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

Abstract: We relate two recent enrichments of the Hochschild theory of a finite-dimensional algebra $\Lm$: the $τ$-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 $σ<em>\Lm$ of the Tamarkin--Tsygan calculus. We show that the Nakayama functor of the enveloping algebra transforms Happel's minimal resolution into a complex representing $\D\Lm\Ltimes</em>\Lm \D\Lm$, the square of the Serre bimodule whose shift generates $σ<em>\Lm$, and that the $τ$-translates $τ_n\Lm$ are precisely the cycle bimodules of this complex. This produces extensions $0\to \B_n\to τ_n\Lm\to \Tor_n^{\Lm(\D\Lm,\D\Lm)\to} 0$ whose outer term is dual to $\Extⁿ</em>{\Lme}(\Lm,\Lme)$ and whose inner term is a strictly Morita-theoretic residue of the minimal model. In top degree $d=\gldim\Lm$ the residue vanishes and $τ<em>d\Lm$ is the dual of the degree-one component of the $(d+1)$-preprojective algebra of Iyama--Oppermann; for $\Lm=\kk Q$ hereditary, $τ</em>{\Lme}\Lm\cong \DΠ(Q)<em>1$ and $\HH¹</em>τ(\kk Q)$ 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 $τ$-cohomology for the Buchweitz--Green--Madsen--Solberg algebras. Taking Euler characteristics in the Cibils--Lanzilotta--Marcos--Solotar dimension formulas recovers Happel's trace formula $\sum_i(-1)<sup>i\dim\HH<sup>i(\Lm)=-\trσ_\Lm$. We prove that the two refinements are transversal, propose the combined Morita invariant, exhibit derived-equivalent algebras of finite global dimension whose $τ$-translates have identical dimension but opposite composition, and pose the problem of derived invariance of $τ$-Hochschild theory over the smooth locus.

• The paper demonstrates that τ-Hochschild (co)homology is linked to the minimal projective bimodule resolution via the Nakayama functor.
• It reveals that the Coxeter automorphism acts as a derived invariant capturing spectral data such as the Coxeter polynomial and trace formula.
• Combined invariants from τ-theory and the Coxeter action distinguish subtle Morita and derived equivalence properties in finite-dimensional algebras.

$\tau$-Hochschild (Co)homology, the Square of the Serre Bimodule, and the Coxeter Automorphism of the Tamarkin–Tsygan Calculus

Introduction and Context

This work establishes a precise structural relationship between two recent extensions of Hochschild theory for finite-dimensional algebras: (i) the $\tau$-Hochschild (co)homology introduced by Cibils–Lanzilotta–Marcos–Solotar (CLMS), and (ii) the Coxeter automorphism $\sigma$ acting on the Tamarkin–Tsygan calculus. The context is over a finite-dimensional $k$-algebra $A$ with separable semisimple quotient, using the conventions of homological algebra and Auslander–Reiten theory.

Main Results

The $\tau$-Hochschild Theory and the Serre Bimodule

The first principal contribution is the demonstration that the $\tau$-Hochschild (co)homology relates intrinsically to the minimal projective bimodule resolution of $A$, viewed under the Nakayama functor $\nu$. Explicitly, the higher Auslander–Reiten translates $\tau_n$ of the regular bimodule correspond to the bimodule cycles of the complex representing the square of the Serre bimodule $\tau$0 in the derived category. The bridge theorem establishes canonical short exact sequences: $\tau$1 where $\tau$2 is a strictly Morita-theoretic residue and the rightmost term is the derived shadow—invariant under derived equivalence only up to conjugation.

Notably, for $\tau$3 selfinjective, the derived shadow vanishes in all positive degrees: the entire $\tau$4-theory is therefore supported in the residue, elucidating the unbounded growth observed by Buchweitz–Green–Madsen–Solberg algebras. In maximal global dimension, the residue vanishes and the top $\tau$5-translate aligns with the dual of the degree-one component of the higher preprojective algebra of Iyama–Oppermann.

