# Comptes Rendus
Mathématique

Mamta Balodi, Abhishek Banerjee and Anita Naolekar

BV-operators and the secondary Hochschild complex

Volume 358, issue 11-12 (2020), p. 1239-1258

Published online: 25 January 2021

https://doi.org/10.5802/crmath.157

<figure><img src="image_1.png" /></figure>

This article is licensed under the
CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL LICENSE.
http://creativecommons.org/licenses/by/4.0/

<figure><img src="image_2.png" /></figure>

Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l'édition scientifique ouverte
www.centre-mersenne.org

e-ISSN : 1778-3569
Algebra, Geometry and Topology / Algèbre, Géométrie et Topologie

BV-operators and the secondary Hochschild
complex

Mamta Balodi<sup>a</sup>, Abhishek Banerjee<sup>*</sup>,<sup>a</sup> and Anita Naolekar<sup>b</sup>

<sup>a</sup> Department of Mathematics, Indian Institute of Science, Bangalore

<sup>b</sup> Stat-Math Unit, Indian Statistical Institute, Bangalore

E-mails: mamta.balodi@gmail.com, abhishekbanerjee1313@gmail.com, anita@isibang.ac.in

**Abstract.** We introduce the notion of a BV-operator $\Delta = \{\Delta^n : V^n \to V^{n-1}\}_{n \ge 0}$ on a homotopy G-algebra $V^*$ such that the Gerstenhaber bracket on $H(V^*)$ is determined by $\Delta$ in a manner similar to the BV-formalism. As an application, we produce a BV-operator on the cochain complex defining the secondary Hochschild cohomology of a symmetric algebra $A$ over a commutative algebra $B$. In this case, we also show that the operator $\Delta^*$ corresponds to Connes' operator.

**2020 Mathematics Subject Classification.** 16E40.

**Funding.** AB was partially supported by SERB Matrics fellowship MTR/2017/000112.

Manuscript received 14th October 2020, revised 21st November 2020, accepted 23rd November 2020.

# 1. Introduction

A Gerstenhaber algebra (see [3]) consists of a graded vector space $W^* = \bigoplus_{n \ge 0} W^n$ equipped with the following two structures:

(a) A dot product $x \cdot y$ of degree zero making $W^*$ into an associative graded commutative algebra.

(b) A bracket $[x, y]$ of degree $-1$ making $W^*$ into a graded Lie algebra satisfying the compatibility property that

$$[x, y \cdot z] = [x, y] \cdot z + (-1)^{(\deg(x)-1)\deg(y)} y \cdot [x, z].$$

Gerstenhaber algebra structures appear in a variety of situations, from Hochschild cohomology of algebras to the exterior algebra of a Lie algebra and the algebra of differential forms on a Poisson manifold.

<footnote>* Corresponding author.</footnote>
An operator $\partial = \{\partial^n : W^n \to W^{n-1}\}_{n \ge 0}$ on $W^*$ of degree $-1$ is said to generate the Gerstenhaber bracket (see Koszul [7, §2] and also [6, Definition 3.2]) if it satisfies

$$[x, y] = (-1)^{(\deg(x)-1)\deg(y)} (\partial(x) \cdot y + (-1)^{\deg(x)} x \cdot \partial(y) - \partial(x \cdot y))$$

In particular, a Batalin-Vilkovisky algebra (or BV-algebra) consists of a Gerstenhaber algebra along with a generator $\partial$ for the bracket such that $\partial^2 = 0$.

In [4], [5], Gerstenhaber and Voronov introduced the notion of a homotopy $G$-algebra, which is a brace algebra equipped with a differential of degree 1 and a dot product of degree 0 satisfying certain conditions. In particular, the cohomology groups $H(V^*)$ of a homotopy $G$-algebra $V^*$ carry the structure of a Gerstenhaber algebra.

In this paper, we introduce the notion of a BV-operator $\Delta = \{\Delta^n : V^n \to V^{n-1}\}_{n \ge 0}$ on a homotopy $G$-algebra $V^*$ such that the Gerstenhaber bracket on $H(V^*)$ is determined by $\Delta$ in a manner similar to the BV-formalism. More explicitly, for classes $\bar{f} \in H^n(V^*)$ and $\bar{g} \in H^m(V^*)$, we have

$$[\bar{f}, \bar{g}] = (-1)^{(n-1)m} \overline{(\Delta(f) \cdot g + (-1)^n f \cdot \Delta(g) - \Delta(f \cdot g))} \in H^{m+n-1}(V^*)$$

where $f \in Z^n(V^*)$, $g \in Z^m(V^*)$ are cocycles representing $\bar{f}$ and $\bar{g}$ respectively. We note that $\Delta$ need not be a morphism of cochain complexes and therefore may not induce any operator on $H(V^*)$. As such, $\Delta$ may not descend to a generator for the Gerstenhaber bracket on $H(V^*)$.

Our motivation is to introduce a BV-operator on the cochain complex defining the secondary Hochschild cohomology of a symmetric algebra $A$ over a commutative algebra $B$. For a datum $(A, B, \varepsilon)$ consisting of an algebra $A$, a commutative algebra $B$ and an extension of rings $\varepsilon : B \to A$ such that $\varepsilon(B) \subseteq Z(A)$, the secondary Hochschild cohomology $H^*(A, B, \varepsilon)$ was introduced by Staic [9] in order to study deformations of algebras $A[[t]]$ having a $B$-algebra structure. More generally, Staic [9] introduced the secondary Hochschild complex $C^*((A, B, \varepsilon); M)$ with coefficients in an $A$-bimodule $M$.

In [10], Staic and Stancu showed that the secondary Hochschild complex $C^*(A, B, \varepsilon) := C^*((A, B, \varepsilon); A)$ with coefficients in $A$ is a non-symmetric operad with multiplication, giving it the structure of a homotopy $G$-algebra. Hence, the secondary cohomology $H^*(A, B, \varepsilon)$ is equipped with a graded commutative cup product and a Lie bracket which makes it a Gerstenhaber algebra. For more on the secondary cohomology, the reader may see, for instance, [1], Corrigan-Salter and Staic [2], Laubacher, Staic and Stancu [8].

