) would remove ambiguity.","section":"Section 2.10 and Proposition 2.11"},{"comment":"In Step 1 the symbol Q_A^G is used both for the equivariant cofibrant replacement functor and for the underlying object Q_A^G X; clarifying that these are the same object-level notation would help the reader follow the long diagram chase.","section":"Section 4, proof of Theorem 4.11"},{"comment":"The proof of Lemma 5.19 is condensed and relies on facts (1) and (2) that are only cited; expanding the verification of these two facts, or at least giving precise references with the exact adjunction isomorphisms, would improve readability.","section":"Section 5.3, Lemma 5.19"},{"comment":"Reference [15] is cited as a preprint without a year; if a published version exists, it should be updated.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"To the editor: The central gap I found is confined to one line in Proposition 2.11 and is easily repaired; the rest of the proof structure is consistent with the stated external results. I could not fully verify every diagram chase in Section 4 and the appendix, but I found no other internal inconsistency. The paper is long but within the scope of the journal; the authors' attribution of imported theorems is clear."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is worth taking seriously. Its real contribution is a transfer theorem for cofibrantly generated hereditary Hovey triples along Frobenius functors, then applied to equivariant categories. Theorem A, comparing the homotopy category of the lifted model structure with the equivariantization of the original homotopy category up to retracts, is a genuine extension of the Chen/Elagin/Sun results from derived and stable categories to abelian model structures. Theorem B, the homotopy square compatibility for derived functors, is also new and is the kind of result people will want to cite. The PGF module corollaries give concrete payoff, and the authors are honest about which results are imported from Gillespie, Sun, and Chen. The stress-test note is correct. In Prop. 2.11, proof of the inclusion ^\\perp H^{-1}(D) ⊆ H^{-1}(C), the displayed chain uses the identity ^\\perp((^\\perp F(D))^\\perp) = ^\\perp F(D). That identity is not valid in general; the double left orthogonal can be strictly larger. However, the repair is immediate: condition (a) already gives F(S) ⊆ ^\\perp F(D), so for X in ^\\perp(F(S)^\\perp) one directly gets Ext^1_B(X, F(D))=0 for all D in D, hence X in ^\\perp F(D). The reverse inclusion is fine. So the central claim of Prop. 2.11 is likely true, but the written proof is incomplete at exactly the point that feeds Theorem 2.16 and Theorem A(1). A referee should ask for that fix. Other soft spots are proportionate. The split counit in Lemma 3.1 depends on |G| being invertible; that is a real restriction, but it is stated plainly and is also needed for Sun's triangulated equivariant structure. Theorem B's hypotheses are heavy, especially the condition on coker(eta_X) and ker(epsilon_Y) and the faithfulness assumptions, but the statements are explicit and the applications to Frobenius bimodules and stable equivalences of adjoint type show the conditions are not empty. The proofs are long and depend on many imported results, so I could not fully check every diagram. The citation pattern looks healthy; I did not see self-citation inflation or fitting. Who gets value: anyone working on abelian model structures, Gorenstein homological algebra, or equivariant categories. The paper deserves a serious referee, not a desk reject. I would send it to someone who can check Section 2 carefully and the compatibility square in Theorem B. With the Prop. 2.11 repair, I expect the central results to stand.","headline":"A substantial transfer theorem for Hovey triples to equivariant categories, with a genuinely new comparison result; the main ideas are sound, but one step in Prop. 2.11 is wrong as written and needs a small repair before the proof is complete.","tokens_in":750,"tokens_out":907,"would_cite":true,"duration_ms":28552,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18G25","18G80","18G65"],"pacs":[],"model":"deepseek-v4-flash","headline":"Finite group actions with invertible order transfer abelian model structures to equivariant categories, and the homotopy categories agree up to retracts.","keywords":["abelian model structure","equivariant category","Hovey triple","Frobenius functor","triangle equivalence up to retracts","PGF modules","skew group ring","homotopy square"],"falsifier":"Take A = Mod(Z) with G = C2 acting by the sign automorphism, so |G| = 2 is not invertible, and let M be the PGF Hovey triple on Mod(Z). Compute the lifted triple (C^G, W^G, F^G) on Mod(Z)^G and test whether it satisfies the completeness of the two cotorsion pairs required for a Hovey triple; if it does, the |G|-invertible hypothesis is unnecessary. A more direct test is to find one equivariant object X for which the counit short exact sequence 0 → ker ε_X → Ind U X → X → 0 does not split; Lemma 3.1, and hence the proof of the