{"id":"1aad3c30-8cf3-44d3-83b3-3742338733d8","arxiv_id":"2607.10913","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"τ-translates of the regular bimodule are the cycle modules of the Nakayama-twisted Happel resolution of the square of the Serre bimodule, linking τ-Hochschild theory to the Coxeter automorphism.","lead":"Two independent refinements of Hochschild theory for finite-dimensional algebras are shown to come from one complex: the Nakayama twist of Happel's minimal resolution. The result gives a single Morita invariant that separates algebras neither refinement alone can, and identifies top τ-translates with higher preprojective data.","discovery_kind":"unification","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the Bridge Theorem as the strongest claim and separability of E as the main structural hypothesis. That hypothesis is standard in the literature the paper works in (Happel, Butler–King, CLMS) and is explicitly stated; under it the argument is a clean, natural identification with tracked bimodule structures. The subsequent structural consequences (self-injective pure residue, top-degree preprojective identification, Euler recovery of the Coxeter trace, transversality examples) follow without additional free parameters or data fitting. The open question on derived invariance of the τ-groups over the smooth locus is well-posed and does not undermine the proved claims. No load-bearing gap that would move the verdict was found; the recommended action is therefore UNCHANGED (ACCEPT).","tokens_in":20897,"tokens_out":603,"duration_ms":5356,"concrete_test":"Independently recompute the four-term sequence and dim τ_Λ^e (kK_2) for the Kronecker algebra (Example 4.7) from Happel’s resolution and from the preprojective description of Π(K_2)_1; both must give dimension 12 and the same HH^1_τ ≅ k^3. Agreement confirms the cycle identification and dualities of Thm 3.2(1–2) and Thm 4.5.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central Bridge Theorem (Thm 3.2) rests on standard facts: evaluation for projectives gives ν(P) ≅ DΛ ⊗_Λ P ⊗_Λ DΛ (Lemma 3.1), Happel’s resolution is left-projective so DΛ ⊗_Λ P• resolves DΛ as a right module, and the AR four-term sequence applied to the minimal presentation of Ω^{n-1}Λ yields τ_n Λ = ker ν(d_n). Separability of E (the reader’s weakest assumption) is the usual hypothesis that makes Happel’s form P_n = Λ ⊗_E T_n ⊗_E Λ available with the stated projectivity; it is stated in §2.1 and used consistently. The residue/shadow split, top-degree identification with Π_{d+1}(Λ)_1, and Euler recovery of Happel’s trace follow directly. No internal inconsistency or hidden gap in the tracked bimodule structures appears.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper identifies the higher Auslander–Reiten translates τ_n Λ of the regular bimodule (used by Cibils–Lanzilotta–Marcos–Solotar to define τ-Hochschild (co)homology) with the cycle bimodules of the complex obtained by applying the Nakayama functor of Λ^e to Happel’s minimal resolution. That complex represents the square of the Serre bimodule ω_Λ ⊗^L_Λ ω_Λ, whose shift generates the Coxeter automorphism σ_Λ of the Tamarkin–Tsygan calculus. The resulting short exact sequences 0 → B_n → τ_n Λ → Tor_n^Λ(DΛ, DΛ) → 0 split each translate into a derived-invariant shadow and a Morita-theoretic residue. Consequences include: vanishing of the residue in top degree and identification of τ_d Λ with the dual of the degree-one piece of the (d+1)-preprojective algebra; pure residue for self-injective algebras (explaining growth for BGMS algebras); recovery of Happel’s trace formula as the Euler characteristic of the CLMS dimension formulas; and explicit examples showing the two refinements are transversal, together with a proposed combined Morita invariant and an open question on derived invariance of the τ-groups over the smooth locus.","tokens_in":21197,"tokens_out":824,"duration_ms":6986,"significance":"If the bridge holds, the paper supplies a single geometric object (the Nakayama twist of Happel’s resolution) that simultaneously organises two independent enrichments of Hochschild theory and makes their numerical and structural relations transparent. The top-degree identification with higher preprojective data and the structural explanation of the BGMS growth are concrete payoffs. The examples (K_2, A_n, the Xi pair, A_3 versus A_3/rad^{2}) are computed both by hand and by machine and match known dimension formulas, giving reproducible evidence for the transversality