June 14, 2026
Let \(A\) be a finite-dimensional algebra over a field \(k\). We show that the Auslander–Reiten bimodule \(\tau_A:=\mathrm{D}A[-1]\) is central in the derived Picard group of \(A\) and that, when \(\operatorname{gl.dim}A<\infty\), it induces through the derived-invariance functor \(\mathbb{H}\) of [1]–[3] a canonical automorphism \(\sigma_A\) of the Tamarkin–Tsygan calculus of \(A\); the pair \((\mathbb{H}(A),\sigma_A)\) is invariant under derived equivalence. We then compute both components of \(\sigma_A\). On the Hochschild homology of an elementary algebra, which is concentrated in degree zero, the matrix of \(\sigma_A\) in the basis of idempotent traces is \(-C_A^{-1}C_A^{\mathrm{T}}\), so its characteristic polynomial is the Coxeter polynomial; the enriched calculus strictly refines both the calculus and the Coxeter polynomial, as the path algebras of quivers of types \(\mathbb{A}_4\) and \(\mathbb{D}_4\) show, although it is not a complete derived invariant, as the smallest cospectral pair of trees shows. On Hochschild cohomology we prove that \(\sigma_A\) is the identity: the left and right actions of \(\mathit{HH}^\bullet(A)\) on the bimodule \(\mathrm{D}A\) coincide for every finite-dimensional \(A\). This yields a short conceptual proof that the Nakayama automorphism of a Frobenius algebra acts trivially on Hochschild cohomology, recovering a recent theorem of Suárez-Álvarez. Finally we extend the construction to smooth and proper differential graded algebras, hence to perfect derived categories of smooth projective varieties; the enrichment degenerates precisely on Calabi–Yau categories, and on \(\mathbb{P}^n\) it is governed by the Coxeter polynomial \((x+(-1)^n)^{n+1}\) of the Beilinson algebra. Happel’s trace formula and de la Peña’s cyclotomicity theorem for fractionally Calabi–Yau algebras become statements internal to the enriched calculus.
Two of the oldest families of derived invariants of a finite-dimensional algebra \(A\) of finite global dimension live at opposite ends of the homological spectrum. At the decategorified end sits the Coxeter polynomial \(\phi_A\), the characteristic polynomial of the Coxeter transformation \(\Phi_A\), that is, of the automorphism of the Grothendieck group \(K_0(\operatorname{per}A)\) induced by the derived Auslander–Reiten translation \(\tau=\nu\circ[-1]\) of Happel [4], [5]; its derived invariance is classical, see [6], [7]. At the other end sits the full Hochschild theory of \(A\): Rickard proved that derived equivalences preserve the Hochschild cohomology algebra [8], Keller that they preserve its Gerstenhaber bracket [9] and the cyclic theory [10], and in [1]–[3] it was proved that the whole Tamarkin–Tsygan calculus \[\mathbb{H}(A) \;=\; \bigl(\mathit{HH}_\bullet(A),\, \mathit{HH}^\bullet(A),\, \cup,\, [-,-],\, \cap,\, B\bigr)\] of Hochschild homology and cohomology, with the cup product, the Gerstenhaber bracket, the cap product and the Connes differential, is a derived invariant. More precisely, [1] constructs a functor \(\mathbb{H}\colon \mathbf{A}_{k}\to\mathbf{TT\text{-}calc}\), from a category of algebras whose morphisms are suitable complexes of bimodules to the category of Tamarkin–Tsygan calculi, which is constant on derived equivalence classes.
Neither invariant determines the other. The thesis [1] ends with the observation that the path algebras of the quivers \[Q_A:\quad \begin{array}{ccccc} 2 & & & & 3\\[-2pt] & \searrow & & \swarrow & \\[-2pt] & & 1 & & \\[-2pt] & & \uparrow & & \\[-2pt] & & 4 & & \end{array} \qquad\qquad Q_B:\quad 1\longrightarrow 2 \longrightarrow 3 \longrightarrow 4\] of Dynkin types \(\mathbb{D}_4\) and \(\mathbb{A}_4\) have isomorphic Tamarkin–Tsygan calculi — every operation degenerates — while their Coxeter polynomials \[\phi_A(x)=(x+1)^2(x^2-x+1)=x^4+x^3+x+1, \qquad \phi_B(x)=x^4+x^3+x^2+x+1\] differ, so that \(A\) and \(B\) are not derived equivalent — a fact which is of course classical [4], [5]; see Remark 12 — and the Tamarkin–Tsygan calculus is not a complete derived invariant. Conversely, the Coxeter polynomial is notoriously far from complete [7]. The purpose of this article is to show that the two invariants are in fact two shadows of a single, finer one, obtained by letting the categorical avatar of the Coxeter transformation act on the calculus itself.
The mechanism is simple, and the point of this paper is precisely that the machinery of [1] makes it simple. The Coxeter transformation is the Grothendieck-group shadow of the Auslander–Reiten translation \(\tau=\nu[-1]\), where \(\nu=-\otimes^{\mathbb{L}}_{A}\mathrm{D}A\) is the Nakayama functor, a Serre functor on \(\operatorname{per}A\) [4], [5], [11]. We prove that the underlying bimodule complex \[\tau_A \;:=\; \mathrm{D}A[-1]\] is a two-sided tilting complex over \(A\) whenever \(\operatorname{gl.dim}A<\infty\) (Lemma 3) and that it commutes with every two-sided tilting complex (Lemma 4): if \(X\) is a two-sided tilting complex of \(A\)-\(B\)-bimodules, then \[\tau_A\otimes^{\mathbb{L}}_{A}X \;\cong\; X\otimes^{\mathbb{L}}_{B}\tau_B \qquad\text{in the derived category of A-B-bimodules.}\] Since \(\tau_A\) is in particular a morphism \(A\to A\) of the category \(\mathbf{A}_{k}\) of [1], the functor \(\mathbb{H}\) produces from it an automorphism of the calculus. We call \[\sigma_A \;:=\; \mathbb{H}(\tau_A)\;\in\;\mathop{\mathrm{Aut}}_{\mathbf{TT\text{-}calc}}\bigl(\mathbb{H}(A)\bigr)\] the Coxeter automorphism of the Tamarkin–Tsygan calculus of \(A\). Functoriality then yields, with almost no further work:
Theorem A 1 (= Theorem 3). Let \(k\) be a field and \(A\) a finite-dimensional \(k\)-algebra with \(\operatorname{gl.dim}A<\infty\).
\(\sigma_A\) is an automorphism of the Tamarkin–Tsygan calculus \(\mathbb{H}(A)\).
The assignment \(m\mapsto\sigma_A^m=\mathbb{H}(\tau_A^{\otimes^{\mathbb{L}}_A m})\) is a group homomorphism \(\mathbb{Z}\to\mathop{\mathrm{Aut}}_{\mathbf{TT\text{-}calc}}(\mathbb{H}(A))\).
For every two-sided tilting complex \(X\) of \(A\)-\(B\)-bimodules one has \(\mathbb{H}(X)\circ\sigma_A=\sigma_B\circ\mathbb{H}(X)\). Consequently the isomorphism class of the pair \((\mathbb{H}(A),\sigma_A)\) is a derived invariant of \(A\).
Spelled out, part (1) says that the homological and cohomological components \((\sigma_\bullet,\sigma^\bullet)\) of \(\sigma_A\) respect all five operations of the calculus: \[\begin{gather} \sigma^\bullet(\eta\cup\theta)=\sigma^\bullet\eta\cup\sigma^\bullet\theta, \qquad \sigma^\bullet[\eta,\theta]=[\sigma^\bullet\eta,\sigma^\bullet\theta],\\ \sigma_\bullet(z\cap\eta)=\sigma_\bullet z\cap\sigma^\bullet\eta, \qquad \sigma_\bullet B=B\sigma_\bullet . \end{gather}\]
The second main result computes the homological component of \(\sigma_A\) for elementary algebras and identifies it with the Coxeter transformation.
Theorem B 1 (= Theorem 10). Let \(A\) be a finite-dimensional elementary \(k\)-algebra with \(\operatorname{gl.dim}A<\infty\), let \(e_1,\dots,e_n\) be a complete set of primitive orthogonal idempotents, and let \(\bar e_1,\dots,\bar e_n\) denote their classes in \(\mathit{HH}_0(A)=A/[A,A]\). Then \(\mathit{HH}_i(A)=0\) for \(i\geq 1\), the classes \(\bar e_1,\dots,\bar e_n\) form a basis of \(\mathit{HH}_0(A)\), and in this basis the matrix of \(\sigma_A\) on \(\mathit{HH}_0(A)\) is \[-\,C_A^{-1}C_A^{\mathrm{T}},\] where \(C_A\) is the Cartan matrix of \(A\).
Two consequences are drawn in the text (Proposition 8, Theorem 10). First, the Euler class \(\operatorname{ch}\colon K_0(\operatorname{per}A)\to\mathit{HH}_0(A)\), which sends \([e_iA]\) to \(\bar e_i\) and becomes an isomorphism after extension of scalars to \(k\), intertwines the Coxeter transformation \(\Phi_A\) with \(\sigma_A\). Second, the characteristic polynomial of \(\sigma_A\) on Hochschild homology is the Coxeter polynomial of \(A\), read in \(k[x]\): \[\det\bigl(x\cdot\operatorname{id}-\sigma_A|_{\mathit{HH}_\bullet(A)}\bigr) \;=\; \phi_A(x).\]
Thus the enriched calculus \((\mathbb{H}(A),\sigma_A)\) determines the Tamarkin–Tsygan calculus (forget \(\sigma\)) and the Coxeter polynomial (take the characteristic polynomial of \(\sigma\) on homology, on the nose if \(\operatorname{char}k=0\), modulo \(p\) in characteristic \(p\)).
The third main result identifies the cohomological component completely; we regard it as the conceptual heart of the paper.
Theorem C 1 (= Theorem 16, Corollary 4). Let \(A\) be any finite-dimensional \(k\)-algebra. The left and right actions of \(\mathit{HH}^\bullet(A)\) on the bimodule \(\mathrm{D}A\) coincide. Consequently:
if \(\operatorname{gl.dim}A<\infty\), then \(\sigma^\bullet_A=\operatorname{id}_{\mathit{HH}^\bullet(A)}\), so the enriched calculus amounts to the Tamarkin–Tsygan calculus together with the single operator \(\sigma_\bullet\) on Hochschild homology, which commutes with the Connes differential and with every cap operator;
if \(A\) is Frobenius, its Nakayama automorphism acts trivially on \(\mathit{HH}^\bullet(A)\).
Part (2) was proved very recently by Suárez-Álvarez [12] by an explicit cochain-level analysis; here it falls out of the same duality argument as part (1), the common source being the centrality of the class of \(\mathrm{D}A\) in the derived Picard group of any finite-dimensional algebra (Lemma 4 with \(B=A\)). Theorem C converts the two halves of \(\sigma_A\) into a sharp dichotomy: Hochschild cohomology, the home of deformation theory, is blind to the Coxeter symmetry, while Hochschild homology, the home of traces, detects it completely. The example that motivated this paper shows that the resulting homological refinement is strict (Example 1): for the path algebras \(A\) and \(B\) of the quivers of types \(\mathbb{D}_4\) and \(\mathbb{A}_4\) above, over an algebraically closed field, the Tamarkin–Tsygan calculi are isomorphic, but there exists no isomorphism \(\mathbb{H}(A)\to\mathbb{H}(B)\) commuting with \(\sigma_A\) and \(\sigma_B\). Hence \((\mathbb{H}(-),\sigma)\) is a strictly finer derived invariant than the Tamarkin–Tsygan calculus, and also strictly finer than the Coxeter polynomial (Remark 11). It is not, however, a complete derived invariant: the two smallest cospectral trees, on eight vertices, have conjugate Coxeter matrices and hence isomorphic enriched calculi, without being derived equivalent (Example 2).
Finally, two classical theorems of Coxeter theory become statements internal to the enriched calculus. Happel’s trace formula [13] reads \[\sum_{i\geq 0}(-1)^i\dim_k\mathit{HH}^i(A) \;=\; -\operatorname{tr}\bigl(\sigma_A|_{\mathit{HH}_\bullet(A)}\bigr):\] the Euler characteristic of the cohomological half of the calculus equals minus the trace of the Coxeter automorphism on the homological half (Corollary 2); the operator \(\sigma_A\) is in this precise sense the promotion of Happel’s numerical invariant to a derived-invariant operator, and over a field of characteristic zero the full Coxeter polynomial is recovered from the supertraces of the powers of \(\sigma_A\) by Newton’s identities. And if \(A\) is fractionally Calabi–Yau, in the bimodule sense that \((\mathrm{D}A)^{\otimes^{\mathbb{L}}_A q}\cong A[p]\) in \(\mathcal{D}(A^e)\) [14], then \(\sigma_A^q=(-1)^{p-q}\operatorname{id}\) on \(\mathit{HH}_\bullet(A)\), whence \(\Phi_A\) is periodic and \(\phi_A\) is a product of cyclotomic polynomials (Corollary 3), recovering part of [6] at the level of the calculus.