Let $k$ be a field. It is well known (see Tradler [11]) that the Hochschild cohomology of a finite dimensional $k$-algebra $A$ equipped with a symmetric, non-degenerate, invariant bilinear form $\langle \cdot, \cdot \rangle : A \times A \to k$ carries the structure of a BV-algebra. For the terms $C^n((A, B, \varepsilon)) = \text{Hom}_k(A^{\otimes n} \otimes B^{\otimes \frac{n(n-1)}{2}}, A)$ in the secondary Hochschild complex, we define the BV-operator $\Delta = \sum_{i=1}^{n+1} (-1)^{in} \Delta_i : C^{n+1}(A, B, \varepsilon) \to C^n(A, B, \varepsilon)$ by the condition (see Section 3)

$$ \begin{aligned} & \left\langle \Delta_i f \left( \otimes \begin{pmatrix} a_1 & b_{1,2} & b_{1,3} & \dots & b_{1,n} \\ 1 & a_2 & b_{2,3} & \dots & b_{2,n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & 1 & \dots & b_{n-1,n} \\ 1 & 1 & 1 & \dots & a_n \end{pmatrix} \right), a_{n+1} \right\rangle \\ &= \left\langle f \left( \otimes \begin{pmatrix} a_i & b_{i,i+1} & b_{i,i+2} & \dots & b_{i,n} & 1 & b_{1,i} & b_{2,i} & \dots & b_{i-1,i} \\ 1 & a_{i+1} & b_{i+1,i+2} & \dots & b_{i+1,n} & 1 & b_{1,i+1} & b_{2,i+1} & \dots & b_{i-1,i+1} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \dots & a_n & 1 & b_{1,n} & b_{2,n} & \dots & b_{i-1,n} \\ 1 & 1 & \dots & 1 & a_{n+1} & 1 & \dots & b_{1,2} & \dots & b_{1,i-1} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \dots & \dots & \dots & \dots & \dots & \dots & a_{i-1} \end{pmatrix} \right), 1 \right\rangle \quad (1) \end{aligned} $$
We then show that the Gerstenhaber bracket on the secondary Hochschild cohomology of $(A, B, \varepsilon)$ is determined by $\Delta$ in a manner similar to the BV-formalism.

From Tradler [11], we also know that the operator $\Delta^\bullet : C^\bullet(A, A) \to C^{\bullet-1}(A, A)$ on usual Hochschild cochains inducing the BV-structure on $H^\bullet(A, A)$ corresponds to the operator $N\mathcal{S}$ on duals of Hochschild chains, where $N$ is the "norm operator" and $\mathcal{S}$ is the "extra degeneracy" (see (16)). The isomorphism between the two complexes is induced by the $k$-module isomorphism $A^* \cong A$ determined by the non-degenerate bilinear form $\langle \cdot, \cdot \rangle : A \times A \to k$. If we pass to the cohomology and take the normalized Hochschild complex which is a quasi-isomorphic subcomplex of $C^\bullet(A, A)$, it follows that Tradler's $\Delta^\bullet$ operator corresponds to Connes' operator on Hochschild cohomology with coefficients in $A$.

However, in the case of secondary cohomology, we have mentioned that the operator $\Delta^\bullet$ defined in (1) is not a morphism of complexes and we cannot pass to cohomology. Accordingly, we show that the operator $\Delta^\bullet$ defined in (1) fits into a commutative diagram (see Theorem 10)

$$ \begin{tikzcd} \bar{C}^\bullet(A, B, \varepsilon) \arrow[r, "B"] & \bar{C}^{\bullet-1}(A, B, \varepsilon) \arrow[d] \\ C^\bullet((A, B, \varepsilon); A) \arrow[u, "\Delta^\bullet"] \arrow[r, "\Delta^\bullet"] & C^{\bullet-1}((A, B, \varepsilon); A) \arrow[u] \arrow[d] \end{tikzcd} \qquad (2) $$

where $B$ is Connes' operator. Here, $\bar{C}^\bullet(A, B, \varepsilon)$ is the normalization of the co-simplicial module $\bar{C}^\bullet(A, B, \varepsilon)$ introduced by Laubacher, Staic and Stancu [8], which is used to compute the secondary Hochschild cohomology associated to the triple $(A, B, \varepsilon)$. It should be noted (see [8, Remark 4.7]) that despite similar names, the complex $\bar{C}^\bullet(A, B, \varepsilon)$ cannot be expressed as a secondary Hochschild complex with coefficients in some $A$-bimodule. The vertical morphisms in (2) are induced by composing the canonical morphisms $(A \otimes B^{\otimes n})^* \to A^*$ for each $n \ge 0$, the isomorphism $A^* \cong A$ as well as the inclusion of the quasi-isomorphic subcomplex $\bar{C}^\bullet(A, B, \varepsilon) \hookrightarrow \bar{C}^\bullet(A, B, \varepsilon)$.

## 2. Main Result: BV-operator on homotopy $G$-algebra

We begin by recalling the notion of a homotopy $G$-algebra from [5]. A brace algebra (see [5, Definition 1]) is a graded vector space $V = \bigoplus_{n \ge 0} V^n$ with a collection of multilinear operators (braces) $x\{x_1, \dots, x_n\}$ satisfying the following conditions (with $x\{\}$ understood to be $x$):

$$ (1) \operatorname{deg}(x\{x_1, \dots, x_n\}) = \operatorname{deg}(x) + \sum_{i=1}^{n} \operatorname{deg}(x_i) - n $$

(2) For homogeneous elements $x, x_1, \dots, x_m, y_1, \dots, y_n$, we have

$$ x\{x_1, \dots, x_m\}\{y_1, \dots, y_n\} = \sum_{0 \le i_1 \le j_1 \le \dots \le i_m \le j_m \le n} (-1)^e x\{y_1, \dots, y_{i_1}, x_1\{y_{i_1+1}, \dots, y_{j_1}\}, y_{j_1+1}, \dots, y_{i_m}, x_m\{y_{i_m+1}, \dots, y_{j_m}\}, y_{j_m+1}, \dots, y_n\} $$

$$ \text{where } \epsilon = \sum_{p=1}^{m} |x_p| (\sum_{q=1}^{i_p} |y_q|) \text{ and } |x| := \operatorname{deg}(x) - 1. $$

**Definition 1 (see [5, Definition 2]).** A homotopy $G$-algebra consists of the following data:

$$ (1) \textit{A brace algebra } V = \bigoplus_{n \ge 0} V^n. $$

(2) A dot product of degree zero

$$ V^m \otimes V^n \to V^{m+n} \quad x \otimes y \to x \cdot y $$

for all $m, n \ge 0$.

(3) A differential $d: V^\bullet \to V^{\bullet+1}$ of degree one making $V$ into a DG-algebra with respect to the dot product.
(4) The dot product satisfies the following compatibility conditions

$$ (x_1 \cdot x_2)\{y_1, \dots, y_n\} = \sum_{k=0}^{n} (-1)^{\epsilon_k} (x_1\{y_1, \dots, y_k\}) \cdot (x_2\{y_{k+1}, \dots, y_n\}) $$

where $\epsilon_k = |x_2| \sum_{p=1}^{k} |y_p|$ and

$$ \begin{aligned} d(x\{x_1, \dots, x_{n+1}\}) - (dx)\{x_1, \dots, x_{n+1}\} - (-1)^{|x|} \sum_{i=1}^{n+1} (-1)^{|x_1|+\dots+x_{i-1}|} x\{x_1, \dots, dx_i, \dots, x_{n+1}\} \\ = (-1)^{|x||x_1|+1} x_1 \cdot x\{x_2, \dots, x_{n+1}\} + (-1)^{|x|} \sum_{i=1}^{n} (-1)^{|x_1|+\dots+x_{i-1}|} x\{x_1, \dots, x_i \cdot x_{i+1}, \dots, x_{n+1}\} - x\{x_1, \dots, x_n\} \cdot x_{n+1} \end{aligned} $$

In particular, a homotopy G-algebra is equipped with a graded Lie bracket which descends to the cohomology of the corresponding cochain complex $(V^*, d)$ (see [5])

$$ [\cdot, \cdot] : H^m(V^*) \otimes H^n(V^*) \to H^{m+n-1}(V^*) \quad (3) $$

The dot product also descends to the cohomology and the bracket with an element becomes a graded derivation for the induced dot product on $H(V^*) = \bigoplus_{n \ge 0} H^n(V^*)$. In other words, the cohomology $(H(V^*), [\cdot, \cdot], \cdot)$ of a homotopy G-algebra $V^*$ is canonically equipped with the structure of a Gerstenhaber algebra.

