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

We relate two recent enrichments of the Hochschild theory of a finite-dimensional algebra $\text{Lm}$: 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 ${\sigma }_{\text{Lm}}$ of the Tamarkin–Tsygan calculus. We show that the Nakayama functor of the enveloping algebra transforms Happel’s minimal resolution into a complex representing $\text{D}\text{Lm}{\text{Ltimes}}_{\text{Lm}}\text{D}\text{Lm}$, the square of the Serre bimodule whose shift generates ${\sigma }_{\text{Lm}}$, and that the $\tau$-translates ${\tau }_n\text{Lm}$ are precisely the cycle bimodules of this complex. This produces extensions $0\to {\text{B}}_n\to {\tau }_n\text{Lm}\to {\text{Tor}}_n^{\text{Lm}}(\text{D}\text{Lm},\text{D}\text{Lm})\to 0$ whose outer term is dual to ${\text{Ext}}_{\text{Lme}}^n(\text{Lm},\text{Lme})$ and whose inner term is a strictly Morita-theoretic residue of the minimal model. In top degree $d=\text{gldim}\text{Lm}$ the residue vanishes and ${\tau }_d\text{Lm}$ is the dual of the degree-one component of the $(d+1)$-preprojective algebra of Iyama–Oppermann; for $\text{Lm}=\text{kk}Q$ hereditary, ${\tau }_{\text{Lme}}\text{Lm}\cong \text{D}\mathrm{\Pi }(Q{)}_1$ and ${\text{HH}}_{\tau }^1(\text{kk}Q)$ 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 ${\sum }_i(-1{)}^i\dim {\text{HH}}^i(\text{Lm})=-\text{tr}{\sigma }_{\text{Lm}}$. 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.