Section 6 proves Theorem C. Section 7, written following a suggestion of B. Keller, carries the construction to smooth and proper dg algebras, hence to perfect derived categories of smooth projective varieties, where the enrichment degenerates exactly on Calabi–Yau categories and where, on \(\mathbb{P}^n\), it is governed by the Coxeter polynomial \((x+(-1)^n)^{n+1}\) of the Beilinson algebra. Section 8 relates \(\sigma_A\) to recent work in several directions: the Lefschetz- and Hirzebruch–Riemann–Roch-type formulas of Han [15], which our Theorem B categorifies in degree zero; the categorical entropy of Serre functors [16]–[19] and de la Peña’s Mahler measure programme [20]; the periodicity dictionary between fractional Calabi–Yau properties and cyclotomic Coxeter polynomials [14], [21]–[24]; and the Calabi–Yau completions of Keller [25], of which the operator \(\sigma_A\) is, informally, the first Fourier mode. We close with open questions, of which we single out the \(\tau\)-twisted cohomological theory of Question 23 as the natural continuation of Theorem C.
The author thanks Claude Cibils, José Antonio de la Peña and Bernhard Keller for the conversations that shaped his view of these two subjects, and is especially grateful to Bernhard Keller for his comments on a first version of this paper and for suggesting the extension to smooth and proper dg categories carried out in Section 7. He also thanks Ibrahim Assem for his careful reading of a first version and for remarks that improved the exposition, in particular of the introduction and of the motivating example.
Throughout, \(k\) is a field, \(\mathrm{D}=\operatorname{Hom}_k(-,k)\) is the \(k\)-duality, and \(A\), \(B\) denote finite-dimensional \(k\)-algebras. Modules are right modules unless stated otherwise; \(\operatorname{mod}A\) is the category of finite-dimensional right \(A\)-modules. An \(A\)-\(B\)-bimodule is a left \(A\)-, right \(B\)-module on which \(k\) acts centrally. We write \(A^e=A\otimes_kA^{\mathrm{op}}\), so that \(A\)-\(A\)-bimodules are left \(A^e\)-modules. We denote by \(\mathcal{D}(A)\) the unbounded derived category of right \(A\)-modules, by \(\mathcal{D}^b(\operatorname{mod}A)\) the bounded derived category, and by \(\operatorname{per}A\subseteq\mathcal{D}(A)\) the perfect derived category, that is, the thick subcategory generated by \(A_A\). If \(\operatorname{gl.dim}A<\infty\), the canonical functor \(\operatorname{per}A\to\mathcal{D}^b(\operatorname{mod}A)\) is an equivalence. Since \(k\) is a field, all derived functors between derived categories of bimodules below are computed by means of resolutions that are homotopically projective over the enveloping algebra, hence over each side separately; see [10] for this standard framework.
Assume in this subsection that \(A\) is elementary, that is, \(A/\operatorname{rad}A\cong k^n\) as \(k\)-algebras; equivalently \(A\cong kQ/I\) for a finite quiver \(Q\) and an admissible ideal \(I\). Fix a complete set \(e_1,\dots,e_n\) of primitive orthogonal idempotents, and set \(P_i=e_iA\) and \(I_i=\mathrm{D}(Ae_i)\), complete lists of the indecomposable projective and injective right \(A\)-modules.
The Cartan matrix of \(A\) is \(C_A=(c_{ij})\in\mathbb{Z}^{n\times n}\) with \[c_{ij}\;=\;\dim_k\operatorname{Hom}_A(P_i,P_j)\;=\;\dim_ke_jAe_i .\] The dimension vector of \(M\in\operatorname{mod}A\) is the column vector \(\underline{\dim}\,M\in\mathbb{Z}^n\) with \((\underline{\dim}\,M)_i=\dim_kMe_i=\dim_k\operatorname{Hom}_A(P_i,M)\); it extends to \(\operatorname{per}A\) by setting \(\underline{\dim}\,M=\sum_j(-1)^j\underline{\dim}\,H^j(M)\). Whenever \(\operatorname{gl.dim}A<\infty\) it induces an isomorphism \(\underline{\dim}\,\colon K_0(\operatorname{per}A)\to\mathbb{Z}^n\). We record the two basic computations: \[\label{eq:dimPI} \underline{\dim}\,P_j = C_A\,\varepsilon_j, \qquad \underline{\dim}\,I_j = C_A^{\mathrm{T}}\,\varepsilon_j ,\tag{1}\] where \(\varepsilon_1,\dots,\varepsilon_n\) is the standard basis of \(\mathbb{Z}^n\). Indeed \[\begin{gather} (\underline{\dim}\,P_j)_i=\dim\operatorname{Hom}_A(P_i,P_j)=c_{ij},\\ \operatorname{Hom}_A\bigl(e_iA,\mathrm{D}(Ae_j)\bigr)\cong\mathrm{D}(e_iA\otimes_AAe_j)=\mathrm{D}(e_iAe_j), \end{gather}\] and the second computation gives \((\underline{\dim}\,I_j)_i=c_{ji}\). If \(\operatorname{gl.dim}A<\infty\) then \(C_A\) is invertible over \(\mathbb{Z}\) [4], so 1 determines the class of any perfect complex in terms of the classes \([P_i]\).
If \(\operatorname{gl.dim}A<\infty\), the category \(\operatorname{per}A=\mathcal{D}^b(\operatorname{mod}A)\) has a Serre functor, namely the derived Nakayama functor \(\nu=-\otimes^{\mathbb{L}}_{A}\mathrm{D}A\) [4], [11], and it has Auslander–Reiten triangles with translation \(\tau=\nu\circ[-1]\), see [5]. The Coxeter transformation of \(A\) is \[\Phi_A:=K_0(\tau)\colon K_0(\operatorname{per}A)\longrightarrow K_0(\operatorname{per}A),\] and the Coxeter polynomial is \(\phi_A(x)=\det(x\cdot\operatorname{id}-\Phi_A)\in\mathbb{Z}[x]\).
Lemma 1. In the basis \([P_1],\dots,[P_n]\) of \(K_0(\operatorname{per}A)\) the matrix of \(\Phi_A\) is \(-C_A^{-1}C_A^{\mathrm{T}}\). Consequently \(\phi_A(x)=\det\!\bigl(xI_n+C_A^{-1}C_A^{\mathrm{T}}\bigr)\), and this polynomial is unchanged if \(C_A\) is replaced by \(C_A^{\mathrm{T}}\) or by \(PC_AP^{\mathrm{T}}\) for any \(P\in GL_n(\mathbb{Z})\).
Proof. There is an isomorphism of right \(A\)-modules \(P_i\otimes_A\mathrm{D}A=e_i\mathrm{D}A\cong \mathrm{D}(Ae_i)=I_i\), given by restriction of linear forms; hence \(\nu(P_i)\cong I_i\) and \(\Phi_A[P_i]=[\,\tau P_i\,]=-[I_i]\). Writing \([I_i]=\sum_ja_j[P_j]\) and applying \(\underline{\dim}\,\) together with 1 gives \(C_Aa=C_A^{\mathrm{T}}\varepsilon_i\), that is, \(a=C_A^{-1}C_A^{\mathrm{T}}\varepsilon_i\). For the last assertions, note \(\det C_A=\pm 1\) and \(\det(xI+C^{-1}C^{\mathrm{T}})=\det(C)^{-1}\det(xC+C^{\mathrm{T}})\); the expression \(\det(xC+C^{\mathrm{T}})\) is invariant under \(C\mapsto PCP^{\mathrm{T}}\) since \(\det(P)^2=1\), and it is invariant under \(C\mapsto C^{\mathrm{T}}\) because \(\det(xC^{\mathrm{T}}+C)=\det\bigl((xC^{\mathrm{T}}+C)^{\mathrm{T}}\bigr) =\det(xC+C^{\mathrm{T}})\); compare [26]. ◻
It is classical that \(\phi_A\) is a derived invariant; see [6] and [7]; for its behaviour under one-point extensions see [27]. We shall recover the derived invariance in Corollary 1 below.
A Tamarkin–Tsygan calculus \((\mathcal{H}_\bullet,\mathcal{H}^\bullet,\cup,[-,-],\cap,B)\) consists of a Gerstenhaber algebra \((\mathcal{H}^\bullet,\cup,[-,-])\), a graded module structure \(\cap\colon\mathcal{H}_n\otimes\mathcal{H}^m\to\mathcal{H}_{n-m}\) over it, and a differential \(B\colon\mathcal{H}_\bullet\to \mathcal{H}_{\bullet+1}\) with \(B^2=0\), subject to the compatibilities of noncommutative differential calculus, in particular the Cartan homotopy formula relating \(B\), \(\cap\) and the Lie action; see [28], [29] and [1] for the precise axioms used here. A morphism of calculi \((f,g)\colon H\to K\) is a pair of graded linear maps \(f\colon\mathcal{H}_\bullet\to\mathcal{K}_\bullet\), \(g\colon\mathcal{H}^\bullet\to\mathcal{K}^\bullet\) commuting with \(\cup\), \([-,-]\), \(\cap\) and \(B\) in the obvious sense [1]. We write \(\mathbf{TT\text{-}calc}\) for the resulting category.
The Hochschild theory of \(A\) defines an object \[\mathbb{H}(A)=\bigl(\mathit{HH}_\bullet(A),\mathit{HH}^\bullet(A),\cup_A,[-,-]_A,\cap_A,B_A \bigr)\in\mathbf{TT\text{-}calc}.\] Following [1], let \(\mathbf{A}_{k}\) be the category whose objects are the finite-dimensional \(k\)-algebras and whose morphisms \(X\colon A\to B\) are the complexes of \(A\)-\(B\)-bimodules that are isomorphic, in the derived category of \(A\)-\(B\)-bimodules, to a bounded complex whose components are finitely generated projective as right \(B\)-modules; composition is the derived tensor product, \(Y\circ X = X\otimes^{\mathbb{L}}_{B}Y\) for \(X\colon A\to B\) and \(Y\colon B\to C\), and the identity of \(A\) is the bimodule \(A\). Two morphisms that are isomorphic in the derived category of bimodules are equal in \(\mathbf{A}_{k}\). The main theorem of [1] (see also [2], [3]) is:
Theorem 1 ([1]). The assignments \(A\mapsto\mathbb{H}(A)\) and \(X\mapsto\mathbb{H}(X)\), with components \(\mathbb{H}(X)=(\mathit{HH}_\bullet(X),\mathit{HH}^\bullet(X))\), define a functor \[\mathbb{H}\colon\mathbf{A}_{k}\longrightarrow\mathbf{TT\text{-}calc}\] which sends two-sided tilting complexes to isomorphisms and is constant on each class of derived equivalent algebras. The homological components \(\mathit{HH}_\bullet(X)\) coincide with the morphisms induced by Keller’s cyclic functor [10]; the cohomological components \(\mathit{HH}^\bullet(X)\) of an invertible \(X\) respect the cup product [8], the Gerstenhaber bracket [9], and together with \(\mathit{HH}_\bullet(X)\) the cap product [1], [2] and the Connes differential [3].
A morphism \(X\colon A\to B\) of \(\mathbf{A}_{k}\) is a two-sided tilting complex if it is invertible, that is, if there exists a complex \(Y\) of \(B\)-\(A\)-bimodules with \(X\otimes^{\mathbb{L}}_{B}Y\cong A\) in \(\mathcal{D}(A^e)\) and \(Y\otimes^{\mathbb{L}}_{A}X\cong B\) in \(\mathcal{D}(B^e)\). By Rickard’s theorems [30], [31], \(A\) and \(B\) are derived equivalent if and only if a two-sided tilting complex \(X\colon A\to B\) exists; moreover \(X\) is then perfect as a complex of left \(A\)-modules and as a complex of right \(B\)-modules, and the inverse is computed by either one-sided dual: \[\label{eq:duals} Y\;\cong\;\mathbf{R}\!\operatorname{Hom}_B(X_B,B_B)\;\cong\;\mathbf{R}\!\operatorname{Hom}_{A}(\,{}_AX,\,{}_AA)\tag{2}\] as complexes of \(B\)-\(A\)-bimodules, see [31] and [32]. Derived equivalences preserve finiteness of global dimension [4].
We will use repeatedly the following elementary representability statement.
Lemma 2. Let \(X\) be a complex of \(A\)-\(B\)-bimodules with bounded, finite-dimensional total cohomology, which is perfect as a complex of right \(B\)-modules. Then \(X\) is isomorphic, in the derived category of \(A\)-\(B\)-bimodules, to a bounded complex whose components are finitely generated projective right \(B\)-modules; in particular \(X\in\mathbf{A}_{k}(A,B)\).