We now introduce the notion of a BV-operator on a homotopy G-algebra.

**Definition 2.** Let $V^* = \bigoplus_{n \ge 0} V^n$ be a homotopy G-algebra, let $d: V^* \to V^{*+1}$ be its differential and let $[\cdot, \cdot]: V^n \otimes V^m \to V^{m+n-1}$ be its Lie bracket. We will say that a family $\Delta = \{\Delta^n : V^n \to V^{n-1}\}_{n \ge 0}$ is a BV-operator on $V^*$ if it satisfies

$$ [f, g] - (-1)^{(n-1)m} (\Delta(f) \cdot g + (-1)^n f \cdot \Delta(g) - \Delta(f \cdot g)) \in d(V^{m+n-2}) $$

for any cocycles $f \in Z^n(V^*)$, $g \in Z^m(V^*)$.

If $V^*$ is a homotopy G-algebra equipped with a BV-operator $\Delta$, we now show that the bracket on the Gerstenhaber algebra $H(V^*)$ is determined by $\Delta$ in a manner similar to the BV-formalism.

**Theorem 3.** Let $V^* = \bigoplus_{n \ge 0} V^n$ be a homotopy G-algebra equipped with a BV-operator $\Delta = \{\Delta^n : V^n \to V^{n-1}\}_{n \ge 0}$. Consider $\bar{f} \in H^n(V^*)$ and $\bar{g} \in H^m(V^*)$ and choose cocycles $f \in Z^n(V^*)$ and $g \in Z^m(V^*)$ corresponding respectively to $\bar{f}$ and $\bar{g}$. Then, we have

$$ (\Delta(f) \cdot g + (-1)^n f \cdot \Delta(g) - \Delta(f \cdot g)) \in Z^{m+n-1}(V^*) $$

The Gerstenhaber bracket on the cohomology of $V^*$ is now determined by

$$ [\bar{f}, \bar{g}] = (-1)^{(n-1)m} (\Delta(f) \cdot g + (-1)^n f \cdot \Delta(g) - \Delta(f \cdot g)) \in H^{m+n-1}(V^*) $$

In particular, the right hand side does not depend on the choice of representatives $f$ and $g$.

**Proof.** We know that $f \in Z^n(V^*)$ and $g \in Z^m(V^*)$. Since the bracket $[\cdot, \cdot]: V^n \otimes V^m \to V^{m+n-1}$ descends to a bracket on the cohomology, it follows that $[f, g] \in Z^{m+n-1}(V^*)$. Since $\Delta = \{\Delta^n : V^n \to V^{n-1}\}_{n \ge 0}$ is a BV-operator, it follows from Definition 2 that

$$ [f, g] - (-1)^{(n-1)m} (\Delta(f) \cdot g + (-1)^n f \cdot \Delta(g) - \Delta(f \cdot g)) \in d(V^{m+n-2}) \quad (4) $$

Let us put $z_1 = [f, g]$ and $z_2 = (-1)^{(n-1)m} (\Delta(f) \cdot g + (-1)^n f \cdot \Delta(g) - \Delta(f \cdot g))$. Since $z_1 - z_2 \in d(V^{m+n-2})$, we must have $z_1 - z_2 \in Z^{m+n-1}(V^*)$. We have already seen that $z_1 \in Z^{m+n-1}(V^*)$. Hence, $z_2 \in Z^{m+n-1}(V^*)$. By (4), we know that $z_1 - z_2$ is a coboundary and hence the cohomology classes $\bar{z}_1 = \bar{z}_2$. The result is now clear. $\square$
### 3. Application : BV-operator on secondary Hochschild cohomology

Let $k$ be a field and $A$ be an algebra over $k$. Let $B$ be a commutative $k$-algebra and $\varepsilon: B \to A$ be a morphism of $k$-algebras such that $\varepsilon(B) \subseteq Z(A)$, where $Z(A)$ denotes the center of $A$. Let $M$ be an $A$-bimodule such that $\varepsilon(b)m = m\varepsilon(b)$ for all $b \in B$ and $m \in M$. Following [9, §4.2], we consider the complex $(C^*((A, B, \varepsilon); M), \delta^*)$ whose terms are given by

$$C^n((A, B, \varepsilon); M) = \operatorname{Hom}_k(A^{\otimes n} \otimes B^{\otimes \frac{n(n-1)}{2}}, M)$$

An element in $A^{\otimes n} \otimes B^{\otimes \frac{n(n-1)}{2}}$ will be expressed as a “tensor matrix” of the form

$$\otimes \begin{pmatrix} a_1 & b_{1,2} & b_{1,3} & \dots & b_{1,n-1} & b_{1,n} \\ 1 & a_2 & b_{2,3} & \dots & b_{2,n-1} & b_{2,n} \\ 1 & 1 & a_3 & \dots & b_{3,n-1} & b_{3,n} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & 1 & \dots & a_{n-1} & b_{n-1,n} \\ 1 & 1 & 1 & \dots & 1 & a_n \end{pmatrix}$$