The Coxeter Automorphism and Numerical Invariants

On the other axis, the work situates the Coxeter automorphism $\tau$6, which acts on the full Tamarkin–Tsygan calculus (cup/cap products, Gerstenhaber brackets, etc.), as a strictly derived invariant, thereby codifying spectral data such as the Coxeter polynomial within Hochschild theory. The automorphism arises from the action of the two-sided tilting complex $\tau$7 and is shown to be reflected in dimension formulas and Euler characteristics.

A remarkable consequence is the identification, via the dimension formulas of CLMS, that the classical trace formula of Happel $\tau$8 is precisely the Euler characteristic of the $\tau$9 correction terms. This triangulates the relation between $\sigma$0-Hochschild theory, Happel’s trace formula, and operator-level derived invariants.

Theoretical and Computational Implications

Several further results amplify the theoretical structure:

• In top degree, $\sigma$1-Hochschild (co)homology collapses to ordinary Hochschild (co)homology for any coefficients, and for hereditary algebras, the theory computes the necklace space of the degree-one part of the preprojective algebra.
• Derived equivalence preserves the Tamarkin–Tsygan calculus (and the Coxeter automorphism) but does not in general preserve the $\sigma$2-Hochschild (co)homology. The paper gives explicit examples, including derived-equivalent algebras where $\sigma$3-theory separates the objects but the Coxeter data does not, and vice versa.
• The combined invariant

$\sigma$4

is strictly finer than either constituent alone and strictly Morita-invariant.

• Over the singular locus, i.e. for selfinjective (but not semisimple) algebras, the $\sigma$5-dimension is controlled entirely by minimal residues, providing a clear dichotomy with the “shadow” terms vanishing completely.
• Complementarity is established: There are algebras ($\sigma$6 vs.\ $\sigma$7) which cannot be separated by $\sigma$8-theory but are by the Coxeter automorphism; conversely, there are derived-equivalent singular algebras with identical Coxeter structure yet distinct $\sigma$9-theory.

Structural and Conceptual Implications

This analysis exposes the transversal nature of the two refinements:

• $k$0-Hochschild (co)homology is invariant under Morita equivalence but not under derived equivalence; thus, it encodes “finer” residue data relevant in the study of singularities and minimal resolutions.
• The Coxeter automorphism is stable under derived equivalence but insensitive to certain Morita-level distinctions; its spectral data categorifies numerical invariants such as the trace and Coxeter polynomial.
• The identification of the shadow-residue decomposition provides an organizing principle for further investigation of the (non-)invariance properties of invariants across derived/Morita equivalence classes.

The theoretical framework described gives rise to precise questions about the derived invariance of $k$1-theory in the smooth locus, the internal structure of the extension classes in $k$2, and possible module or cap/cup product actions extending the calculus structure functorially over the new invariants.

Connections and Future Directions

The approach links to recent advances in higher Auslander–Reiten theory, $k$3-representation-finite/infinite algebras, the structure and homology of preprojective algebras, and more generally to the application of bimodule and derived functors in the classification and characterization of finite-dimensional algebras.

The methodology suggests a pathway toward new invariants and structural theorems at the interface of representation theory, noncommutative geometry, and homological algebra. Open problems highlighted include:

• The derived invariance of $k$4-Hochschild groups on the smooth locus
• The module/cap structure for $k$5-theory over the ordinary calculus
• The extension of the Coxeter action to the derived-shadow terms functorially
• The relations with singular/Tate Hochschild cohomology, which is derived invariant and complementary to the $k$6 theory
• The effect of minimal residue and shadow terms for higher Calabi–Yau completions and entropy computations in the sense of Han

Conclusion

The work provides a unified structural theory tying together $k$7-Hochschild (co)homology and the Coxeter automorphism, elucidating their articulated roles as dual refinements of Hochschild theory with distinctive invariance and distinguishing properties. The results clarify long-standing structural phenomena in representation and homological algebra and outline conjectures and open directions for further investigation, particularly in the context of derived equivalence and the structure of singular and smooth loci in the representation theory of algebras.

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Reference:

"$k$8-Hochschild (co)homology, the square of the Serre bimodule, and the Coxeter automorphism of the Tamarkin–Tsygan calculus" (2607.10913)