claim. The work therefore converts a coincidence of foundational ingredients into a theorem and poses a clean open problem on derived invariance over finite global dimension.","major_comments":[],"minor_comments":[{"comment":"The disclosure statement at the end of the manuscript should be moved to an acknowledgements or methods footnote so that it does not appear as part of the mathematical text.","section":"Disclosure"},{"comment":"In Example 4.7 the dimension count for Π(K_2)_1 is given as 2+3+3+4=12; a one-line reference to the explicit basis (or to the machine verification mentioned later) would make the arithmetic easier to check.","section":"Example 4.7"},{"comment":"The phrase “strictly Morita-theoretic residue” is used repeatedly; a single formal definition of B_n as the image of ν(d_{n+1}) early in §3 would avoid any ambiguity when the term reappears in later sections.","section":"§3"},{"comment":"Question 6.7 is well-posed, but a short remark on whether the excess dim HH^1_τ − dim HH^1 is already known to be derived-invariant for gl.dim ≤ 2 (or a pointer to the literature) would help the reader gauge the difficulty of the first open case.","section":"Question 6.7"}],"recommendation":"accept","confidential_remarks":"The heavy self-citation of Ar1–Ar5 is natural given that the Coxeter side is the author’s own recent work; the bridge itself is independent and does not rely on circular reasoning. The manuscript is a clean fit for a representation-theory or homological-algebra journal. No load-bearing gaps were found."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The new content is the identification that τ_n Λ is exactly the n-cycle bimodule of ν(P•), where P• is Happel’s minimal resolution and ν is the Nakayama functor of Λe. That single fact produces the residue/shadow SES, recovers Happel’s trace as the Euler characteristic of the CLMS dimension formulas, and identifies the top translate with D(Π_{d+1}(Λ)_1). The self-injective vanishing of the derived shadow also gives a structural reason for the unbounded τ-cohomology of the BGMS algebras.\n\nThe proofs are standard homological algebra done carefully: evaluation for projectives, left-projectivity of Happel’s terms so that DΛ ⊗_Λ P• resolves DΛ on the right, and the AR four-term sequence applied to the minimal presentation of Ω^{n-1}Λ. Bimodule structures and naturality are tracked. The examples (K_2, A_n, the Xi pair, A_3 versus A_3/rad^{2}) match known dimension formulas and illustrate transversality cleanly, including the finite-global-dimension case where the two translates have the same total dimension but opposite composition.\n\nSoft spots are minor. Separability of E is the usual hypothesis that makes Happel’s form available; it is stated and used consistently. Self-citation of Ar1–Ar5 and CLMS is dense, but the bridge itself is an independent identification rather than a redefinition. Machine computations are asserted without code, which is normal for this literature. The open question on derived invariance of the τ-groups over the smooth locus is well-posed and already reduced to the excess in low degrees.\n\nThis is for people working on Hochschild theory, higher AR theory, or derived invariants of finite-dimensional algebras. The math is solid, the citations are appropriate, and the contribution is a genuine unification with concrete structural consequences. I would send it to peer review and would cite the bridge and the top-degree identification myself.","headline":"Clean bridge between τ-Hochschild theory and the Coxeter automorphism via cycles of the Nakayama twist of Happel’s resolution; solid and worth engaging.","tokens_in":21874,"tokens_out":589,"would_cite":true,"duration_ms":5863,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16E40","16G70","18G80","16E35"],"pacs":[],"model":"grok-4.5","headline":"Two enrichments of Hochschild theory meet on the square of the Serre bimodule, with τ-translates as its cycle modules.","keywords":["τ-Hochschild homology","τ-Hochschild cohomology","Serre bimodule","Coxeter automorphism","Tamarkin–Tsygan calculus","higher preprojective algebra","Happel resolution","derived invariance"],"falsifier":"Exhibit a finite-dimensional algebra with separable semisimple quotient for which the n-cycles of the Nakayama-twisted Happel complex fail to recover the higher Auslander–Reiten translate of the regular bimodule, or for which the resulting short exact sequences do not split the translate into the claimed Tor term dual to Ext^n and a residue of the