where $a_i \in A$ and $b_{i,j} \in B$. The differentials

$$\delta^n : C^n((A, B, \varepsilon); M) \to C^{n+1}((A, B, \varepsilon); M)$$

may be described as follows

$$\begin{aligned} \delta^n(f) &= \left( \otimes \begin{pmatrix} a_1 & b_{1,2} & b_{1,3} & \dots & b_{1,n-1} & b_{1,n} \\ 1 & a_2 & b_{2,3} & \dots & b_{2,n-1} & b_{2,n} \\ 1 & 1 & a_3 & \dots & b_{3,n-1} & b_{3,n} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & 1 & \dots & a_n & b_{n,n+1} \\ 1 & 1 & 1 & \dots & 1 & a_{n+1} \end{pmatrix} \right) \\ &= a_1 \varepsilon(b_{1,2} b_{1,3} \dots b_{1,n+1}) f \left( \otimes \begin{pmatrix} a_2 & b_{2,2} & \dots & b_{2,n} & b_{2,n+1} \\ 1 & a_3 & \dots & b_{3,n} & b_{3,n+1} \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \dots & a_n & b_{n,n+1} \\ 1 & 1 & \dots & 1 & a_{n+1} \end{pmatrix} \right) \\ &\quad + \sum_{i=1}^n (-1)^i f \left( \otimes \begin{pmatrix} a_1 & b_{1,2} & \dots & b_{1,i} & b_{1,i+1} & \dots & b_{1,n} & b_{1,n+1} \\ 1 & a_2 & \dots & b_{2,i} & b_{2,i+1} & \dots & b_{2,n} & b_{2,n+1} \\ \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \dots & \varepsilon(b_{i,i+1}) a_i a_{i+1} & \dots & b_{i,n} b_{i+1,n} & b_{i,n+1} b_{i+1,n+1} \\ \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \dots & \dots & \dots & a_n & b_{n,n+1} & a_{n+1} \\ 1 & 1 & \dots & \dots & \dots & 1 & a_{n+1} & a_{n+1} \end{pmatrix} \right) \\ &\quad + (-1)^{n+1} f \left( \otimes \begin{pmatrix} a_1 & b_{1,2} & \dots & b_{1,n-1} & b_{1,n} \\ 1 & a_2 & \dots & b_{2,n-1} & b_{2,n} \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \dots & a_{n-1} & b_{n-1,n} \\ 1 & 1 & \dots & 1 & a_n \end{pmatrix} \right) \varepsilon(b_{1,n+1} b_{2,n+1} \dots b_{n,n+1}) a_{n+1} \end{aligned}$$
for $f \in C^n((A, B, \varepsilon); M)$, $a_i \in A$, $b_{i,j} \in B$. The cohomology groups of $(C^•((A, B, \varepsilon); M), \delta^•)$ are known as the secondary Hochschild cohomologies $H^n((A, B, \varepsilon); M)$ of the triple $(A, B, \varepsilon)$ with coefficients in $M$ (see [9]).

From [10, Proposition 3.1], we know that the secondary Hochschild complex $C^•(A, B, \varepsilon) := C^•((A, B, \varepsilon); A)$ carries the structure of a homotopy $G$-algebra. This induces a graded Lie bracket

$$ [\cdot, \cdot] : H^m(A, B, \varepsilon) \otimes H^n(A, B, \varepsilon) \to H^{m+n-1}(A, B, \varepsilon) \quad (5) $$

on the secondary cohomology. It follows (see [10, Corollary 3.2]) that the secondary cohomology $H^•(A, B, \varepsilon)$ carries the structure of a Gerstenhaber algebra in the sense of [3].

From now onwards, we always let $A$ be a finite dimensional $k$-algebra equipped with a symmetric, non-degenerate, invariant bilinear form $\langle \cdot, \cdot \rangle : A \times A \to k$. In particular, $\langle a_1, a_2 \rangle = \langle a_2, a_1 \rangle$, $\langle a_1 a_2, a_3 \rangle = \langle a_1, a_2 a_3 \rangle$ for any $a_1, a_2, a_3 \in A$. For $i \in \{1, \dots, n+1\}$, we define the maps $\Delta_i : C^{n+1}(A, B, \varepsilon) \to C^n(A, B, \varepsilon)$ as follows:

$$ \begin{aligned} \left\langle \Delta_i f \otimes \begin{pmatrix} a_1 & b_{1,2} & b_{1,3} & \dots & b_{1,n} \\ 1 & a_2 & b_{2,3} & \dots & b_{2,n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & 1 & \dots & b_{n-1,n} \\ 1 & 1 & 1 & \dots & a_n \end{pmatrix} \right\rangle, & a_{n+1} \\ = \left\langle f \otimes \begin{pmatrix} a_i & b_{i,i+1} & b_{i,i+2} & \dots & b_{i,n} & 1 & b_{1,i} & b_{2,i} & \dots & b_{i-1,i} \\ 1 & a_{i+1} & b_{i+1,i+2} & \dots & b_{i+1,n} & 1 & b_{1,i+1} & b_{2,i+1} & \dots & b_{i-1,i+1} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \ddots & \vdots \\ 1 & 1 & \dots & a_n & 1 & b_{1,n} & b_{2,n} & \dots & b_{i-1,n} & \vdots \\ 1 & 1 & \dots & 1 & a_{n+1} & 1 & \dots & 1 & \dots & \vdots \\ 1 & 1 & \dots & 1 & 1 & a_1 & b_{1,2} & \dots & b_{1,i-1} & \vdots \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \dots & \dots & \dots & \dots & \dots & \dots & a_{i-1} \end{pmatrix} \right\rangle, & 1 \end{aligned} $$

To clarify the above operator, let us express

$$ \otimes \begin{pmatrix} a_1 & b_{1,2} & b_{1,3} & \dots & b_{1,n} \\ 1 & a_2 & b_{2,3} & \dots & b_{2,n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & 1 & \dots & b_{n-1,n} \\ 1 & 1 & 1 & \dots & a_n \end{pmatrix} = \begin{pmatrix} U(i-1) & X_{12} \\ 1 & U(n-i-1) \end{pmatrix} $$

where $U(k)$ is a square matrix of dimension $k$. Then, we have

$$ \left\langle \Delta_i f \begin{pmatrix} U(i-1) & X_{12} \\ 1 & U(n-i-1) \end{pmatrix}, a_{n+1} \right\rangle = \left\langle f \begin{pmatrix} U(n-i-1) & 1 & X_{12}^t \\ 1 & a_{n+1} & 1 \\ 1 & 1 & U(i) \end{pmatrix}, 1 \right\rangle $$

where $X_{12}^t$ denotes the transpose of $X_{12}$. The operator $\Delta : C^{n+1}(A, B, \varepsilon) \to C^n(A, B, \varepsilon)$ is then defined as

$$ \Delta := \sum_{i=1}^{n+1} (-1)^{in} \Delta_i. $$

Following [10, §3], we know that the complex $C^•(A, B, \varepsilon)$ carries a dot product of degree 0, i.e., for $f \in C^n(A, B, \varepsilon)$, $g \in C^m(A, B, \varepsilon)$, we have $f \cdot g \in C^{m+n}(A, B, \varepsilon)$. We also consider the operations

$$ o_i : C^n(A, B, \varepsilon) \otimes C^m(A, B, \varepsilon) \to C^{m+n-1}(A, B, \varepsilon) $$
and set $f \circ g := \sum_{i=1}^{n} (-1)^{(i-1)(m-1)} f \circ_i g$ as in [10, §3]. We also set

$$ \rho^1, \rho^2 : C^n(A, B, \varepsilon) \otimes C^m(A, B, \varepsilon) \to C^{n+m-1}(A, B, \varepsilon) $$

$$ \rho^1(f \otimes g) := \sum_{i=1}^{m} (-1)^{i(n+m-1)} \Delta_i(f \cdot g) \quad \rho^2(f \otimes g) := \sum_{i=m+1}^{m+n} (-1)^{i(n+m-1)} \Delta_i(f \cdot g) $$

for $f \in C^n(A, B, \varepsilon)$, $g \in C^m(A, B, \varepsilon)$. It is clear that $\rho^1(f \otimes g) + \rho^2(f \otimes g) = \Delta(f \cdot g)$.

