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$\tau$-Hochschild (co)homology, the square of the Serre bimodule, and the Coxeter automorphism of the Tamarkin--Tsygan calculus

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Pith's one-line read Two enrichments of Hochschild theory meet on the square of the Serre bimodule, with τ-translates as its cycle modules.

desk verdict Clean bridge between τ-Hochschild theory and the Coxeter automorphism via cycles of the Nakayama twist of Happel’s resolution; solid and worth engaging. read the letter →

arxiv 2607.10913 v1 pith:WJ4K7S2F submitted 2026-07-12 math.RT

Marco Armenta This is my paper
classification math.RT MSC 16E4016G7018G8016E35
keywords τ-HochschildhomologycohomologySerrebimoduleCoxeterautomorphismTamarkin–TsygancalculushigherpreprojectivealgebraHappelresolutionderivedinvariance
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

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.

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.

minor comments (4)
  1. [Disclosure] 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.
  2. [Example 4.7] 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.
  3. [§3] 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.
  4. [Question 6.7] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the Bridge Theorem is an independent identification of AR cycles with those of the Nakayama-twisted Happel complex; self-citations supply definitions and context only.

full rationale

The paper is pure algebra relating two pre-existing constructions (τ-translates of CLMS via Iyama AR operators on Happel’s resolution, and the Coxeter automorphism σ of the author’s prior Tamarkin–Tsygan work). Theorem 3.2 (Bridge) is proved from first principles: Lemma 3.1 (evaluation isomorphism ν(P) ≅ DΛ ⊗_Λ P ⊗_Λ DΛ for projective bimodules), left-projectivity of Happel’s terms so that DΛ ⊗_Λ P• resolves DΛ on the right, and the classical AR four-term sequence applied to the minimal presentation of Ω^{n-1}Λ, yielding τ_n Λ = ker ν(d_n) and the SES with Tor_n(DΛ,DΛ). None of these steps is defined in terms of the target, fitted, or imported as a uniqueness theorem from the author’s own prior papers. Self-citations (Ar1–Ar5 for the existence and derived-invariance of σ, CLMS1–2 for the definition and dimension formulas of τ-groups) are used only to set the scene, recover Happel’s known trace formula as an Euler characteristic, and exhibit transversality via examples; they are not load-bearing for the new identification or its structural consequences (top-degree preprojective duality, vanishing of the derived shadow on self-injectives, residue/shadow split). No fitted parameters, no ansatz smuggled via citation, no renaming of a known empirical pattern. The derivation chain is therefore self-contained against external benchmarks once the standard AR and resolution facts are granted. Score 1 only for the volume of author self-citation that is normal but non-circular.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

The paper works entirely inside standard finite-dimensional algebra and AR theory. Load-bearing background is separability of E (to get Happel’s resolution in the stated form), classical dualities Tor/Ext, and the existence of higher AR translates τ_n = τΩ^{n−1}. No free parameters are fitted. Invented entities are organizational (combined invariant I(Λ), residue B_n) rather than new physical or algebraic objects postulated without definition.