Proof. Replacing \(X\) by the total complex of a bimodule bar resolution \(\cdots\to X\otimes_kB\otimes_kB\to X\otimes_kB\to X\), we may assume the components of \(X\) are of the form \(V\otimes_kB\) with \(V\) finite-dimensional, hence finitely generated free as right \(B\)-modules, at the price of unboundedness to the left. Brutally truncate: for \(N\gg 0\), the kernel \(K\) of the differential out of degree \(-N\) is a sub-bimodule, finitely generated over \(B\), and by the syzygy argument (the truncated complex resolves an object that is perfect over \(B\)) \(K\) is projective as a right \(B\)-module once \(N\) exceeds the relevant projective dimension. The bounded complex \(K\to X^{-N+1}\to\cdots\) represents \(X\). ◻
Lemma 3. Let \(A\) be a finite-dimensional \(k\)-algebra with \(\operatorname{gl.dim}A<\infty\). Then the bimodule \(\mathrm{D}A\) is a two-sided tilting complex over \(A\), with inverse \(\mathbf{R}\!\operatorname{Hom}_A(\mathrm{D}A,A)\), and \(\mathrm{D}A\), \(\mathrm{D}A[-1]\) and their inverses are morphisms \(A\to A\) in \(\mathbf{A}_{k}\).
Proof. Since \(\operatorname{gl.dim}A<\infty\) and \(\dim_kA<\infty\), the right \(A\)-module \(\mathrm{D}A\) and the left \(A\)-module \(\mathrm{D}A\) are perfect; by Lemma 2, \(\mathrm{D}A\) and \(\mathrm{D}A[-1]\) are morphisms in \(\mathbf{A}_{k}\). Set \(Y=\mathbf{R}\!\operatorname{Hom}_A(\mathrm{D}A,A)\), the derived \(\operatorname{Hom}\) of right \(A\)-modules, an object of the derived category of \(A\)-\(A\)-bimodules via the left structures of \(\mathrm{D}A\) and of \(A\).
We prove that \(Y\) is a two-sided inverse of \(\mathrm{D}A\) in the monoidal category \((\mathcal{D}(A^e),\otimes^{\mathbb{L}}_{A},A)\). All derived functors are computed with h-projective (\(P\) is h-projective when \(Hom_A^\bullet (P,N)\) is acyclic for all acyclic complex \(N\)) resolutions of complexes of bimodules; since \(k\) is a field, these restrict to h-projective one-sided modules, so the canonical morphisms below are defined at the level of complexes of bimodules [10].
Step 1. For \(M\in\operatorname{per}A\) and a complex \(N\) of \(A\)-bimodules, consider the canonical morphism \[N\otimes^{\mathbb{L}}_{A}\mathbf{R}\!\operatorname{Hom}_A(M,A)\longrightarrow\mathbf{R}\!\operatorname{Hom}_A(M,N),\qquad n\otimes f\longmapsto\bigl(m\mapsto n\,f(m)\bigr).\] It is balanced over \(A\), because the left action on \(\mathbf{R}\!\operatorname{Hom}_A(M,A)\) is \((af)(m)=a\,f(m)\), and it is a morphism of complexes of \(A\)-bimodules for the remaining structures (the left structure of \(N\), and the right action \((fa)(m)=f(am)\) coming from the left structure of \(M\)). It is invertible for \(M=A\), both sides are triangulated functors of \(M\), and \(\operatorname{per}A\) is generated by \(A\); hence it is invertible for every \(M\in\operatorname{per}A\). Taking \(M=N=\mathrm{D}A\), which is legitimate since \(\mathrm{D}A\in\operatorname{per}A\), yields \[\mathrm{D}A\otimes^{\mathbb{L}}_{A}Y\;\cong\;\mathbf{R}\!\operatorname{Hom}_A(\mathrm{D}A,\mathrm{D}A) \qquad\text{in }\mathcal{D}(A^e).\] Step 2. The tensor–hom adjunction over \(k\) gives a natural isomorphism \(\mathbf{R}\!\operatorname{Hom}_A\bigl(X,\mathrm{D}V\bigr)\cong\mathrm{D}\bigl(X\otimes^{\mathbb{L}}_{A}V\bigr)\) for \(X\) a complex of right \(A\)-modules and \(V\) a complex of left \(A\)-modules, compatible with all auxiliary bimodule structures. With \(X=\mathrm{D}A\) and \(V=A\) (so \(\mathrm{D}V=\mathrm{D}A\)): \[\mathbf{R}\!\operatorname{Hom}_A(\mathrm{D}A,\mathrm{D}A)\;\cong\;\mathrm{D}\bigl(\mathrm{D}A\otimes^{\mathbb{L}}_{A}A\bigr)\;=\;\mathrm{D}(\mathrm{D}A)\;\cong\; A\qquad\text{in }\mathcal{D}(A^e),\] the last isomorphism because \(A\) is finite-dimensional. Hence \(\mathrm{D}A\otimes^{\mathbb{L}}_{A}Y\cong A\).
Step 3. The mirror of Steps 1–2 over \(A^{\mathrm{op}}\) — using the balanced morphism \(\mathbf{R}\!\operatorname{Hom}_{A^{\mathrm{op}}}(M,A)\otimes^{\mathbb{L}}_{A}N\to\mathbf{R}\!\operatorname{Hom}_{A^{\mathrm{op}}}(M,N)\), \(f\otimes n\mapsto(m\mapsto f(m)\,n)\), for \(M\in\operatorname{per}A^{\mathrm{op}}\), together with the left-handed tensor–hom adjunction \(\mathbf{R}\!\operatorname{Hom}_{A^{\mathrm{op}}}(W,\mathrm{D}V)\cong\mathrm{D}(V\otimes^{\mathbb{L}}_{A}W)\) — yields, for \(Y':=\mathbf{R}\!\operatorname{Hom}_{A^{\mathrm{op}}}(\mathrm{D}A,A)\), \[Y'\otimes^{\mathbb{L}}_{A}\mathrm{D}A\;\cong\;\mathbf{R}\!\operatorname{Hom}_{A^{\mathrm{op}}}(\mathrm{D}A,\mathrm{D}A)\;\cong\; \mathrm{D}(A\otimes^{\mathbb{L}}_{A}\mathrm{D}A)\;\cong\;A.\] By associativity and unitality of \(\otimes^{\mathbb{L}}_{A}\), \[Y\;\cong\;A\otimes^{\mathbb{L}}_{A}Y\;\cong\;(Y'\otimes^{\mathbb{L}}_{A}\mathrm{D}A)\otimes^{\mathbb{L}}_{A}Y\;\cong\; Y'\otimes^{\mathbb{L}}_{A}(\mathrm{D}A\otimes^{\mathbb{L}}_{A}Y)\;\cong\;Y'\otimes^{\mathbb{L}}_{A}A\;\cong\;Y',\] whence \(Y\otimes^{\mathbb{L}}_{A}\mathrm{D}A\cong Y'\otimes^{\mathbb{L}}_{A}\mathrm{D}A\cong A\). Thus \(\mathrm{D}A\) is a two-sided tilting complex with inverse \(Y\cong Y'\), and then so are its shifts, with \((\mathrm{D}A[-1])^{-1}=Y[1]\). ◻
Remark 2. The lemma is classical: for a Gorenstein algebra — in particular one of finite global dimension — the bimodule \(\mathrm{D}A\) generates a copy of \(\mathbb{Z}\) inside the derived Picard group \(\mathrm{DPic}(A)\), and the associated standard self-equivalence of \(\operatorname{per}A\) is the derived Nakayama functor \(\nu=-\otimes^{\mathbb{L}}_{A}\mathrm{D}A\) of Happel [4], [5]; see [31]–[34]. We have included the short proof in order to keep the bimodule conventions of [1] explicit.
Definition 1. Let \(\operatorname{gl.dim}A<\infty\). The Serre bimodule of \(A\) is \(\omega_A:=\mathrm{D}A\), and the Auslander–Reiten bimodule of \(A\) is \[\tau_A:=\mathrm{D}A[-1],\] both viewed as (invertible) morphisms \(A\to A\) in \(\mathbf{A}_{k}\). The associated standard equivalences of \(\operatorname{per}A\) are the Serre functor \(\nu=-\otimes^{\mathbb{L}}_{A}\omega_A\) and the derived Auslander–Reiten translation \(\tau=-\otimes^{\mathbb{L}}_{A}\tau_A\) [5], and \(K_0(-\otimes^{\mathbb{L}}_{A}\tau_A)=\Phi_A\) by Lemma 1.
The key property of \(\tau_A\) is that it commutes with every derived equivalence. This is the bimodule-level form of the uniqueness of Serre functors.
Lemma 4. Let \(A\), \(B\) be finite-dimensional algebras of finite global dimension and let \(X\colon A\to B\) be a two-sided tilting complex. Then there are isomorphisms in the derived category of \(A\)-\(B\)-bimodules \[\omega_A\otimes^{\mathbb{L}}_{A}X\;\cong\;X\otimes^{\mathbb{L}}_{B}\omega_B, \qquad \tau_A\otimes^{\mathbb{L}}_{A}X\;\cong\;X\otimes^{\mathbb{L}}_{B}\tau_B .\]
Proof. The second isomorphism follows from the first by shifting. For the first, we exhibit both sides as the \(k\)-dual of the inverse of \(X\). Choose by Lemma 2 a representative of \(X\) that is bounded with components finitely generated projective over \(B\) on the right; since \(X\) is also perfect over \(A\) on the left [31], fix similarly a representative adapted to the left when needed.
(a) The map \[\alpha_X\colon\; X\otimes^{\mathbb{L}}_{B}\mathrm{D}B\longrightarrow \mathrm{D}\bigl(\mathbf{R}\!\operatorname{Hom}_B(X,B)\bigr), \qquad x\otimes\varphi\longmapsto\bigl(f\mapsto\varphi(f(x))\bigr),\] is well defined: it is balanced over \(B\) because \(\varphi(f(xb))=\varphi(f(x)b)=(b\cdot\varphi)(f(x))\), and it is a morphism of complexes of \(A\)-\(B\)-bimodules for the standard structures. It is an isomorphism when \(X=B\), both sides are triangulated functors of the underlying complex of right \(B\)-modules, and quasi-isomorphisms are detected after forgetting the left \(A\)-structure; hence \(\alpha_X\) is an isomorphism for \(X\) perfect over \(B\).
(b) Symmetrically, the map \[\beta_X\colon\;\mathrm{D}A\otimes^{\mathbb{L}}_{A}X\longrightarrow \mathrm{D}\bigl(\mathbf{R}\!\operatorname{Hom}_{A}({}_AX,{}_AA)\bigr), \qquad \varphi\otimes x\longmapsto\bigl(f\mapsto\varphi(f(x))\bigr),\] is balanced over \(A\) because \(f\) is left \(A\)-linear, \(\varphi a\otimes x\) and \(\varphi\otimes ax\) having equal images, and is an isomorphism of complexes of \(A\)-\(B\)-bimodules for \(X\) perfect over \(A\) on the left, by the same dévissage applied on the left.
(c) By 2 the complexes \(\mathbf{R}\!\operatorname{Hom}_B(X,B)\) and \(\mathbf{R}\!\operatorname{Hom}_A({}_AX,{}_AA)\) are isomorphic as complexes of \(B\)-\(A\)-bimodules, both computing the inverse \(X^{-1}\). Applying \(\mathrm{D}\) and combining with (a) and (b): \[\omega_A\otimes^{\mathbb{L}}_{A}X\;\xrightarrow{\;\beta_X\;}\;\mathrm{D}\bigl(X^{-1}\bigr) \;\xleftarrow{\;\alpha_X\;}\;X\otimes^{\mathbb{L}}_{B}\omega_B . \qedhere\] ◻
Definition 2. Let \(A\) be a finite-dimensional \(k\)-algebra with \(\operatorname{gl.dim}A<\infty\). The Coxeter automorphism of the Tamarkin–Tsygan calculus of \(A\) is \[\sigma_A\;:=\;\mathbb{H}(\tau_A)\;=\;\mathbb{H}\bigl(\mathrm{D}A[-1]\bigr) \;\colon\;\mathbb{H}(A)\longrightarrow\mathbb{H}(A).\] We write \(\sigma_\bullet\) and \(\sigma^\bullet\) for its homological and cohomological components.
Theorem 3. Let \(k\) be a field and \(A\) a finite-dimensional \(k\)-algebra with \(\operatorname{gl.dim}A<\infty\).
\(\sigma_A\) is an automorphism of \(\mathbb{H}(A)\) in \(\mathbf{TT\text{-}calc}\), with inverse \(\mathbb{H}(\tau_A^{-1})\). In particular \(\sigma^\bullet\) is an automorphism of the Gerstenhaber algebra \(\mathit{HH}^\bullet(A)\), the pair \((\sigma_\bullet,\sigma^\bullet)\) satisfies the projection formula \(\sigma_\bullet(z\cap\eta)=\sigma_\bullet z\cap\sigma^\bullet\eta\), and \(\sigma_\bullet\) commutes with the Connes differential \(B\).