**Lemma 4.** $\rho^1(f \otimes g) = (-1)^{nm} \rho^2(g \otimes f)$ for all $f \in C^n(A, B, \varepsilon)$ and $g \in C^m(A, B, \varepsilon)$.

**Proof.** This may be verified by direct computation.

<figure><img src="image_3.png" /></figure>

**Lemma 5.** Let $f \in Z^n(A, B, \varepsilon)$, $g \in Z^m(A, B, \varepsilon)$. Then $f \circ g = (-1)^{(n-1)m} \Delta(f) \cdot g + (-1)^{(n-1)m} \rho^2(f \otimes g)$ is a coboundary. In fact, if we define $H$

$$ H = \sum_{i,j \ge 1, i+j \le n} (-1)^{(j-1)(m-1)+i(n+m)+1} \Delta_i(f \circ_j g), $$

then,

$$ \delta H = f \circ g - (-1)^{(n-1)m} \Delta(f) \cdot g + (-1)^{(n-1)m} \rho^2(f \otimes g). $$

**Proof.** We set, for $k \ge 0, p \ge 0$:

$$ T_{k+p}^k = \otimes \begin{pmatrix} a_{k+1} & \dots & b_{k+1,k+p} \\ \vdots & \vdots & \vdots \\ 1 & \dots & a_{k+p} \end{pmatrix} $$

We see that

$$ \left\langle \delta(\Delta_i(f \circ_j g)) \right\rangle \left( \otimes \begin{pmatrix} a_1 & b_{1,2} & b_{1,3} & \dots & b_{1,n+m-1} \\ 1 & a_2 & b_{2,3} & \dots & b_{2,n+m-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & 1 & \dots & b_{n+m-2,n+m-1} \\ 1 & 1 & 1 & \dots & a_{n+m-1} \end{pmatrix} \right), a_{n+m} \rangle $$

$$ = \left\langle f \left( \begin{array}{ccccccc} a_{i+1} & \dots & b_{i+1,i+j-1} & \prod_{k=0}^{m-1} b_{i+1,i+j+k} & \dots & 1 & b_{2,i+1} & \dots & b_{i-1,i+1} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 1 & \dots & a_{i+j-1} & \prod_{k=0}^{m-1} b_{i+j-1,i+j+k} & \dots & 1 & b_{2,i+j-1} & \dots & b_{i-1,i+j-1} \\ 1 & \dots & 1 & g(T_{i+j+m-1}^{i+j-1}) & \prod_{k=0}^{m-1} b_{i+j+k,i+j+m} & 1 & \dots & \dots & \prod_{k=0}^{m-1} b_{i-1,i+j+k} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 1 & \dots & 1 & 1 & 1 & \alpha & 1 & \dots & 1 \\ 1 & \dots & 1 & 1 & 1 & a_2 & \dots & b_{2,i} & \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 1 & \dots & 1 & 1 & 1 & 1 & 1 & \dots & a_i \end{array} \right) , 1 \right\rangle $$

$$ + \sum_{\lambda=1}^{i-1} (-1)^{\lambda} \left\langle f \left( \begin{array}{ccccccc} a_{i+1} & \dots & \prod_{k=0}^{m-1} b_{i+1,i+j+k} & \dots & 1 & b_{1,i+1} & b_{\lambda,i+1} & b_{1,i+1} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 1 & \dots & \prod_{k=0}^{m-1} b_{i+j-1,i+j+k} & \dots & 1 & b_{1,i+j-1} & b_{\lambda,i+j-1} & b_{1,i+j-1} \\ 1 & \dots & g(T_{i+j+m-1}^{i+j-1}) & \prod_{k=0}^{m-1} b_{i+j+k,i+j+m} & \dots & \prod_{k=0}^{m-1} b_{1,i+j+k} & \prod_{k=0}^{m-1} b_{\lambda,i+j+k} & \prod_{k=0}^{m-1} b_{1,i+j+k} \\ \vdots & \vdots & \vdots & \vdots & 1 & \vdots & \vdots & \vdots \\ 1 & 1 & \dots & \dots & a_{n+m} & 1 & \dots & 1 \\ 1 & 1 & \dots & \dots & 1 & a_1 & \dots & b_{1,i} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 1 & 1 & \dots & 1 & 1 & 1 & \beta_{\lambda} & \dots b_{\lambda,i} b_{\lambda+1,i} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 1 & 1 & \dots & 1 & 1 & 1 & 1 & a_i \end{array} \right) , 1 \right\rangle $$
We write the entire expression of (6) as

$$
\left\langle \delta(\Delta_i(f \circ_j g)) \otimes \begin{pmatrix} a_1 & b_{1,2} & b_{1,3} & \dots & b_{1,n+m-1} \\ 1 & a_2 & b_{2,3} & \dots & b_{2,n+m-1} \\ \vdots & \vdots & \vdots & & \vdots \\ 1 & 1 & 1 & \dots & b_{n+m-2,n+m-1} \\ 1 & 1 & 1 & \dots & a_{n+m-1} \end{pmatrix} \right\rangle, a_{n+m} = E_1 + E_2 + E_3 + E_4 + E_5 + E_6,
$$

where $E_k$ denotes the $k$-th term in the expression.
We set for $i, j \ge 1$ and $i + j \le n$,

$$
A_{i,j} := (-1)^{i+1} \left\langle a'_i f \left( \begin{array}{ccccccc} a_{i+1} & \cdots & b_{i+1,i+j-1} & \prod_{k=0}^{m-1} b_{i+1,i+j+k} & \cdots & 1 & \cdots & b_{i-1,i+1} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 1 & \cdots & a_{i+j-1} & \prod_{k=0}^{m-1} b_{i+j-1,i+j+k} & \cdots & 1 & \cdots & b_{i-1,i-j-1} \\ 1 & \cdots & g(T_{i+j+m-1}^{i+j-1}) & \prod_{k=0}^{m-1} b_{i+j+k,i+j+m} & \cdots & 1 & \cdots & \prod_{k=0}^{m-1} b_{i-1,i+j+k} \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\ 1 & \cdots & \cdots & \cdots & \cdots & a_{n+m} & 1 & 1 \\ 1 & 1 & \cdots & \cdots & \cdots & a_1 & \cdots & b_{1,i-1} \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\ 1 & 1 & \cdots & \cdots & \cdots & 1 & \cdots & a_{i-1} \end{array} \right) \right\rangle, 1
$$

$$
+ E_3 + (-1)^{i+j-1} \left\langle f \left( \begin{array}{ccccccc} a_i & \cdots & b_{i,i+j-2} & \prod_{k=1}^{m-2} b_{i,i+j+k} & \cdots & 1 & \cdots & b_{i-1,i} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 1 & \cdots & a_{i+j-2} & \prod_{k=1}^{m-2} b_{i+j-2,i+j+k} & \cdots & 1 & \cdots & b_{i-1,i-j-2} \\ 1 & \cdots & \eta & \prod_{k=0}^{m-1} b_{i+j+k,i+j+m} & \cdots & 1 & \cdots & \prod_{k=0}^{m-1} b_{i-1,i+j+k} \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\ 1 & \cdots & \cdots & \cdots & \cdots & a_{n+m} & 1 & 1 \\ 1 & 1 & \cdots & \cdots & \cdots & a_1 & \cdots & b_{1,i-1} \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\ 1 & 1 & 1 & \cdots & \cdots & 1 & \cdots & a_{i-1} \end{array} \right) \right\rangle, 1
$$