transfer theorem, would fail at that point.","tokens_in":45471,"feed_emoji":"🔄","tokens_out":7476,"duration_ms":101109,"temperature":0.7,"pith_summary":"This paper establishes a lifting theorem: when a finite group G acts on a Grothendieck category A with enough projectives and |G| is invertible, any cofibrantly generated hereditary Hovey triple whose classes are suitably G-invariant induces the same kind of Hovey triple on the equivariant category A^G. It then proves that the homotopy category of the lifted model structure is triangle equivalent, up to retracts, to the equivariantization of the original homotopy category, with a genuine triangle equivalence when a certain stable category is idempotent complete. The paper also shows that derived functors of Quillen adjunctions are compatible with this comparison, producing a commutative square of homotopy categories. This gives a general mechanism for passing abelian model structures, and their homotopy theories, through equivariantization, with concrete consequences for modules over skew group rings and PGF modules.","feed_headline":"Group actions lift model structures to equivariant categories","feed_subtitle":"A transfer theorem makes homotopy categories agree up to retracts when the group order is invertible.","key_machinery":"The engine is the transfer theorem for Hovey triples along a faithful Frobenius functor (Theorem 2.16), specialised to the Frobenius pair Ind: A ⇄ A^G : U. A Frobenius pair is an adjunction whose right adjoint is also a left adjoint; the symmetry lets one control both cotorsion pairs of a Hovey triple simultaneously. In the equivariant setting, |G|-invertibility makes the counit Ind∘U → id split, which turns the epimorphism condition of the transfer criterion into a tautology, so the transferred classes (C^G, W^G, F^G) inherit cofibrant generation and heredity. The comparison of homotopy categories is carried by factoring Ho(γ_A^G) through the stable categories St_{ω^G}(C^G ∩ F^G) and St_ω(C ∩ F)^G, where the comparison is a triangle equivalence up to retracts.","core_discovery":"The central claim is Theorem A: under the |G|-invertible hypothesis, if any two of the three classes C, W, F of a cofibrantly generated hereditary Hovey triple are G-invariant, then (C^G, W^G, F^G) is again a cofibrantly generated hereditary Hovey triple on A^G, and the induced functor Ho(γ_A^G): Ho(M^G) → Ho(M)^G is a triangle equivalence up to retracts, becoming a genuine triangle equivalence when St_{ω^G}(C^G ∩ F^G) is idempotent complete. Theorem B adds that for a two-sided Quillen adjunction satisfying faithfulness, invariance, and weak-equivalence conditions on unit and counit cokernels and kernels, the square comparing L(F^G) with L(F)^G commutes up to natural isomorphism and the horizontal functors are triangle equivalences. The paper verifies the hypotheses for PGF Hovey triples, obtaining a comparison functor Ho(PGF(RG)) → Ho(PGF(R))^G that is a triangle equivalence up to retracts, and constructs homotopy squares from Frobenius bimodules and from stable equivalences of adjoint type.","pith_inferences":["The |G|-invertibility condition appears to be the real dividing line: without it, the split counit used in the transfer proof is unavailable, so one should expect genuinely different behaviour, and testing whether some weaker averaging condition suffices would be a natural next step.","The \"up to retracts\" caveat is not cosmetic; any failure of idempotent completeness in the equivariant stable category would make the comparison fail to be a full equivalence, so examples with non-idempotent-complete stable categories could separate the two notions.","The transfer machinery is stated for Grothendieck categories, but its core transfer theorem is purely about Frobenius functors; applying it to other Frobenius adjunctions, such as induction between module categories over Frobenius extensions, may yield analogous lifting results without any group action.","The PGF illustration suggests the same square should exist for Gorenstein projective, injective, and flat Hovey triples; the paper states the verification is analogous, so those cases are a promising place to test whether the hypotheses can be relaxed."],"forward_implications":["For any finite group action with |G| invertible, a cofibrantly generated hereditary abelian model structure whose classes are G-invariant transfers to the equivariant category, so equivariant homological algebra inherits the model structure.","The homotopy categories Ho(M^G) and Ho(M)^G coincide up to retracts; when the relevant stable category is idempotent complete, they are genuinely triangle equivalent.","Total derived functors commute with equivariantization: L(F^G) and L(F)^G sit in a commutative square, so equivariant derived functors can be computed either before or after passing to fixed points.","For rings, the right orthogonal class PGF(RG)^⊥ equals (PGF(R)^⊥)^G, and Ho(PGF(RG)) is triangle equivalent up to retracts to Ho(PGF(R))^G.","Stable equivalences of adjoint type between base rings lift to the