minimal model.","tokens_in":21762,"feed_emoji":"⧖","tokens_out":886,"duration_ms":6329,"temperature":0.7,"pith_summary":"The paper unifies two recent refinements of Hochschild theory for finite-dimensional algebras. One refinement, τ-Hochschild (co)homology, is built from higher Auslander–Reiten translates of the regular bimodule and is Morita-invariant but not derived-invariant. The other is the Coxeter automorphism of the full Tamarkin–Tsygan calculus, which is derived-invariant when the global dimension is finite. The author shows that the Nakayama functor applied to Happel’s minimal resolution produces a complex representing the derived square of the Serre bimodule (whose shift generates the Coxeter automorphism), and that the higher τ-translates are exactly the cycle bimodules of this complex. This yields short exact sequences that split each τ-translate into a derived shadow and a strictly Morita-theoretic residue of the minimal model. In top degree the residue vanishes and the top translate recovers the dual of the degree-one piece of the higher preprojective algebra. Taking Euler characteristics recovers Happel’s trace formula as the common numerical core of the two refinements. The paper then proves the refinements are transversal, proposes their combination as a finer Morita invariant, and exhibits derived-equivalent algebras of finite global dimension whose τ-translates have the same dimension but opposite composition, leaving open whether the groups themselves are derived-invariant over the smooth locus.","feed_headline":"τ-translates are cycles of the Serre-bimodule square","feed_subtitle":"Two Hochschild refinements meet: one Morita, one derived, glued by the Nakayama twist of Happel’s resolution","key_machinery":"The bridge identification τ_nΛ=Z_n(ν P_•)=ker ν(d_n) of Theorem A, which realises the higher translates as cycles of the Nakayama twist of Happel’s resolution and thereby realises them as extensions of the derived square of the Serre bimodule by a minimal-model residue.","core_discovery":"The Nakayama functor of the enveloping algebra sends Happel’s minimal resolution of the regular bimodule to a complex representing the derived square of the Serre bimodule. For each n≥1 the higher Auslander–Reiten translate τ_n of the regular bimodule is precisely the module of n-cycles of that complex, giving short exact sequences that split τ_n into a derived-invariant shadow Tor_n(DΛ,DΛ) dual to Ext^n(Λ,Λ^e) and a strictly Morita-theoretic residue of the minimal model.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Nakayama sends Happel resolution to Serre square whose cycles are the τ_n","τ_n bimodules are exactly the cycle modules of DΛ Ltimes_Λ DΛ","τ-translates split into Morita residue of the minimal model plus Tor dual to Ext","Serre-bimodule square cycles recover the higher Auslander–Reiten translates","Two Hochschild refinements meet as Nakayama image of Happel’s resolution"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The semisimple quotient of the algebra by its radical must be separable over the ground field, so that Happel’s minimal resolution exists in the form used throughout.","fun_headline_variants_meta":{"raw":{"variants":["Nakayama sends Happel resolution to Serre square whose cycles are the τ_n","τ_n bimodules are exactly the cycle modules of DΛ Ltimes_Λ DΛ","τ-translates split into Morita residue of the minimal model plus Tor dual to Ext","Serre-bimodule square cycles recover the higher Auslander–Reiten translates","Two Hochschild refinements meet as Nakayama image of Happel’s resolution"]},"model":"grok-4.5","effort":"low","cost_usd":0.007746,"raw_usage":{"total_tokens":2049,"prompt_tokens":1089,"num_sources_used":0,"completion_tokens":115,"cost_in_usd_ticks":77460000,"prompt_tokens_details":{"text_tokens":1089,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":845,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":1089,"tokens_out":115,"duration_ms":9692,"temperature":1.0,"reasoning_tokens":845,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T08:20:07.149199+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a finite-dimensional algebra with separable semisimple quotient for which the n-cycles of the Nakayama-twisted Happel complex fail to recover the higher Auslander–Reiten translate of the regular bimodule, or for which the resulting short exact sequences do not split the translate into the claimed Tor term dual to Ext^n and a residue of the minimal model.","supporting_citations":[],"review_version":1}