where $a'_i = a_i \epsilon(b_{i,i+1} \dots b_{i,n+m-1} b_{1,i} \dots b_{i-1,i})$. We also set

$$
B_{i,j} := (-1)^{i+j+m-2} \left\langle f \left( \begin{array}{ccccccc} a_i & \cdots & b_{i,i+j-2} & \prod_{k=1}^{m-2} b_{i,i+j+k} & \cdots & 1 & \cdots & b_{i-1,i} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 1 & \cdots & a_{i+j-2} & \prod_{k=1}^{m-2} b_{i+j-2,i+j+k} & \cdots & 1 & \cdots & b_{i-1,i+j-2} \\ 1 & \cdots & \zeta & \prod_{k=1}^{m-2} b_{i+j+k,i+j+m-1} & \cdots & 1 & \cdots & \prod_{k=1}^{m-2} b_{i-1,i+j+k} \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\ 1 & \cdots & \cdots & \cdots & \cdots & a_{n+m} & 1 & 1 \\ 1 & 1 & \cdots & \cdots & \cdots & a_1 & \cdots & b_{1,i-1} \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\ 1 & 1 & 1 & \cdots & \cdots & 1 & \cdots & a_{i-1} \end{array} \right) \right\rangle + E_5 + E_6
$$

where

$$
\eta = a_{i+j-1} \epsilon (b_{i+j-1, i+j} \dots b_{i+j-1, n+m-1}) g \left( T_{i+j+m-1}^{i+j-1} \right)
$$

$$
\zeta = g \left( T_{i+j+m-2}^{i+j-2} \right) \epsilon \left( \prod_{k=-1}^{m-2} b_{i+j+k, i+j+m-1} \right),
$$
and

$$
C_{i,j} := (-1)^i \left\langle f \begin{pmatrix}
a_{i+1} \dots b_{i+1,i+j-1} & \prod_{k=0}^{m-1} b_{i+1,i+j+k} & \dots & 1 & \dots & b_{i-1,i+1} \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
1 & \dots & a_{i+j-1} & \prod_{k=0}^{m-1} b_{i+j-1,i+j+k} & \dots & 1 & \dots & b_{i-1,i+j-1} \\
1 & \dots & \dots & g(T_{i+j-m-1}^{i+j-1}) & \prod_{k=0}^{m-1} b_{i+j+k,i+j+m} & \dots & 1 & \dots & \prod_{k=0}^{m-1} b_{i-1,i+j+k} \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
1 & \dots & \dots & \dots & \dots & a_{n+m} & 1 & \dots & 1 \\
1 & 1 & \dots & \dots & \dots & \dots & a_1 & \dots & b_{1,i-1} \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
1 & 1 & \dots & \dots & \dots & \dots & \dots & a_{i-1} & \dots \\
\end{pmatrix} \middle\rangle a'_i, 1 \right\rangle + E_1 + E_2
$$

The first term of $A_{i,j}$ and that of $C_{i,j}$ are the same modulo a sign. Using the fact that $\delta g = 0$, the third term of $A_{i,j}$ and the first term of $B_{i,j}$ add up to give $E_4$. Thus, we have

$$
\left\langle \delta(\Delta_i(f \circ_j g)) \right\rangle \left( \otimes \left( \begin{pmatrix} a_1 & b_{1,2} & b_{1,3} & \dots & b_{1,n+m-1} \\ 1 & a_2 & b_{2,3} & \dots & b_{2,n+m-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & 1 & \dots & b_{n+m-2,n+m-1} \\ 1 & 1 & 1 & \dots & a_{n+m-1} \end{pmatrix} \right) \right), a_{n+m} \rangle = A_{i,j} + B_{i,j} + C_{i,j} \quad (7)
$$

It may be verified that

$$
(-1)^{i+1} A_{i,j-1} + (-1)^{i+m} B_{i,j} + (-1)^{i+n} C_{i-1,j} \\
= \left\langle \Delta_i((\delta f) \circ_j g) \right\rangle \left( \otimes \left( \begin{pmatrix} a_1 & b_{1,2} & b_{1,3} & \dots & b_{1,n+m-1} \\ 1 & a_2 & b_{2,3} & \dots & b_{2,n+m-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & 1 & \dots & a_{m+n-1} \end{pmatrix} \right) \right), a_{n+m} \rangle = 0 \quad (8)
$$

for $2 \le i \le n$, $2 \le j \le n-1$ and $i+j \le n$. The second equality in (8) uses the fact that $\delta f = 0$.
For $i,j \in \{1, \dots, n+1\}$, we define

$$
A_{i,0} := (-1)^{i+1} \left\langle g \left( T_{i+m-1}^{i-1} \right) f \begin{pmatrix} a_{i+m} & b_{i+m,i+m+1} & \dots & b_{i+m,n+m-1} & 1 & b_{1,i+m} & b_{2,i+m} & \dots & b_{i-1,i+m} \\ 1 & a_{i+m+1} & \dots & b_{i+m+1,n+m-1} & 1 & b_{2,i+m+1} & b_{2,i+m+1} & \dots & b_{i-1,i+m+1} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & \dots & a_{n+m} & 1 & \dots & 1 & \dots & 1 \\ 1 & 1 & \dots & \dots & 1 & a_1 & b_{1,2} & \dots & b_{1,i-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & \dots & \dots & \dots & \dots & \dots & a_{i-1} & \dots \end{pmatrix} \middle\rangle, 1 \right\rangle,
$$

$$
C_{0,j} = \left\langle f \begin{pmatrix} a_1 \dots b_{1,j-1} & \prod_{k=0}^{m-1} b_{1,j+k} & b_{1,j+m} & \dots & b_{1,n+m-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 \dots a_{j-1} & \prod_{k=0}^{m-1} b_{j-1,j+k} & b_{j-1,j+m} & \dots & b_{j-1,n+m-1} \\ 1 \dots 1 & g(T_{j+m-1}^{j-1}) & \prod_{k=0}^{m-1} b_{j+k,j+m} & \dots & \prod_{k=0}^{m-1} b_{j+k,n+m-1} \\ 1 \dots 1 & 1 & a_{j+m} & \dots & b_{j+m,n+m-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 \dots \dots & \dots & \dots & \dots & a_{n+m-1} \end{pmatrix} \middle\rangle, a_{n+m}, 1 \right\rangle,
$$
and for $i \in \{1, \dots, n\}$, define

$$ B_{i,n-i+1} := (-1)^{n+m+1} \left\langle f \left( \begin{array}{ccccccc} a_i & \cdots & b_{i,n-1} & \prod_{k=1}^{m} b_{i,n-1+k} & b_{1,i} & \cdots & b_{i-1,i} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & \cdots & a_{n-1} & \prod_{k=1}^{m} b_{n-1,n-1+k} & b_{1,n-1} & \cdots & b_{i-1,n-1} \\ 1 & \cdots & 1 & g(T_{n+m-1}^{n-1}) \cdot a_{n+m} & \prod_{k=0}^{m-1} b_{1,n+k} & \cdots & \prod_{k=0}^{m-1} b_{i-1,n+k} \\ 1 & \cdots & 1 & 1 & a_1 & \cdots & b_{1,i-1} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & \cdots & \cdots & \cdots & \cdots & \cdots & a_{i-1} \end{array} \right) , 1 \right\rangle $$