skew group rings, giving triangle equivalences of the associated equivariant PGF homotopy categories."],"supporting_citations":[{"why":"Hovey's correspondence converts complete cotorsion pairs and a thick class into an abelian model structure, which is the structure being transferred.","marker":"[26]"},{"why":"Supplies the localization theorem, the triangulated structure of homotopy categories, and the stable-category model for the homotopy category of a hereditary Hovey triple.","marker":"[21]"},{"why":"Sun's theorem gives the canonical triangulated structure on equivariant categories and the comparison functor that is an equivalence up to retracts, underpinning Theorem A(2)-(3).","marker":"[36]"},{"why":"Provides the induction and forgetful functors Ind ⊣ U as a Frobenius pair for equivariant categories, the adjunction to which the transfer theorem is specialised.","marker":"[13]"},{"why":"Establishes the PGF Hovey triple (PGF(R), PGF(R)⊥, Mod(R)) as cofibrantly generated and hereditary, used in the module-theoretic illustrations.","marker":"[37]"},{"why":"Provides the formalism of G-functors and equivariant categories, including the induced equivariant functor and adjunction lemmas used throughout.","marker":"[32]"},{"why":"Shows that Frobenius pairs between module categories arise from Frobenius bimodules, the source of the homotopy square in the first application of Theorem B.","marker":"[8]"}],"fun_headline_variants":["Group actions lift model structures when order is invertible","Equivariant model categories from group actions with invertible order","Lifting theorem for abelian model structures to equivariant categories","Homotopy squares from lifted model structures under group actions","Transfer of model structures to equivariant categories up to retracts"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole argument depends on the order of the group being invertible in the category: that single hypothesis makes the counit of the induction-forgetful adjunction split, and without the split the lifted triple is not guaranteed to satisfy the completeness conditions of a Hovey triple.","fun_headline_variants_meta":{"raw":{"variants":["Group actions lift model structures when order is invertible","Equivariant model categories from group actions with invertible order","Lifting theorem for abelian model structures to equivariant categories","Homotopy squares from lifted model structures under group actions","Transfer of model structures to equivariant categories up to retracts"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000701,"raw_usage":{"total_tokens":3177,"prompt_tokens":968,"completion_tokens":2209,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":584,"completion_tokens_details":{"reasoning_tokens":2126}},"tokens_in":584,"tokens_out":2209,"duration_ms":14383,"temperature":1.0,"reasoning_tokens":2126,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T00:20:48.396978+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take A = Mod(Z) with G = C2 acting by the sign automorphism, so |G| = 2 is not invertible, and let M be the PGF Hovey triple on Mod(Z). Compute the lifted triple (C^G, W^G, F^G) on Mod(Z)^G and test whether it satisfies the completeness of the two cotorsion pairs required for a Hovey triple; if it does, the |G|-invertible hypothesis is unnecessary. A more direct test is to find one equivariant object X for which the counit short exact sequence 0 → ker ε_X → Ind U X → X → 0 does not split; Lemma 3.1, and hence the proof of the transfer theorem, would fail at that point.","supporting_citations":[{"cited_title":"Z.241(2002), no","cited_arxiv_id":null,"evidence_quote":"Hovey's correspondence converts complete cotorsion pairs and a thick class into an abelian model structure, which is the structure being transferred."},{"cited_title":"215, Cam- bridge University Press, Cambridge, 2025","cited_arxiv_id":null,"evidence_quote":"Supplies the localization theorem, the triangulated structure of homotopy categories, and the stable-category model for the homotopy category of a hereditary Hovey triple."},{"cited_title":"Algebra 534(2019), 483–530","cited_arxiv_id":null,"evidence_quote":"Sun's theorem gives the canonical triangulated structure on equivariant categories and the comparison functor that is an equivalence up to retracts, underpinning Theorem A(2)-(3)."},{"cited_title":"I, Selecta Math","cited_arxiv_id":null,"evidence_quote":"Provides the induction and forgetful functors Ind ⊣ U as a Frobenius pair for equivariant categories, the adjunction to which the transfer theorem is specialised."},{"cited_title":"(N.S.)26(2020), no","cited_arxiv_id":null,"evidence_quote":"Establishes the PGF Hovey triple (PGF(R), PGF(R)⊥, Mod(R)) as cofibrantly generated and hereditary, used in the module-theoretic illustrations."},{"cited_title":"Casta˜ no Iglesias, J","cited_arxiv_id":null,"evidence_quote":"Shows that Frobenius pairs between module categories arise from Frobenius bimodules, the source of the homotopy square in the first application of Theorem B."}],"review_version":1}