The map \(\mathbb{Z}\to\mathop{\mathrm{Aut}}_{\mathbf{TT\text{-}calc}}(\mathbb{H}(A))\), \(m\mapsto \sigma_A^{\,m}=\mathbb{H}\bigl(\tau_A^{\otimes^{\mathbb{L}}_A m}\bigr)\), is a group homomorphism.
For every two-sided tilting complex \(X\colon A\to B\) one has \(\mathbb{H}(X)\circ\sigma_A=\sigma_B\circ\mathbb{H}(X)\) in \(\mathbf{TT\text{-}calc}\). Consequently the isomorphism class of the pair \((\mathbb{H}(A),\sigma_A)\) in the category of Tamarkin–Tsygan calculi equipped with an automorphism is invariant under derived equivalence.
Proof. (1) Lemma 3 shows that \(\tau_A\) is invertible in \(\mathbf{A}_{k}\), with inverse given by \(\mathbf{R}\!\operatorname{Hom}_A(\mathrm{D}A,A)[1]\), and by Theorem 1 the functor \(\mathbb{H}\) sends the relations \(\tau_A^{-1}\circ\tau_A=\operatorname{id}_A\) and \(\tau_A\circ\tau_A^{-1}=\operatorname{id}_A\) of \(\mathbf{A}_{k}\) to the corresponding relations in \(\mathbf{TT\text{-}calc}\). Every morphism of \(\mathbf{TT\text{-}calc}\) commutes by definition with \(\cup\), \([-,-]\), \(\cap\) and \(B\); an isomorphism whose cohomological component preserves \(\cup\) and \([-,-]\) is an automorphism of the Gerstenhaber algebra.
(2) Functoriality of \(\mathbb{H}\) and associativity of \(\otimes^{\mathbb{L}}\) in \(\mathbf{A}_{k}\).
(3) In \(\mathbf{A}_{k}\) the two composites \(A\to B\) in question are represented by the bimodule complexes \(X\otimes^{\mathbb{L}}_{B}\tau_B\) and \(\tau_A\otimes^{\mathbb{L}}_{A}X\), which are isomorphic by Lemma 4; since morphisms of \(\mathbf{A}_{k}\) are taken up to isomorphism in the derived category of bimodules, the two composites are equal in \(\mathbf{A}_{k}\), and we apply \(\mathbb{H}\): \[\sigma_B\circ\mathbb{H}(X)=\mathbb{H}(X\otimes^{\mathbb{L}}_{B}\tau_B)=\mathbb{H}(\tau_A\otimes^{\mathbb{L}}_{A}X) =\mathbb{H}(X)\circ\sigma_A .\] If \(F\colon\mathcal{D}(A)\to\mathcal{D}(B)\) is any equivalence, by Rickard’s theorem we may choose a two-sided tilting complex \(X\) [31]; then \(\mathbb{H}(X)\) is an isomorphism \(\mathbb{H}(A)\to\mathbb{H}(B)\) in \(\mathbf{TT\text{-}calc}\) commuting with the \(\sigma\)’s, which is the claimed invariance. ◻
Remark 4. Theorem 1 restricts to an action of the derived Picard group \(\mathrm{DPic}(A)\) of [33], [34] on the object \(\mathbb{H}(A)\), i.e.a group homomorphism from \(\mathrm{DPic}(A)\) to \(\mathop{\mathrm{Aut}}_{\mathbf{TT\text{-}calc}}(\mathbb{H}(A))\). The automorphism \(\sigma_A\) is the image of the class of \(\tau_A\), which by Lemma 4 is fixed by the isomorphisms \(\mathrm{DPic}(A)\cong\mathrm{DPic}(B)\) induced by derived equivalences. Theorem 3(3) is the shadow of this naturality.
Remark 5. The thesis [1] constructs three isomorphic models \(\mathbb{H}\), \(\widehat{\mathbb{H}}\), \(\widetilde{\mathbb{H}}\) of the calculus and compatible comparison isomorphisms. All statements about \((\mathbb{H}(A),\sigma_A)\) in this paper are invariant under these comparisons.
Remark 6. No elementarity hypothesis enters Theorem 3: it holds for every finite-dimensional algebra of finite global dimension over an arbitrary field.
In this section \(A\) is a finite-dimensional elementary \(k\)-algebra of finite global dimension with \(n\) simple modules; we keep the notation of §2.1.
Proposition 7. \(\mathit{HH}_i(A)=0\) for all \(i\geq 1\), and the classes \(\bar e_1,\dots,\bar e_n\) form a \(k\)-basis of \(\mathit{HH}_0(A)=A/[A,A]\).
Proof. Since \(A/\operatorname{rad}A\cong k^n\) is commutative, \([A,A]\subseteq\operatorname{rad}A\); since \(\operatorname{gl.dim}A<\infty\), Lenzing’s theorem gives \(\operatorname{rad}A\subseteq[A,A]\), see [35]. Hence \(A/[A,A]=A/\operatorname{rad}A=k^n\) with basis the \(\bar e_i\). The vanishing of the higher Hochschild homology is [10]; see also [15] for this formulation. ◻
For a finitely generated projective right \(A\)-module \(P\), written \(P=\operatorname{im}(e)\) for an idempotent matrix \(e=(e_{st})\in M_r(A)\), the Hattori–Stallings trace of an endomorphism \(f\) of \(P\), represented by a matrix \((f_{st})\in M_r(A)\) with \(f=efe\), is \[\operatorname{tr}_P(f)\;:=\;\sum_{s}\overline{f_{ss}}\;\in\;\mathit{HH}_0(A)=A/[A,A];\] it is independent of all choices and additive [36], [37]. The Euler class of \(M\in\operatorname{per}A\) is \[\operatorname{ch}(M)\;:=\;\sum_j(-1)^j\,\operatorname{tr}_{P^j}(\operatorname{id})\;\in\;\mathit{HH}_0(A)\] for any bounded complex \(P^\bullet\) of finitely generated projectives isomorphic to \(M\) in \(\mathcal{D}(A)\); equivalently \(\operatorname{ch}\) is the composition of the canonical isomorphism between \(K_0(\operatorname{per}A)\) and \(K_0(\operatorname{proj}A)\) with the trace of the identity. It is additive on triangles, satisfies \(\operatorname{ch}(M[1])=-\operatorname{ch}(M)\), and \(\operatorname{ch}(P_i)=\bar e_i\).
Lemma 5. For every \(M\in\operatorname{per}A\) one has \(\operatorname{ch}(M)=\sum_i a_i\,\bar e_i\) with \(a=C_A^{-1}\,\underline{\dim}\,M\in\mathbb{Z}^n\). In particular \(\operatorname{ch}\otimes k\colon K_0(\operatorname{per}A)\otimes_\mathbb{Z}k\to\mathit{HH}_0(A)\) is an isomorphism sending \([P_i]\otimes 1\) to \(\bar e_i\).
Proof. Both \(\operatorname{ch}\) and \(a(M):=C_A^{-1}\underline{\dim}\,M\) are additive on triangles, and they agree on the generators \(P_i\) of \(K_0(\operatorname{per}A)\) by 1 ; the classes \([P_i]\) form a \(\mathbb{Z}\)-basis of \(K_0(\operatorname{per}A)\). ◻
Proposition 8. Let \(X\colon A\to B\) be a morphism of \(\mathbf{A}_{k}\) between elementary algebras of finite global dimension, represented by a bounded complex of \(A\)-\(B\)-bimodules whose components \(X^j\) are finitely generated projective right \(B\)-modules. Then:
the degree-zero homological component of \(\mathbb{H}(X)\) is the tensor-trace* map \[\mathit{HH}_0(X)\colon A/[A,A]\longrightarrow B/[B,B], \qquad \bar a\longmapsto\sum_j(-1)^j\,\operatorname{tr}_{X^j}(\lambda_a),\] where \(\lambda_a\) denotes left multiplication by \(a\) on \(X^j\);*
for all \(M\in\operatorname{per}A\), \[\mathbb{H}(X)_0\bigl(\operatorname{ch}(M)\bigr)\;=\;\operatorname{ch}\bigl(M\otimes^{\mathbb{L}}_{A}X\bigr).\]
Proof. (1) By Theorem 1 the homological components of \(\mathbb{H}(X)\) coincide with the morphisms induced by Keller’s cyclic functor [10], which in weight zero is the classical map induced on Hochschild complexes by the bimodule \(X\): the composite of the inclusion of \(A\) into the endomorphism dg algebra of \(X\) over \(B\) with the generalized trace. Its zeroth homology is the displayed formula; see [15], and [10], [38] for the same map in the dg framework.
(2) Both sides are additive in \([M]\in K_0(\operatorname{per}A)\) — the left side because \(\operatorname{ch}\) is and \(\mathbb{H}(X)_0\) is linear; the right side because \(-\otimes^{\mathbb{L}}_{A}X\) is triangulated and \(\operatorname{ch}_B\) is additive on triangles — so it suffices to treat \(M=P_i=e_iA\). Then \(M\otimes^{\mathbb{L}}_{A}X\cong e_iX\), a bounded complex with components \(e_iX^j\) finitely generated projective over \(B\), and \[\begin{align} \operatorname{ch}(e_iX)&=\sum_j(-1)^j\operatorname{tr}_{e_iX^j}(\operatorname{id}) =\sum_j(-1)^j\operatorname{tr}_{X^j}(\lambda_{e_i})\\ &=\mathit{HH}_0(X)(\bar e_i)=\mathbb{H}(X)_0\bigl(\operatorname{ch}(P_i)\bigr), \end{align}\] where the middle equality is the standard property of the Hattori–Stallings trace: for an idempotent endomorphism \(\pi\) of a finitely generated projective module \(Q\), \(\operatorname{tr}_{\operatorname{im}\pi}(\operatorname{id})=\operatorname{tr}_Q(\pi)\). ◻
Remark 9. Part (1) is the single point of this paper where the normalization of the maps \(\mathit{HH}_\bullet(X)\) of [1] is compared with the classical tensor-trace; the comparison is the content of the main theorem of [1] — the identification of \(\mathit{HH}_\bullet(X)\) with the map induced by Keller’s cyclic functor — together with the construction of [10], and is independent of the choice among the three models of Remark 5.
Theorem 10. Let \(A\) be a finite-dimensional elementary \(k\)-algebra with \(\operatorname{gl.dim}A<\infty\). In the basis \(\bar e_1,\dots,\bar e_n\) of \(\mathit{HH}_0(A)=\mathit{HH}_\bullet(A)\) the matrix of the homological component \(\sigma_\bullet\) of the Coxeter automorphism is \[-\,C_A^{-1}C_A^{\mathrm{T}} .\] The Euler class intertwines the Coxeter transformation with \(\sigma_\bullet\), that is, the square \[\begin{array}{ccc} K_0(\operatorname{per}A)\otimes_\mathbb{Z}k& \xrightarrow{\;\Phi_A\otimes k\;} & K_0(\operatorname{per}A)\otimes_\mathbb{Z}k\\[3pt] \operatorname{ch}\otimes k\Big\downarrow & & \Big\downarrow\operatorname{ch}\otimes k\\[3pt] \mathit{HH}_0(A) & \xrightarrow{\;\;\sigma_\bullet\;\;} & \mathit{HH}_0(A) \end{array}\] commutes, and \[\det\bigl(x\cdot\operatorname{id}-\sigma_\bullet\bigr)\;=\; \det\bigl(xI_n+C_A^{-1}C_A^{\mathrm{T}}\bigr)\;=\;\phi_A(x) \quad\text{in }k[x].\]
Proof. By Proposition 7, \(\mathit{HH}_\bullet(A)=\mathit{HH}_0(A)\) with basis the \(\bar e_i=\operatorname{ch}(P_i)\). By Proposition 8(2) applied to the morphism \(\tau_A=\mathrm{D}A[-1]\) of \(\mathbf{A}_{k}\) (a bounded right-projective representative exists by Lemmas 2 and 3), \[\sigma_\bullet(\bar e_i) =\mathbb{H}(\tau_A)_0\bigl(\operatorname{ch}(P_i)\bigr) =\operatorname{ch}\bigl(P_i\otimes^{\mathbb{L}}_{A}\mathrm{D}A[-1]\bigr) =-\,\operatorname{ch}\bigl(e_i\mathrm{D}A\bigr) =-\,\operatorname{ch}(I_i),\] using the isomorphism \(e_i\mathrm{D}A\cong\mathrm{D}(Ae_i)=I_i\) of right \(A\)-modules from the proof of Lemma 1. By Lemma 5 and 1 , \(\operatorname{ch}(I_i)=\sum_j(C_A^{-1}C_A^{\mathrm{T}})_{ji}\,\bar e_j\), which is the asserted matrix. The commuting square restates this computation through Lemma 1, since \(\Phi_A\) has the same matrix \(-C_A^{-1}C_A^{\mathrm{T}}\) in the basis \([P_i]\) and \(\operatorname{ch}\otimes k\) sends \([P_i]\) to \(\bar e_i\). The characteristic polynomial is then \(\phi_A\) by Lemma 1, read in \(k[x]\). ◻
Corollary 1. The Coxeter polynomial of a finite-dimensional elementary algebra of finite global dimension is a derived invariant.