Thus, $A_{i,j}$, $B_{i,j+1}$, $C_{i-1,j}$ are defined for all the values of $i, j$ with $i, j \ge 1$ and $i + j \le n + 1$. Moreover, it may be verified that

$$ A_{1,j-1} + (-1)^{m+1} B_{1,j} + (-1)^{n+1} C_{0,j} = \left\langle (\delta f) \left( \begin{array}{ccccccc} a_1 & \cdots & b_{1,j-1} & \prod_{k=0}^{m-1} b_{1,j+k} & b_{1,j+m} & \cdots & b_{1,n+m-1} & 1 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & 1 \\ 1 & \cdots & a_{j-1} & \prod_{k=0}^{m-1} b_{j-1,j+k} & b_{j-1,j+m} & \cdots & b_{j-1,n+m-1} & 1 \\ 1 & \cdots & 1 & g(T_{j+m-1}^{j-1}) & \prod_{k=0}^{m-1} b_{j+k,j+m} & \cdots & \prod_{k=0}^{m-1} b_{j+k,n+m-1} & 1 \\ 1 & \cdots & 1 & 1 & a_{j+m} & \cdots & b_{j+m,n+m-1} & 1 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & 1 \\ 1 & \cdots & \cdots & \cdots & \cdots & \cdots & a_{n+m-1} & 1 \\ 1 & \cdots & 1 & 1 & 1 & \cdots & 1 & a_{n+m} \end{array} \right) , 1 \right\rangle = 0 $$

We also have

$$ \begin{aligned} & (-1)^{i+1} A_{i,0} + (-1)^{i+m} B_{i,1} + (-1)^{i+n} C_{i-1,1} \\ &= \left\langle (\delta f) \left( \begin{array}{ccccccc} g(T_{i+m-1}^{i-1}) & \prod_{k=0}^{m-1} b_{i+k,i+m} & \cdots & \prod_{k=0}^{m-1} b_{i+k,n+m-1} & 1 & \prod_{k=0}^{m-1} b_{1,i+k} & \cdots & \prod_{k=0}^{m-1} b_{i-1,i+k} \\ 1 & a_{i+m} & \cdots & b_{i+m,n+m-1} & 1 & b_{1,i+m} & \cdots & \prod_{k=0}^{m-1} b_{i-1,i+m} \\ \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \ddots & \vdots \\ \cdots & \cdots & \cdots & a_{n+m-1} & 1 & b_{1,n+m-1} & \cdots & b_{i-1,n+m-1} \\ \cdots & \cdots & \cdots & 1 & a_{n+m} & 1 & \cdots & 1 \\ \cdots & \cdots & \cdots & 1 & 1 & a_1 & \cdots & b_{1,i-1} \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \cdots & \cdots & \cdots & 1 & 1 & 1 & \cdots & a_{i-1} \end{array} \right) , 1 \right\rangle = 0 \end{aligned} $$

and

$$ \begin{aligned} & (-1)^{i+1} A_{i,n-i} + (-1)^{i+m} B_{i,n-i+1} + (-1)^{i+n} C_{i-1,n-i+1} \\ &= \left\langle (\delta f) \left( \begin{array}{ccccccc} a_i & \cdots & b_{i,n-1} & \prod_{k=0}^{m-1} b_{i,n+k} & 1 & b_{1,i} & \cdots & b_{i-1,i} \\ 1 & a_{i+1} & \cdots & b_{i+1,n-1} & \prod_{k=0}^{m-1} b_{i+1,n+k} & 1 & b_{1,i+1} & \cdots & b_{i-1,i+1} \\ \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\ 1 & \cdots & a_{n-1} & \prod_{k=0}^{m-1} b_{n-1,n+k} & 1 & \cdots & \cdots & b_{i-1,n-1} \\ 1 & \cdots & \cdots & g(T_{n+m-1}^{n-1}) & 1 & \prod_{k=0}^{m-1} b_{1,n+k} & \cdots & \prod_{k=0}^{m-1} b_{i-1,n+k} \\ 1 & \cdots & \cdots & \cdots & a_{n+m} & 1 & \cdots & 1 \\ 1 & 1 & \cdots & \cdots & \cdots & a_1 & \cdots & b_{1,i-1} \\ \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ 1 & 1 & \cdots & \cdots & \cdots & \cdots & \cdots & a_{i-1} \end{array} \right) , 1 \right\rangle \end{aligned} $$
Thus, we obtain

$$0 = \sum_{1 \le i \le n, 1 \le j \le n-1, i+j \le n+1} (-1)^{(j-1)(m-1)+i(n+m-1)} \left( (-1)^{i+1} A_{i,j-1} + (-1)^{i+m} B_{i,j} + (-1)^{i+n} C_{i-1,j} \right) \quad (9)$$

Rearranging the terms in the above sum, and using equation (7), we get,

$$0 = \sum_{1 \le i \le n, 1 \le j \le n, i+j \le n} (-1)^{(j-1)(m-1)+i(n+m)+1} (A_{i,j} + B_{i,j} + C_{i,j}) \\ - \sum_{i=1}^{n} (-1)^{m-1+i(n+m)} A_{i,0} - \sum_{i=1}^{n} (-1)^{n(m+1)+i(n+1)} B_{i,n-i+1} - \sum_{j=1}^{n} (-1)^{(j-1)(m-1)} C_{0,j}$$

$$= \left\langle \delta(H) \begin{pmatrix} a_1 & b_{1,2} & b_{1,3} & \dots & b_{1,n+m-1} \\ 1 & a_2 & b_{2,3} & \dots & b_{2,n+m-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & 1 & \dots & b_{n+m-2,n+m-1} \\ 1 & 1 & 1 & \dots & a_{n+m-1} \end{pmatrix}, a_{n+m} \right\rangle$$

$$-(-1)^{m(n+1)} \langle \rho^2(f \otimes g) \begin{pmatrix} a_1 & b_{1,2} & b_{1,3} & \dots & b_{1,n+m-1} \\ 1 & a_2 & b_{2,3} & \dots & b_{2,n+m-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & 1 & \dots & b_{n+m-2,n+m-1} \\ 1 & 1 & 1 & \dots & a_{n+m-1} \end{pmatrix}, a_{n+m} \rangle$$

$$-(-1)^{m(n+1)} \langle (\Delta(f) \cdot g) \begin{pmatrix} a_1 & b_{1,2} & b_{1,3} & \dots & b_{1,n+m-1} \\ 1 & a_2 & b_{2,3} & \dots & b_{2,n+m-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & 1 & \dots & b_{n+m-2,n+m-1} \\ 1 & 1 & 1 & \dots & a_{n+m-1} \end{pmatrix}, a_{n+m} \rangle$$

$$- \langle (f \circ g) \begin{pmatrix} a_1 & b_{1,2} & b_{1,3} & \dots & b_{1,n+m-1} \\ 1 & a_2 & b_{2,3} & \dots & b_{2,n+m-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & 1 & \dots & b_{n+m-2,n+m-1} \\ 1 & 1 & 1 & \dots & a_{n+m-1} \end{pmatrix}, a_{n+m} \rangle.$$