assumptions (6)
  • domain assumption E = Λ/rad Λ is separable over k, so Happel’s minimal resolution P_n = Λ ⊗_E T_n ⊗_E Λ exists with the stated projectivity and minimality.
    Stated in §2.1 and used for all identifications of ν(P_n) and for pd_{Λ^e} Λ = gl.dim Λ.
  • standard math Classical duality D Tor_n^Λ(M,N) ≅ Ext^n_Λ(N, DM) as bimodules (Cartan–Eilenberg).
    Invoked in Thm 3.2(1)–(3) to identify H_n(νP•) with D Ext^n_{Λ^e}(Λ, Λ^e).
  • standard math Four-term Auslander–Reiten sequence 0 → τM → νQ_1 → νQ_0 → νM → 0 for minimal projective presentations over Λ^e.
    §2.5 and Thm 3.2(2); standard AR theory (ARS Ch. IV.2).
  • domain assumption ω_Λ[−1] is a two-sided tilting complex when gl.dim Λ < ∞, and its conjugation class is a derived invariant.
    Taken from Ar5 Lemmas 3.1 and 3.4; used to identify the derived shadow as conjugation-invariant and to define σ_Λ.
  • domain assumption Han–Keller vanishing: HH^n(Λ) = 0 for n ≥ 1 when Λ is elementary of finite global dimension (in particular HH^d(Λ) = 0).
    Used in Prop 4.1–4.3, Thm 4.5(3), and Thm 5.2; cited from Ke1, Han1, Len.
  • domain assumption Dimension formulas for τ-Hochschild groups of elementary algebras (CLMS2 Thm 4.5 and Cor 4.10).
    Used to recover Happel’s trace formula by Euler characteristic (Thm 5.2) and for numerical checks in examples.
invented entities (2)
  • Minimal residue B_n = im ν(d_{n+1}) inside τ_n Λ
    purpose: Isolate the strictly Morita-theoretic part of the τ-translate after removing the derived shadow Tor_n(DΛ, DΛ).
    Defined in Thm 3.2(2) as the image of the next differential; not previously named as a residue of the minimal model.
  • Combined Morita invariant I(Λ) = (H(Λ), σ_Λ; H^•_τ(Λ,−), H_τ_•(Λ,−))
    purpose: Package the two transversal refinements into a single invariant strictly finer than each constituent.
    Definition 6.4; justified by Ex 6.1 and 6.3 showing each refinement separates pairs the other cannot.

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Pith. "Pith review of $\tau$-Hochschild (co)homology, the square of the Serre bimodule, and the Coxeter automorphism of the Tamarkin--Tsygan calculus." pith.science (2026). https://pith.science/paper/WJ4K7S2F

@misc{pith2026260710913,
  author       = {Pith},
  title        = {Pith review of: $\tau$-Hochschild (co)homology, the square of the Serre bimodule, and the Coxeter automorphism of the Tamarkin--Tsygan calculus},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WJ4K7S2F}},
  note         = {Machine review of arXiv:2607.10913}
}
abstract

We relate two recent enrichments of the Hochschild theory of a finite-dimensional algebra $\Lm$: the $\tau$-Hochschild (co)homology of Cibils, Lanzilotta, Marcos and Solotar, built from Iyama's higher Auslander--Reiten translates of the regular bimodule, and the Coxeter automorphism $\sigma_\Lm$ of the Tamarkin--Tsygan calculus. We show that the Nakayama functor of the enveloping algebra transforms Happel's minimal resolution into a complex representing $\D\Lm\Ltimes_\Lm \D\Lm$, the square of the Serre bimodule whose shift generates $\sigma_\Lm$, and that the $\tau$-translates $\tau_n\Lm$ are precisely the cycle bimodules of this complex. This produces extensions $0\to \B_n\to \tau_n\Lm\to \Tor_n^\Lm(\D\Lm,\D\Lm)\to 0$ whose outer term is dual to $\Ext^n_{\Lme}(\Lm,\Lme)$ and whose inner term is a strictly Morita-theoretic residue of the minimal model. In top degree $d=\gldim\Lm$ the residue vanishes and $\tau_d\Lm$ is the dual of the degree-one component of the $(d+1)$-preprojective algebra of Iyama--Oppermann; for $\Lm=\kk Q$ hereditary, $\tau_{\Lme}\Lm\cong \D\Pi(Q)_1$ and $\HH^1_\tau(\kk Q)$ is the degree-one part of the zeroth Hochschild homology of the preprojective algebra. For self-injective algebras, the derived part vanishes identically, which explains structurally the growth of $\tau$-cohomology for the Buchweitz--Green--Madsen--Solberg algebras. Taking Euler characteristics in the Cibils--Lanzilotta--Marcos--Solotar dimension formulas recovers Happel's trace formula $\sum_i(-1)^i\dim\HH^i(\Lm)=-\tr\sigma_\Lm$. We prove that the two refinements are transversal, propose the combined Morita invariant, exhibit derived-equivalent algebras of finite global dimension whose $\tau$-translates have identical dimension but opposite composition, and pose the problem of derived invariance of $\tau$-Hochschild theory over the smooth locus.

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