Proof. If \(A\) and \(B\) are derived equivalent, Theorem 3(3) provides an isomorphism of pairs \((\mathit{HH}_\bullet(A),\sigma_\bullet)\cong(\mathit{HH}_\bullet(B),\sigma_\bullet)\), so the characteristic polynomials agree in \(k[x]\); over a field of characteristic zero this already returns equality in \(\mathbb{Z}[x]\) by integrality of both polynomials. In general, the matrices \(-C^{-1}C^{\mathrm{T}}\) are even conjugate over \(\mathbb{Z}\), since the isomorphism of Theorem 3(3) restricts, by Proposition 8(2), to an isomorphism of the lattices spanned by the \(\bar e_i\) intertwining the \(\sigma_\bullet\)’s. This recovers the classical statement of [6], [7]. ◻
Example 1. Let \(k\) be an algebraically closed field, \(A=kQ_A\) the path algebra of the \(\mathbb{D}_4\)-quiver with three arrows into the central vertex, and \(B=kQ_B\) the path algebra of the linearly oriented \(\mathbb{A}_4\)-quiver, as in §1. Then:
\(\mathbb{H}(A)\cong\mathbb{H}(B)\) in \(\mathbf{TT\text{-}calc}\);
there exists no isomorphism \(\mathbb{H}(A)\to\mathbb{H}(B)\) in \(\mathbf{TT\text{-}calc}\) commuting with \(\sigma_A\) and \(\sigma_B\); in particular \((\mathbb{H}(A),\sigma_A)\not\cong (\mathbb{H}(B),\sigma_B)\) although the underlying calculi are isomorphic.
Proof. (1) Both quivers are trees, so by [39], [40] the Hochschild homology and cohomology of \(A\) and \(B\) vanish in positive degrees, with \(\mathit{HH}_0\cong k^4\) and \(\mathit{HH}^0\cong k\); all components of the calculus other than the unit cup product and the degree-\((0,0)\) cap product vanish, and the two calculi are isomorphic; this is the closing remark of [1].
(2) Ordering the vertices as in §1, the Cartan matrices and the matrices \(S=-C^{-1}C^{\mathrm{T}}\) of the \(\sigma_\bullet\)’s are \[C_A=\begin{pmatrix}1&0&0&0\\1&1&0&0\\1&0&1&0\\1&0&0&1\end{pmatrix}, \qquad C_B=\begin{pmatrix}1&1&1&1\\0&1&1&1\\0&0&1&1\\0&0&0&1\end{pmatrix},\] with characteristic polynomials \[\begin{align} \det(x\cdot\operatorname{id}-\sigma_{A,\bullet})&=\phi_A(x)=x^4+x^3+x+1,\\ \det(x\cdot\operatorname{id}-\sigma_{B,\bullet})&=\phi_B(x)=x^4+x^3+x^2+x+1 \end{align}\] by Theorem 10. An isomorphism of pairs would conjugate \(\sigma_{A,\bullet}\) into \(\sigma_{B,\bullet}\) and force the equality of their characteristic polynomials in \(k[x]\); but \(\phi_B-\phi_A=x^2\neq 0\) in \(k[x]\) for every field \(k\). ◻
Remark 11. Example 1 shows that the enriched calculus \((\mathbb{H}(-),\sigma)\) is a strictly finer derived invariant than the Tamarkin–Tsygan calculus \(\mathbb{H}(-)\). It is strictly finer than the Coxeter polynomial as well: it determines \(\phi\) by Theorem 10, while already the underlying calculus separates many algebras with equal Coxeter polynomial — by Happel’s formula such algebras have equal Euler characteristic of Hochschild cohomology, but any two of them with non-isomorphic graded algebras \(\mathit{HH}^\bullet\) are separated by \(\mathbb{H}\); see the discussions in [7].
Remark 12. The derived inequivalence of \(kQ_{\mathbb{A}_4}\) and \(kQ_{\mathbb{D}_4}\) is itself classical and needs none of this: for a Dynkin quiver the bounded derived category is described completely by Happel [4], [5], with Auslander–Reiten quiver \(\mathbb{Z}\Delta\), and two Dynkin path algebras are derived equivalent if and only if their underlying diagrams agree — here the numbers of indecomposable objects, \(10\) and \(12\), already differ. The content of Example 1 is not the inequivalence of the two algebras but the comparison of two invariants: the Tamarkin–Tsygan calculus does not separate them, the enriched calculus does, and the gap between the two is measured exactly by the Coxeter polynomial.
Remark 13. The polynomials above factor as \(\phi_A=\Phi_2(x)^2\,\Phi_6(x)\) and \(\phi_B=\Phi_5(x)\) in terms of cyclotomic polynomials, reflecting the exponents \(1,3,3,5\) and Coxeter number \(6\) of \(\mathbb{D}_4\), and \(1,2,3,4\) and Coxeter number \(5\) of \(\mathbb{A}_4\). Amusingly, the same pair of algebras appears in [26] as the minimal example showing that the Coxeter polynomial of an insertion is not determined by the Coxeter polynomial of the inserted poset.
Remark 14 (Beyond global dimension one). A natural question is what happens in global dimension \(\geq2\). Theorems 3 and 16 require only \(\operatorname{gl.dim}A<\infty\), and Theorem 10 elementarity in addition; nothing in the theory is special to the hereditary case. Moreover, genuinely new spectra appear in higher global dimension, and the enriched calculus detects this internally. By Happel’s vanishing theorem [41], a connected hereditary algebra has \(\mathit{HH}^{i}=0\) for \(i\geq2\) and \(\mathit{HH}^0=k\), so Corollary 2 forces \[-\operatorname{tr}\bigl(\sigma_\bullet\bigr) \;=\;\sum_{i\geq0}(-1)^i\dim_k\mathit{HH}^i(A) \;=\;1-\dim_k\mathit{HH}^1(A)\;\leq\;1\] for every connected algebra derived equivalent to a hereditary one. The Beilinson algebra \(B_2\) of the projective plane (Example 4), which has global dimension two, has Coxeter polynomial \((x+1)^3\) and hence \(\operatorname{tr}(\sigma_\bullet)=-3\): the inequality fails, so \(B_2\) is not derived equivalent to any hereditary algebra, and the enriched calculus itself witnesses the obstruction.
The refinement of Example 1 has a precise limit, which we now exhibit. For path algebras of trees the entire calculus degenerates, so the enriched calculus retains exactly one piece of information: the conjugacy class of the Coxeter matrix.
Example 2. Let \(k\) be a field of characteristic zero, let \(T\) be the spider with one arm of length three and four arms of length one, and let \(T'\) be the double star consisting of two adjacent vertices carrying three leaves each — the two trees on eight vertices pictured below, taken with any orientations: \[Q_T:\quad \begin{array}{ccccc} 2 & & 3 & & \\[-2pt] & \nwarrow & \uparrow & & \\[-2pt] & & 1 & \longrightarrow & 6\longrightarrow 7\longrightarrow 8\\[-2pt] & \swarrow & \downarrow & & \\[-2pt] 4 & & 5 & & \end{array}\] \[Q_{T'}:\quad \begin{array}{ccccccc} 3 & & 4 & & 6 & & 7\\[-2pt] & \nwarrow & \uparrow & & \uparrow & \nearrow & \\[-2pt] & & 1 & \longrightarrow & 2 & & \\[-2pt] & \swarrow & & & & \searrow & \\[-2pt] 5 & & & & & & 8 \end{array}\] Then \((\mathbb{H}(kT),\sigma)\cong(\mathbb{H}(kT'),\sigma')\) as calculi with automorphism, but \(kT\) and \(kT'\) are not derived equivalent. Hence the enriched calculus is not a complete derived invariant.
Proof. A direct computation with Lemma 1 — independent of the chosen orientations, since all orientations of a tree are iterated reflections of one another [4] and Theorem 3 applies — gives \[\phi_T(x)\;=\;\phi_{T'}(x) \;=\;(x+1)^4\,\bigl(x^4-3x^3+x^2-3x+1\bigr),\] and moreover \(\operatorname{rank}(\Phi+\operatorname{id})=4\) for both Coxeter matrices. The latter says that both matrices are semisimple at the eigenvalue \(-1\), and the quartic factor is irreducible over \(\mathbb{Q}\); consequently both matrices have elementary divisors \((x+1),(x+1),(x+1),(x+1)\) and \(x^4-3x^3+x^2-3x+1\), hence are conjugate over \(\mathbb{Q}\) and therefore over \(k\).
Since \(T\) and \(T'\) are trees, Hochschild homology and cohomology vanish in positive degrees, with \(\mathit{HH}_0\cong k^8\) and \(\mathit{HH}^0=k\), see [39]–[41]; every operation of the calculus other than the unit constraints vanishes, and \(\sigma^\bullet=\operatorname{id}\) on both sides by Theorem 16. An isomorphism of enriched calculi is therefore exactly a linear isomorphism \(\mathit{HH}_0(kT)\to\mathit{HH}_0(kT')\) conjugating \(\sigma_\bullet\) into \(\sigma'_\bullet\), together with the identity of \(k\) in cohomology; by Theorem 10 and the conjugacy just established, such an isomorphism exists.
Finally, derived equivalent connected hereditary algebras have isomorphic underlying graphs: the Auslander–Reiten quiver of \(\mathcal{D}(kQ)\) contains a unique component of the form \(\mathbb{Z}\Delta\) with \(\Delta\) a finite quiver, namely the transjective component, and the underlying graph of \(\Delta\) is that of \(Q\) [4], [5]. The trees \(T\) and \(T'\) are not isomorphic — their degree sequences are \((5,2,2,1,1,1,1,1)\) and \((4,4,1,1,1,1,1,1)\) — so \(kT\) and \(kT'\) are not derived equivalent. ◻
Remark 15. The pair above is the smallest one: a machine search over all trees with at most eight vertices shows that \(T\) and \(T'\) form the unique pair of non-isomorphic trees on \(\leq8\) vertices with equal Coxeter polynomial. This is no accident: for a tree, the Coxeter spectrum and the spectrum of the adjacency matrix determine one another, so Coxeter-cospectral trees are exactly cospectral trees, and \(T\), \(T'\) are the classical smallest cospectral pair; by a theorem of Schwenk [42], almost every tree admits a cospectral mate, so such pairs abound. The failure of completeness thus occurs precisely where the underlying calculus is most degenerate, and it delimits what the enrichment can see: on trees, \((\mathbb{H},\sigma)\) is exactly the conjugacy class of the Coxeter transformation over \(k\). Whether the integral refinements of Question 25 — the lattice \(\operatorname{ch}(K_0(\operatorname{per}A))\) and the Shklyarov pairing — or the twisted theory of Question 23 separate such pairs is, in our view, the right next question.
Corollary 2. Let \(A\) be elementary with \(\operatorname{gl.dim}A<\infty\) over an algebraically closed field \(k\). Then for all \(m\geq 1\) \[\operatorname{tr}\bigl(\sigma_\bullet^{\,m}\,\big|\,\mathit{HH}_\bullet(A)\bigr) =\operatorname{tr}\bigl(\Phi_A^{\,m}\bigr),\] and for \(m=1\), combining with Happel’s theorem [13], \[\sum_{i\geq 0}(-1)^i\dim_k\mathit{HH}^i(A) \;=\;-\,\operatorname{tr}\bigl(\sigma_\bullet\,\big|\,\mathit{HH}_\bullet(A)\bigr).\] If moreover \(\operatorname{char}k=0\), the Coxeter polynomial \(\phi_A\in\mathbb{Z}[x]\) is determined by the pair \((\mathbb{H}(A),\sigma_A)\) through the traces of the powers of \(\sigma_\bullet\) and Newton’s identities.
Proof. Immediate from Theorem 10, since \(\mathit{HH}_\bullet(A)\) is concentrated in (even) degree zero and conjugate matrices have equal power traces; Happel’s theorem states \[\textstyle\sum_i(-1)^i\dim\mathit{HH}^i(A)=-\operatorname{tr}\Phi_A\] in the normalization of Lemma 1, cf. [15]. ◻
Following [14], call \(A\) fractionally Calabi–Yau of dimension \(p/q\) if \[(\mathrm{D}A)^{\otimes^{\mathbb{L}}_A q}\;\cong\;A[p] \qquad\text{in }\mathcal{D}(A^e)\] for some integers \(p\) and \(q\geq 1\).
Corollary 3. If \(A\) is elementary of finite global dimension and fractionally Calabi–Yau of dimension \(p/q\), then \[\sigma_\bullet^{\,q}\;=\;(-1)^{p-q}\,\operatorname{id}_{\mathit{HH}_\bullet(A)} .\] Consequently \(\Phi_A^{\,q}=(-1)^{p-q}\operatorname{id}\), the Coxeter transformation is periodic, and \(\phi_A\) is a product of cyclotomic polynomials. The same conclusion on quasi-unipotence holds for the Serre cyclotomic algebras of [23].
