$\tau$-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 $\tau$-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.