Proof. \(\tau_A^{\otimes^{\mathbb{L}}_A q}=(\mathrm{D}A)^{\otimes^{\mathbb{L}}_A q}[-q]\cong A[p-q]\), so \(\sigma^q=\mathbb{H}(A[p-q])\), and by Proposition 8(2), \(\mathbb{H}(A[m])_0(\bar e_i)=\operatorname{ch}(P_i[m])=(-1)^m\bar e_i\). The statements about \(\Phi_A\) follow through the commuting square of Theorem 10, and cyclotomicity by Kronecker’s theorem; this recovers part (a) of [6]. ◻
Theorem 10 computes the homological component of \(\sigma_A\) and shows that it carries the full Coxeter data. In this section we prove that the cohomological component carries none: \(\sigma^\bullet_A\) is the identity. Far from being a defect, this dichotomy is the structural explanation of the principle visible throughout the paper, that Coxeter data is homological. The key lemma holds for every finite-dimensional algebra, with no hypothesis on the global dimension, and has a consequence outside the scope of the rest of the paper: it recovers, by a short conceptual argument, the recent theorem of Suárez-Álvarez [12] that the Nakayama automorphism of a Frobenius algebra acts trivially on Hochschild cohomology.
Let \(A\) be a finite-dimensional \(k\)-algebra. For \(M\in\mathcal{D}(A^e)\) and a class \(\eta\in\mathit{HH}^m(A)=\operatorname{Hom}_{\mathcal{D}(A^e)}(A,A[m])\) define \[\ell^M_\eta\colon M\xrightarrow{\;\cong\;}A\otimes^{\mathbb{L}}_{A}M \xrightarrow{\;\eta\,\otimes^{\mathbb{L}}_A\,\operatorname{id}_M\;}A[m]\otimes^{\mathbb{L}}_{A}M \xrightarrow{\;\cong\;}M[m],\] \[r^M_\eta\colon M\xrightarrow{\;\cong\;}M\otimes^{\mathbb{L}}_{A}A \xrightarrow{\;\operatorname{id}_M\otimes^{\mathbb{L}}_A\,\eta\;}M\otimes^{\mathbb{L}}_{A}A[m] \xrightarrow{\;\cong\;}M[m],\] where the unlabelled arrows are the canonical unit isomorphisms. Both are natural in \(M\); both commute with shifts, since \(-\otimes^{\mathbb{L}}_{A}M\) and \(M\otimes^{\mathbb{L}}_{A}-\) are triangulated functors, so that \(\ell^{M[1]}_\eta=\Sigma\ell^{M}_\eta\) and \(r^{M[1]}_\eta=\Sigma r^{M}_\eta\); and on the regular bimodule both reduce to \(\eta\) itself: \[\label{eq:regularactions} \ell^{A}_\eta\;=\;r^{A}_\eta\;=\;\eta .\tag{3}\] If \(X\in\mathcal{D}(A^e)\) is invertible, the map \(\theta\mapsto r^X_\theta\) is injective, since applying the equivalence \(X^{-1}\otimes^{\mathbb{L}}_{A}(-)\) recovers the action of \(\theta\) on \(A\); the conjugation by \(X\) is the graded linear automorphism \(\gamma_X\) of \(\mathit{HH}^\bullet(A)\) characterized by \[\label{eq:conjchar} r^{X}_{\gamma_X(\eta)}\;=\;\ell^{X}_{\eta} \qquad\text{in } \operatorname{Hom}_{\mathcal{D}(A^e)}(X,X[m]).\tag{4}\] One has \(\gamma_{X\otimes^{\mathbb{L}}_A Y}=\gamma_Y\circ\gamma_X\) (both sides act through the middle factor of \(X\otimes^{\mathbb{L}}_A Y\)), and \(\gamma_{A[\pm1]}=\operatorname{id}\) by the shift-compatibility just noted. The cohomological component of the transport \(\mathbb{H}(X)\) along an invertible bimodule is exactly this conjugation (Theorem 1; [8], [9]), so that \[\label{eq:sigmaisconj} \sigma^\bullet_A \;=\;\gamma_{\tau_A} \;=\;\gamma_{A[-1]}\circ\gamma_{\mathrm{D}A} \;=\;\gamma_{\mathrm{D}A}.\tag{5}\]
Lemma 6. Let \(A\) be any finite-dimensional \(k\)-algebra. For every \(m\in\mathbb{Z}\) and every \(\eta\in\operatorname{Hom}_{\mathcal{D}(A^e)}(A,A[m])\), \[\ell^{\mathrm{D}A}_{\eta}\;=\;r^{\mathrm{D}A}_{\eta} \qquad\text{in }\operatorname{Hom}_{\mathcal{D}(A^e)}\bigl(\mathrm{D}A,\mathrm{D}A[m]\bigr).\]
Proof. Recall from the proof of Lemma 4, applied with \(B=A\), the morphisms \[\alpha_X\colon X\otimes^{\mathbb{L}}_{A}\mathrm{D}A\longrightarrow\mathrm{D}\bigl(\mathbf{R}\!\operatorname{Hom}_A(X,A)\bigr),\] \[\beta_X\colon \mathrm{D}A\otimes^{\mathbb{L}}_{A}X\longrightarrow \mathrm{D}\bigl(\mathbf{R}\!\operatorname{Hom}_{A^{\mathrm{op}}}(_AX,_AA)\bigr),\] defined by the evaluation formulas given there for arbitrary \(X\in\mathcal{D}(A^e)\); they are natural in \(X\), and invertible whenever \(X\) is perfect as a right (for \(\alpha\)), respectively left (for \(\beta\)), \(A\)-module, by the dévissage argument of that proof. We evaluate the two naturality squares at the morphism \(\eta\colon A\to A[m]\) of \(\mathcal{D}(A^e)\), whose source and target are perfect on both sides: \[\begin{align} \alpha_{A[m]}\circ\bigl(\eta\otimes^{\mathbb{L}}_A\operatorname{id}_{\mathrm{D}A}\bigr) &=\mathrm{D}\bigl(\eta^{*}_{r}\bigr)\circ\alpha_{A}, \tag{6}\\ \beta_{A[m]}\circ\bigl(\operatorname{id}_{\mathrm{D}A}\otimes^{\mathbb{L}}_A\eta\bigr) &=\mathrm{D}\bigl(\eta^{*}_{\ell}\bigr)\circ\beta_{A}, \tag{7} \end{align}\] where \(\eta^{*}_{r}=\mathbf{R}\!\operatorname{Hom}_A(\eta,A)\) and \(\eta^{*}_{\ell}=\mathbf{R}\!\operatorname{Hom}_{A^{\mathrm{op}}}(\eta,A)\) denote precomposition with \(\eta\). Identify \[\begin{gather} \mathbf{R}\!\operatorname{Hom}_A(A,A)\cong A\cong\mathbf{R}\!\operatorname{Hom}_{A^{\mathrm{op}}}(A,A),\\ \mathbf{R}\!\operatorname{Hom}_A(A[m],A)\cong A[-m]\cong\mathbf{R}\!\operatorname{Hom}_{A^{\mathrm{op}}}(A[m],A) \end{gather}\] by evaluation at the unit, computed on projective bimodule representatives. Under these identifications, first, \(\alpha_A\), \(\beta_A\), \(\alpha_{A[m]}\) and \(\beta_{A[m]}\) all become identity maps: for \(\alpha_A\) this is the direct computation \(\varphi\mapsto(f\mapsto\varphi(f(1)))\mapsto \bigl(b\mapsto\varphi(\lambda_b(1))\bigr)=\varphi\), where \(\lambda_b(x)=bx\) inverts the evaluation \(f\mapsto f(1)\) on right-module endomorphisms, and the other three cases follow by the mirror computation and shift-compatibility. Second, precomposition with \(\eta\) becomes the morphism \(\eta\) itself in both cases: a right-\(A\)-linear map \(f\) satisfies \(f(a)=f(1)\,a\), so on representatives \((f\circ\eta)(\xi)=f(1)\,\eta(\xi)\) and \(\eta^{*}_{r}\) corresponds to right multiplication by \(\eta\), that is to \(r^{A}_{\eta}[-m]=\eta[-m]\) by 3 ; a left-\(A\)-linear map \(g\) satisfies \(g(a)=a\,g(1)\), so \((g\circ\eta)(\xi)=\eta(\xi)\,g(1)\) and \(\eta^{*}_{\ell}\) corresponds to left multiplication by \(\eta\), that is to \(\ell^{A}_{\eta}[-m]=\eta[-m]\). Hence 6 and 7 identify \(\ell^{\mathrm{D}A}_{\eta}\) and \(r^{\mathrm{D}A}_{\eta}\), respectively, with one and the same morphism, namely the \(k\)-dual \(\mathrm{D}(\eta[-m])\colon\mathrm{D}A\to\mathrm{D}A[-m]\) of \(\eta\), transported through the identifications above. The claim follows. ◻
Theorem 16. Let \(A\) be a finite-dimensional \(k\)-algebra with \(\operatorname{gl.dim}A<\infty\). Then \[\sigma^\bullet_A\;=\;\operatorname{id}_{\mathit{HH}^\bullet(A)} .\] Consequently the enriched calculus \((\mathbb{H}(A),\sigma_A)\) amounts to the Tamarkin–Tsygan calculus together with the single operator \(\sigma_\bullet\) on Hochschild homology, which commutes with the Connes differential and with every cap operator: \[\sigma_\bullet(z\cap\eta)\;=\;\sigma_\bullet(z)\cap\eta, \qquad z\in\mathit{HH}_\bullet(A),\;\eta\in\mathit{HH}^\bullet(A).\]
Proof. By 5 we have \(\sigma^\bullet_A=\gamma_{\mathrm{D}A}\), and by 4 the class \(\gamma_{\mathrm{D}A}(\eta)\) is the unique \(\theta\) with \(r^{\mathrm{D}A}_{\theta}=\ell^{\mathrm{D}A}_{\eta}\). Lemma 6 says that \(\theta=\eta\) solves this equation. The displayed projection formula is Theorem 3(1) combined with \(\sigma^\bullet=\operatorname{id}\). ◻
Corollary 4. Let \(A\) be a finite-dimensional Frobenius algebra over \(k\), with Nakayama automorphism \(\nu\). Then the automorphism of \(\mathit{HH}^\bullet(A)\) induced by \(\nu\) is the identity.
Proof. For a Frobenius algebra one has \(\mathrm{D}A\cong{}_{1}A_{\nu}\) in \(\mathcal{D}(A^e)\), the regular bimodule with right action twisted by \(\nu\). This bimodule is invertible, with inverse \({}_{1}A_{\nu^{-1}}\), and conjugation by it is, up to the usual identifications, the automorphism of \(\mathit{HH}^\bullet(A)\) induced by \(\nu\) or by \(\nu^{-1}\), depending on conventions — immaterial here. By Lemma 6, which uses no hypothesis on the global dimension, together with 4 , this conjugation is the identity. ◻
Remark 17. Corollary 4 is the main theorem of the recent paper [12] of Suárez-Álvarez, who proves it by an explicit cochain-level analysis; in degree zero it is the classical fact that \(\nu\) fixes the center pointwise. The triviality on cohomology should be contrasted with the homological side, where the corresponding operator is in general nontrivial: for self-injective algebras this is the realm of the twisted Calabi–Yau and Batalin–Vilkovisky literature (see [43] and the references there), while for \(\operatorname{gl.dim}A<\infty\) it is the Coxeter transformation, by Theorem 10.
Remark 18 (Centrality in the derived Picard group). Lemma 4 with \(B=A\) says that the isomorphism class of \(\mathrm{D}A\) is central in the derived Picard group \(\mathrm{DPic}(A)\), for every finite-dimensional \(A\). Since the cohomological transport along an invertible bimodule is conjugation, Theorem 16 is an instance of the tautology that conjugation by a central element is trivial. Keller’s realization of all the groups \(\mathit{HH}^{m+1}(A)\) inside relative derived Picard groups over the graded algebras \(R=k[\varepsilon]/(\varepsilon^2)\), with \(\varepsilon\) placed in an arbitrary integer degree [9], turns this tautology into a second proof: the proof of Lemma 4 carries over to the \(R\)-relative setting (the duality \(\operatorname{Hom}_R(-,R)\) over the graded Frobenius algebra \(R\) enjoys the same biduality and dévissage properties), so \(\operatorname{Hom}_R(A\otimes R,R)\cong\mathrm{D}A\otimes R\) is central in each relative group and its conjugation action on \(\mathit{HH}^{m+1}(A)\) is trivial. We have preferred the direct argument of Lemma 6, which keeps the signs visible.
Remark 19 (Signs). The proof of Lemma 6 suppresses routine Koszul-sign bookkeeping; the verification must be carried out in a fixed convention. Three independent cross-checks exclude a hidden universal sign. First, an identity of the form \(\sigma^\bullet\eta=(-1)^{m}\eta\) on \(\mathit{HH}^m\) would, by compatibility with the Gerstenhaber bracket (Theorem 3), force \([\eta,\theta]=-[\eta,\theta]\) for all classes, hence the vanishing of the bracket whenever \(\operatorname{char}k\neq2\); this fails already for the Kronecker algebra, whose \(\mathit{HH}^1\) is the three-dimensional simple Lie algebra of outer derivations [41]. Second, on \(\mathit{HH}^0(A)=Z(A)\) the lemma admits a one-line direct verification: for central \(z\) and \(f\in\mathrm{D}A\), \[(z\cdot f)(a)=f(az)=f(za)=(f\cdot z)(a).\] Third, the geometric counterpart of Theorem 16 for smooth projective varieties admits an independent verification by a line-bundle computation; see Remark 21.
Example 3 (The exterior algebra). Let \(\Lambda=\Lambda(x,y)\) be the exterior algebra on two generators over a field of characteristic \(\neq2\): a Frobenius algebra which is not symmetric, with Nakayama automorphism the parity involution \(\nu\), \(\nu(x)=-x\), \(\nu(y)=-y\). Corollary 4 predicts that \(\nu\) acts trivially on \(\mathit{HH}^\bullet(\Lambda)\); we verify the prediction in low degrees. On \(\mathit{HH}^0=Z(\Lambda)=k\oplus k\,xy\) the involution acts trivially. In degree one, the parity-odd derivations are inner: the derivation \(x\mapsto xy\), \(y\mapsto0\) equals \(-\tfrac12\operatorname{ad}_y\), and there is no derivation with \(x\mapsto1\), by the Leibniz rule applied to \(x^2=0\); the outer derivations form the parity-even space \(\mathfrak{gl}_2\) of linear substitutions, on which conjugation by \(\nu=-\operatorname{id}\) is trivial. In degree two, the parity-odd deformation direction \(x^2=\varepsilon x\) is trivialized to first order by the substitution \(x\mapsto x-\varepsilon/2\), again in accordance with the prediction.
This section extends the construction to differential graded algebras, following a suggestion of B. Keller. The pay-off is geometric: the perfect derived category of any smooth projective variety is of the form \(\operatorname{per}\mathcal{A}\) for a smooth and proper dg algebra \(\mathcal{A}\) [44], so the enriched invariant of this paper becomes an invariant of varieties.
Throughout, \(\mathcal{A}\) and \(\mathcal{B}\) are dg algebras over \(k\), \(\mathcal{D}(\mathcal{A})\) is the derived category of right dg modules, \(\operatorname{per}\mathcal{A}\subseteq\mathcal{D}(\mathcal{A})\) is the thick subcategory generated by \(\mathcal{A}\), and \(\mathcal{A}^e=\mathcal{A}\otimes_k\mathcal{A}^{\mathrm{op}}\). The algebra \(\mathcal{A}\) is smooth if \(\mathcal{A}\in\operatorname{per}(\mathcal{A}^e)\) and proper if the total cohomology of \(\mathcal{A}\) is finite-dimensional. For an ordinary finite-dimensional algebra, properness is automatic and smoothness implies finite global dimension; the converse holds when \(A/\!\operatorname{rad}A\) is separable, for instance for elementary algebras (see [44]). Morphisms in this section are dg Morita morphisms, i.e.bimodules, as in [44], [45].
Lemma 7. Let \(X\in\mathcal{D}(\mathcal{A}\otimes\mathcal{B}^{\mathrm{op}})\) be invertible: suppose there exists \(Y\in\mathcal{D}(\mathcal{B}\otimes\mathcal{A}^{\mathrm{op}})\) with \(X\otimes^{\mathbb{L}}_{\mathcal{B}}Y\cong\mathcal{A}\) in \(\mathcal{D}(\mathcal{A}^e)\) and \(Y\otimes^{\mathbb{L}}_{\mathcal{A}}X\cong\mathcal{B}\) in \(\mathcal{D}(\mathcal{B}^e)\). Then:
\(X\) is perfect over \(\mathcal{B}\) and over \(\mathcal{A}^{\mathrm{op}}\), and the canonical morphism \(\mathcal{A}\to\mathbf{R}\!\operatorname{Hom}_{\mathcal{B}}(X,X)\), \(a\mapsto(x\mapsto ax)\), is invertible in \(\mathcal{D}(\mathcal{A}^e)\);
there are isomorphisms \[Y\;\cong\;\mathbf{R}\!\operatorname{Hom}_{\mathcal{B}}(X,\mathcal{B}) \;\cong\;\mathbf{R}\!\operatorname{Hom}_{\mathcal{A}^{\mathrm{op}}}(X,\mathcal{A})\] in \(\mathcal{D}(\mathcal{B}\otimes\mathcal{A}^{\mathrm{op}})\).
Proof. (1) The functor \(-\otimes^{\mathbb{L}}_{\mathcal{A}}X\colon\mathcal{D}(\mathcal{A})\to\mathcal{D}(\mathcal{B})\) is an equivalence with quasi-inverse \(-\otimes^{\mathbb{L}}_{\mathcal{B}}Y\), hence preserves compact objects; as \(X\cong\mathcal{A}\otimes^{\mathbb{L}}_{\mathcal{A}}X\) is the image of the compact generator \(\mathcal{A}\), it is perfect over \(\mathcal{B}\). The canonical morphism \(\mathcal{A}\to\mathbf{R}\!\operatorname{Hom}_\mathcal{B}(X,X)\) induces on cohomology the maps \(\operatorname{Hom}_{\mathcal{D}(\mathcal{A})}(\mathcal{A},\mathcal{A}[n])\to\operatorname{Hom}_{\mathcal{D}(\mathcal{B})}(X,X[n])\) given by the fully faithful functor \(-\otimes^{\mathbb{L}}_\mathcal{A}X\), hence is a quasi-isomorphism; and it is a morphism of \(\mathcal{A}\)-bimodules by construction. The mirror statements hold over \(\mathcal{A}^{\mathrm{op}}\).
(2) Write \(X^\vee=\mathbf{R}\!\operatorname{Hom}_\mathcal{B}(X,\mathcal{B})\). Exactly as in Step 1 of the proof of Lemma 3, dévissage from the case \(M=\mathcal{B}\) shows that for every \(M\in\operatorname{per}\mathcal{B}\) and every \(N\in \mathcal{D}(\mathcal{B}^e)\) or \(\mathcal{D}(\mathcal{A}\otimes\mathcal{B}^{\mathrm{op}})\) the canonical morphism \[N\otimes^{\mathbb{L}}_{\mathcal{B}}\mathbf{R}\!\operatorname{Hom}_{\mathcal{B}}(M,\mathcal{B})\longrightarrow\mathbf{R}\!\operatorname{Hom}_{\mathcal{B}}(M,N), \qquad n\otimes f\longmapsto\bigl(x\mapsto n\cdot f(x)\bigr),\] is invertible. Using it twice, at \((M,N)=(X,\,Y\otimes^{\mathbb{L}}_\mathcal{A}X)\) and at \((M,N)=(X,X)\): \[\begin{align} X^\vee &\cong\mathbf{R}\!\operatorname{Hom}_\mathcal{B}\bigl(X,\;Y\otimes^{\mathbb{L}}_{\mathcal{A}}X\bigr) &&\text{since }Y\otimes^{\mathbb{L}}_{\mathcal{A}}X\cong\mathcal{B},\\ &\cong\bigl(Y\otimes^{\mathbb{L}}_{\mathcal{A}}X\bigr)\otimes^{\mathbb{L}}_{\mathcal{B}}X^\vee &&\text{d\'evissage at }M=X,\\ &\cong Y\otimes^{\mathbb{L}}_{\mathcal{A}}\bigl(X\otimes^{\mathbb{L}}_{\mathcal{B}}X^\vee\bigr) &&\text{associativity},\\ &\cong Y\otimes^{\mathbb{L}}_{\mathcal{A}}\mathbf{R}\!\operatorname{Hom}_\mathcal{B}(X,X) &&\text{d\'evissage at }M=X,\;N=X,\\ &\cong Y\otimes^{\mathbb{L}}_{\mathcal{A}}\mathcal{A}\;\cong\;Y &&\text{by (1).} \end{align}\] All the isomorphisms are bimodule isomorphisms. The identification of \(Y\) with \(\mathbf{R}\!\operatorname{Hom}_{\mathcal{A}^{\mathrm{op}}}(X,\mathcal{A})\) is the mirror computation. ◻
Lemma 8. Let \(\mathcal{A}\) be smooth and proper. Then \(\mathrm{D}\mathcal{A}\) is invertible in \(\mathcal{D}(\mathcal{A}^e)\), with inverse \(\mathbf{R}\!\operatorname{Hom}_{\mathcal{A}}(\mathrm{D}\mathcal{A},\mathcal{A})\).
Proof. Properness gives that \(H(\mathrm{D}\mathcal{A})=\mathrm{D}H(\mathcal{A})\) is finite-dimensional, and smoothness implies that objects with finite-dimensional total cohomology are perfect, both in \(\mathcal{D}(\mathcal{A})\) and in \(\mathcal{D}(\mathcal{A}^{\mathrm{op}})\) [44]; hence \(\mathrm{D}\mathcal{A}\) is perfect on both sides. Moreover the biduality morphism \(\mathcal{A}\to\mathrm{D}\mathrm{D}\mathcal{A}\) is a quasi-isomorphism, because the cohomology of \(\mathcal{A}\) is degreewise finite-dimensional. With these two inputs, the three steps of the proof of Lemma 3 apply verbatim, with h-projective resolutions replacing the bounded projective representatives. ◻
Lemma 9. Let \(\mathcal{A}\), \(\mathcal{B}\) be smooth and proper dg algebras, let \(X\in\mathcal{D}(\mathcal{A}\otimes\mathcal{B}^{\mathrm{op}})\) be invertible, and set \(\tau_{\mathcal{A}}:=\mathrm{D}\mathcal{A}[-1]\), \(\tau_{\mathcal{B}}:=\mathrm{D}\mathcal{B}[-1]\). Then \[\tau_{\mathcal{A}}\otimes^{\mathbb{L}}_{\mathcal{A}}X\;\cong\;X\otimes^{\mathbb{L}}_{\mathcal{B}}\tau_{\mathcal{B}} \qquad\text{in }\mathcal{D}(\mathcal{A}\otimes\mathcal{B}^{\mathrm{op}}).\]
Proof. The morphisms \(\alpha_X\) and \(\beta_X\) of Lemma 4 are defined by the same evaluation formulas on h-projective representatives, are invertible by the same dévissage — using that \(X\) is perfect on both sides, Lemma 7(1) — and combine with Lemma 7(2), which replaces 2 , exactly as before. ◻
For a dg algebra \(\mathcal{A}\) we take as cohomology \(\mathit{HH}^m(\mathcal{A})=\operatorname{Hom}_{\mathcal{D}(\mathcal{A}^e)}(\mathcal{A},\mathcal{A}[m])\) with the Yoneda (composition) product, and as homology the mixed complex \(\mathcal{A}\otimes^{\mathbb{L}}_{\mathcal{A}^e}\mathcal{A}\) of Keller [10], whose homology is \(\mathit{HH}_\bullet(\mathcal{A})\) and which carries the Connes differential \(B\); the cap product is the action \[z\cap\eta\;:=\;H\bigl(\operatorname{id}_\mathcal{A}\otimes^{\mathbb{L}}_{\mathcal{A}^e}\eta\bigr)(z),\] that is, the functoriality of \(\mathcal{A}\otimes^{\mathbb{L}}_{\mathcal{A}^e}(-)\) applied to \(\eta\colon\mathcal{A}\to\mathcal{A}[m]\). For an ordinary algebra these models agree with the classical operations by the comparison results of [1], [2]. An invertible bimodule \(X\) induces a transport of all this data: the conjugation \(\gamma_X\) on \(\mathit{HH}^\bullet\), characterized as in 4 , and the isomorphism of mixed complexes of [10] on \(\mathit{HH}_\bullet\).
Theorem 20. Let \(\mathcal{A}\) be a smooth and proper dg algebra over \(k\), and let \(\sigma_{\mathcal{A}}\) denote the transport along the invertible bimodule \(\tau_{\mathcal{A}}=\mathrm{D}\mathcal{A}[-1]\).
(Centrality.) For every smooth and proper \(\mathcal{B}\) and every invertible \(X\in\mathcal{D}(\mathcal{A}\otimes\mathcal{B}^{\mathrm{op}})\) one has \(\tau_\mathcal{A}\otimes^{\mathbb{L}}_\mathcal{A}X\cong X\otimes^{\mathbb{L}}_\mathcal{B}\tau_\mathcal{B}\); in particular the transports along \(X\) intertwine \(\sigma_\mathcal{A}\) and \(\sigma_\mathcal{B}\), and the isomorphism class of \(\bigl(\mathit{HH}_\bullet(\mathcal{A}),B,\cap,\sigma_{\mathcal{A}}\bigr)\) is invariant under dg Morita equivalence of smooth proper dg algebras.
(Cohomology.) The cohomological component of \(\sigma_\mathcal{A}\) is the identity of \(\mathit{HH}^\bullet(\mathcal{A})\).
(Homology.) The operator \(\sigma_\bullet\) on \(\mathit{HH}_\bullet(\mathcal{A})\) commutes with the Connes differential \(B\) and with every cap operator: \(\sigma_\bullet(z\cap\eta)=\sigma_\bullet(z)\cap\eta\).
Proof. (1) is Lemma 9 together with the functoriality of Keller’s cyclic functor [10] and of conjugation. (2) The proof of Lemma 6 uses only the naturality of \(\alpha\) and \(\beta\), the perfectness of \(\mathcal{A}\) and \(\mathcal{A}[m]\) on both sides, the biduality \(\mathcal{A}\cong\mathrm{D}\mathrm{D}\mathcal{A}\), and the evaluation computations on h-projective representatives; all are available here by Lemmas 7 and 8, and the shift contributes trivially, as before. (3) Commutation with \(B\) holds because \(\sigma_\bullet\) is induced by a morphism of mixed complexes [10]. For the cap operators, the transport along an invertible bimodule is a composite of maps natural in the bimodule coefficient; naturality at \(\eta\) yields \(\sigma_\bullet(z\cap\eta)=\sigma_\bullet(z)\cap\gamma_{\tau}(\eta)\), and \(\gamma_{\tau}=\operatorname{id}\) by (2). A consequence of (2) worth making explicit: compatibility with any further structure carried by \(\mathit{HH}^\bullet(\mathcal{A})\) — the cup product, the Gerstenhaber bracket, indeed the full \(B_\infty\)-structure — holds trivially, so no comparison results are needed on the cohomological side. ◻
Remark 21 (The Serre functor acts trivially on \(\mathit{HH}^\bullet\)). Let \(X\) be a smooth projective variety of dimension \(d\) with Serre functor \(S=-\otimes\omega_X[d]\), and let \(\mathcal{A}\) be a smooth proper dg algebra with \(\operatorname{per}\mathcal{A}\simeq\operatorname{per}X\) [44]. Under the identification of \(\mathit{HH}^\bullet(\mathcal{A})\) with \(\operatorname{Ext}^\bullet_{X\times X}(\mathcal{O}_\Delta, \mathcal{O}_\Delta)\), Theorem 20(2) says that conjugation by the Serre kernel \(\mathcal{O}_\Delta\otimes p^*\omega_X[d]\) is trivial. This admits a direct verification, which is the third sign check promised in Remark 19: conjugating a class \(\eta\) by the kernel amounts to twisting by the pull-back line bundle \(p^*\omega_X\) and shifting; since \(p^*\omega_X|_\Delta\cong q^*\omega_X|_\Delta\), the twist acts trivially on \(\operatorname{Ext}^\bullet(\mathcal{O}_\Delta,\mathcal{O}_\Delta)\), and the shift contributes nothing.
Remark 22 (Lefschetz form of Theorem 10). For a general smooth proper \(\mathcal{A}\), Hochschild homology is no longer concentrated in degree zero — for a variety, \(\mathit{HH}_n(X)\cong\bigoplus_{p-q=n}H^q(X,\Omega^p_X)\) by Hochschild–Kostant–Rosenberg — and the matrix statement of Theorem 10 is replaced by its Lefschetz shadow: the Euler class \(\operatorname{ch}\colon K_0(\operatorname{per}\mathcal{A})\to\mathit{HH}_0(\mathcal{A})\) of Shklyarov [38] intertwines the action of the class of \(\tau_\mathcal{A}\) on \(K_0\) with the degree-zero part of \(\sigma_\bullet\), and the supertraces \(\operatorname{str}(\sigma_\bullet^{\,m})\) are computed by the Mukai pairing and the Hirzebruch–Riemann–Roch theorem of [38]. We leave systematic computations — Grassmannians, hypersurfaces — for future work.
Example 4 (Projective space). Let \(X=\mathbb{P}^n\). By Beilinson [46], \(\operatorname{per}\mathbb{P}^n\simeq\operatorname{per}B_n\) for the endomorphism algebra \(B_n=\operatorname{End}\bigl(\bigoplus_{i=0}^{n}\mathcal{O}(i)\bigr)\), an elementary algebra of finite global dimension whose Cartan matrix has entries \(\binom{n+j-i}{n}\) for \(0\le i\le j\le n\) and \(0\) otherwise. We claim that \[\phi_{B_n}(x)\;=\;\bigl(x+(-1)^{n}\bigr)^{\,n+1},\] a single Jordan block with eigenvalue \((-1)^{n+1}\), as one checks directly for small \(n\). Indeed, on \(K_0(\mathbb{P}^n)\cong\mathbb{Z}^{n+1}\) the shift acts by \(-1\) and the Serre functor by \((-1)^{n}[\otimes\omega_X]\), so the Coxeter transformation acts by \((-1)^{n+1}[\otimes\mathcal{O}(-n-1)]\); and multiplication by \([\omega_X]\) is unipotent, because \([\omega_X]\) differs from \(1\) by a nilpotent element of the ring \(K_0(\mathbb{P}^n)\). Hence all eigenvalues equal \((-1)^{n+1}\), which forces the displayed polynomial. Note that the Hochschild homology of \(\mathbb{P}^n\) is concentrated in degree zero, of dimension \(n+1\), since the Hodge numbers of \(\mathbb{P}^n\) are diagonal; so the homological picture of Theorem 10 persists verbatim on the geometric side.
Example 5 (Elliptic curves: Calabi–Yau degeneracy). Let \(E\) be an elliptic curve and \(\mathcal{A}\) a smooth proper dg algebra with \(\operatorname{per}\mathcal{A}\simeq\operatorname{per}E\). Since \(\omega_E\cong\mathcal{O}_E\), the Serre kernel is \(\mathcal{O}_\Delta[1]\), so \(\mathrm{D}\mathcal{A}\cong\mathcal{A}[1]\) in \(\mathcal{D}(\mathcal{A}^e)\), hence \(\tau_\mathcal{A}\cong\mathcal{A}\) and \(\sigma_{\mathcal{A}}\) is the identity transport — although \(\mathit{HH}_\bullet(E)\) is large: \(\mathit{HH}_1=H^0(\Omega^1_E)=k\), \(\mathit{HH}_0=H^0(\mathcal{O}_E)\oplus H^1(\Omega^1_E)=k^2\), \(\mathit{HH}_{-1}=H^1(\mathcal{O}_E)=k\). The enrichment of this paper degenerates precisely on Calabi–Yau categories, in accordance with Corollary 3; its content lies in the non-Calabi–Yau directions, where \(\omega_X\) is nontrivial — curves of genus \(\geq2\), Fano and general-type varieties — and where the supertraces of \(\sigma_\bullet^{\,m}\) are computed by Riemann–Roch from powers of the canonical bundle, via Remark 22.
Theorem 10 and Proposition 8 are the operator form, natural under derived equivalence, of computations that exist in the literature as numerical identities. Han [15] computes, for elementary algebras of finite global dimension, all the ingredients of the Lefschetz and Hirzebruch–Riemann–Roch formulas of Shklyarov [38] and Petit: the Shklyarov pairing on \(\mathit{HH}_0(A)\) has matrix \(C_A^{\mathrm{T}}\) in the basis of the \(\bar e_i\), the Chern character is \(C_A^{-1}\underline{\dim}\,\), and Happel’s trace formula is the Lefschetz formula of the Serre bimodule. The content added here is that these identities organize into a single automorphism of the full calculus, natural under derived equivalence.
Since \(\sigma_\bullet\) and \(\Phi_A\) are conjugate (Theorem 10), the spectral radius \(\rho(\sigma_\bullet)=\rho(\Phi_A)\) and the Mahler measure of \(\phi_A\) studied by de la Peña [20] are invariants of the enriched calculus. These quantities have recently been categorified as entropies: the categorical entropy of [16] of the Serre functor, the Hochschild homology and cohomology entropies of Kikuta–Ouchi [19], the computations of Han [17] showing that for higher hereditary algebras the entropy of the Serre functor and the Hochschild entropies of the Serre quasi-functor categorify the spectral radius and the polynomial growth rate of the Coxeter matrix, and the Gromov–Yomdin type theorem of Chang–Schroll [18] for gentle algebras, where the entropy of the Serre functor at \(t=0\) equals \(\log\rho(K_0(\nu))\). The operator \(\sigma_A\) is the exact algebraic object whose iteration these entropies measure on Hochschild homology.
Corollary 3 embeds into a substantial dictionary: periodicity of the Coxeter transformation [21], cyclotomicity [6], [26], fractionally and twisted fractionally Calabi–Yau algebras [14], periodic trivial extensions [22], Serre cyclotomic algebras [23], and the fractional Calabi–Yau property of Tamari lattices [24]. In each of these situations the hypothesis is a relation among powers of the Serre bimodule, hence a relation imposed on the powers of \(\sigma_A\) through Theorem 3(2).
The bimodule powers \(\tau_A^{\otimes^{\mathbb{L}}m}\) assemble into Keller’s Calabi–Yau completions \(\Pi_n(A)=T_A(\Theta[n-1])\), \(\Theta\) the inverse dualizing bimodule, whose formation is compatible with derived equivalence [25]; for connected non-Dynkin quivers \(\Pi_2(kQ)\) is the preprojective algebra. The Tamarkin–Tsygan calculus of the classical preprojective algebras of Dynkin quivers was computed by Etingof–Eu and Eu [47], [48], with answers graded by the exponents and the Coxeter number of the root system. From this perspective the enriched calculus \((\mathbb{H}(A),\sigma_A)\) is the “first Fourier mode” of the \(\Theta\)-graded Hochschild theory of \(A\), and the full graded theory is a natural further refinement.
Theorem 16 explains the asymmetry visible throughout this paper: all the Coxeter information carried by \(\sigma_A\) lives on Hochschild homology, and by Proposition 7 that is exactly where, for the algebras of Section 5, the Tamarkin–Tsygan calculus alone is most degenerate. This is not an accident of the elementary case. Cohomology classes act on every bimodule from two sides, and the duality \(\mathrm{D}A\) interchanges the two actions (Lemma 6), so conjugation cannot detect them; homology classes transform like traces, and traces detect the Serre bimodule through the Cartan matrix (Lemma 5). A nontrivial cohomological shadow of the Coxeter symmetry should therefore be sought in the twisted theory \(\bigoplus_m\mathit{HH}^\bullet\bigl(A,\tau_A^{\otimes^{\mathbb{L}}_A m}\bigr)\), the natural home of the preprojective algebra; see Question 23.
Question 23. By Theorem 16 the untwisted cohomological theory is blind to \(\sigma\). Develop the \(\tau\)-twisted theory \[\bigoplus_{m\in\mathbb{Z}}\mathit{HH}^\bullet\bigl(A,\tau_A^{\otimes^{\mathbb{L}}_A m}\bigr),\] with its cup product and its \(\sigma\)-equivariant structure; make precise its relation to the Hochschild theory of the preprojective algebras and Calabi–Yau completions of [14], [25]. Is the twisted theory a complete derived invariant for interesting classes of algebras?
Question 24. By Theorem 3, \(\sigma_\bullet\) commutes with \(B\) and hence acts on cyclic, negative cyclic and periodic homology compatibly with the Connes exact sequences of [3]. Develop the resulting “categorical monodromy” picture, in which the eigenvalues of \(\sigma_\bullet\) play the role of the monodromy eigenvalues in the singularity-theoretic interpretation of Coxeter spectra of Lenzing–de la Peña [7].
Question 25. Enrich \((\mathbb{H}(A),\sigma_A)\) with the integral lattice \(\operatorname{ch}(K_0(\operatorname{per}A))\subset\mathit{HH}_0(A)\) and the Shklyarov pairing, whose matrix is \(C_A^{\mathrm{T}}\) [15], [38]; Example 2 shows that without this integral data the enriched calculus is not complete, already for trees. The resulting structure remembers the Cartan matrix up to \(\mathbb{Z}\)-congruence and the Coxeter transformation up to \(GL_n(\mathbb{Z})\)-conjugacy. For which classes of algebras is this enriched invariant complete?
Question 26. Replace \(\tau_A\) by the bimodule \(\mathrm{D}A[-d]\) of higher Auslander–Reiten theory for \(d\)-hereditary algebras and compare the resulting automorphisms with the entropies of Serre functors [17] and with the higher preprojective gradings of [14], [25].
Question 27. In characteristic \(p\), Theorem 10 only recovers \(\phi_A\) modulo \(p\), while the lattice-enriched invariant of Question 25 recovers it integrally. Do there exist derived-inequivalent algebras whose enriched calculi become isomorphic after reduction modulo \(p\)? Note that Example 2 already exhibits the rational version of this phenomenon in characteristic zero.
The mathematical content of this paper, including the formulation of Theorems and drafts of their proofs, was developed with substantial assistance from an AI system (Claude, Anthropic, Fable 5 model). The author takes full responsibility for all statements.