□

**Proposition 6.** The family $\Delta = \{\Delta^* : C^*(A, B, \varepsilon) \to C^{*-1}(A, B, \varepsilon)\}$ determines a BV-operator on the homotopy $G$-algebra $C^*(A, B, \varepsilon)$.

**Proof.** We consider $f \in Z^n(A, B, \varepsilon)$ and $g \in Z^m(A, B, \varepsilon)$. By definition (see [10, §3]), we know that

$$[f, g] = f \circ g - (-1)^{(n-1)(m-1)} g \circ f \in C^{m+n-1}(A, B, \varepsilon) \quad (10)$$

Applying Lemma 5, we know that the cochains

$$f \circ g - (-1)^{(n-1)m} \Delta(f) \cdot g + (-1)^{(n-1)m} \rho^2(f \otimes g)$$

$$g \circ f - (-1)^{(m-1)n} \Delta(g) \cdot f + (-1)^{(m-1)n} \rho^2(g \otimes f)$$

are coboundaries. From (10), it now follows that

$$[f, g] - (-1)^{(n-1)m} \Delta(f) \cdot g + (-1)^{(n-1)m} \rho^2(f \otimes g) + (-1)^{(m-1)} \Delta(g) \cdot f + (-1)^m \rho^2(g \otimes f) \quad (11)$$

is a coboundary. Applying Lemma 4, it follows from (11) that

$$[f, g] - (-1)^{(n-1)m} \Delta(f) \cdot g + (-1)^{m(n-1)} \rho^2(f \otimes g) + (-1)^{(m-1)} \Delta(g) \cdot f + (-1)^{m(n-1)} \rho^1(f \otimes g)$$

is a coboundary. Since $\rho^1(f \otimes g) + \rho^2(f \otimes g) = \Delta(f \cdot g)$, we get

$$[f, g] - (-1)^{(n-1)m} \Delta(f) \cdot g + (-1)^{m(n-1)} \Delta(f \cdot g) + (-1)^{(m-1)} \Delta(g) \cdot f$$
is a coboundary. Using the fact that the dot product is graded commutative, we can put $\Delta(g) \cdot f = (-1)^{n(m-1)} f \cdot \Delta(g)$. The result is now clear. $\square$

**Theorem 7.** For secondary cohomology classes $\bar{f} \in H^n(A, B, \varepsilon)$ and $\bar{g} \in H^m(A, B, \varepsilon)$, the Gersten-haber bracket is determined by

$$[\bar{f}, \bar{g}] = (-1)^{(n-1)m} \overline{(\Delta(f) \cdot g + (-1)^n f \cdot \Delta(g) - \Delta(f \cdot g))} \in H^{m+n-1}(A, B, \varepsilon)$$

Here $f$ and $g$ are any cocycles representing the classes $\bar{f}$ and $\bar{g}$ respectively.

**Proof.** This follows directly by applying Theorem 3 and Proposition 6. $\square$

It is natural to ask whether the BV-operator determined by $\Delta = \{\Delta^\bullet : C^\bullet(A, B, \varepsilon) \to C^{\bullet-1}(A, B, \varepsilon)\}$ induces a BV-algebra structure on the secondary Hochschild cohomology $H^\bullet(A, B, \varepsilon)$. In the special case of $B = k$, we are reduced to ordinary Hochschild cohomology and hence $\Delta$ determines a BV-algebra structure on $H^\bullet(A, B)$. However, this is not true in general because $\Delta^\bullet : C^\bullet(A, B, \varepsilon) \to C^{\bullet-1}(A, B, \varepsilon)$ does not commute with the differentials. For instance, we take $n = 2$ and ask when the following diagram commutes:

$$\begin{tikzcd}
C^2(A, B, \varepsilon) \arrow[r, "\delta^2"] \arrow[d, "\Delta^2"] & C^3(A, B, \varepsilon) \arrow[d, "-\Delta^3"'] \\
C^1(A, B, \varepsilon) \arrow[r, "\delta^1"] & C^2(A, B, \varepsilon)
\end{tikzcd} \qquad (12)$$

For any $a_1, a_2, a_3 \in A$, $b_{1,2} \in B$ and $f \in C^2(A, B, \varepsilon)$, we have

$$\begin{align*}
& \langle (-\Delta^3 \delta^2 f) \begin{pmatrix} a_1 & b_{1,2} \\ a_2 & \end{pmatrix}, a_3 \rangle \\
&= - \langle (\delta^2 f) \begin{pmatrix} a_1 & b_{1,2} & 1 \\ a_2 & 1 & \\ a_3 & & \end{pmatrix}, 1 \rangle - \langle (\delta^2 f) \begin{pmatrix} a_2 & 1 & b_{1,2} \\ a_3 & 1 & \\ a_1 & & \end{pmatrix}, 1 \rangle - \langle (\delta^2 f) \begin{pmatrix} a_3 & 1 & 1 \\ a_1 & b_{1,2} & \\ a_2 & & \end{pmatrix}, 1 \rangle \\
&= - \langle a_1 \varepsilon(b_{1,2}) f \begin{pmatrix} a_2 & 1 \\ a_3 & \end{pmatrix}, 1 \rangle + \langle f \begin{pmatrix} a_1 a_2 \varepsilon(b_{1,2}) & 1 \\ a_3 & \end{pmatrix}, 1 \rangle - \langle f \begin{pmatrix} a_1 & b_{1,2} \\ a_2 a_3 & \end{pmatrix}, 1 \rangle + \langle f \begin{pmatrix} a_1 & b_{1,2} \\ a_2 & \end{pmatrix}, a_3, 1 \rangle \\
&\quad - \langle a_2 \varepsilon(b_{1,2}) f \begin{pmatrix} a_3 & 1 \\ a_1 & \end{pmatrix}, 1 \rangle + \langle f \begin{pmatrix} a_2 a_3 & b_{1,2} \\ a_1 & \end{pmatrix}, 1 \rangle - \langle f \begin{pmatrix} a_2 & b_{1,2} \\ a_3 a_1 & \end{pmatrix}, 1 \rangle + \langle f \begin{pmatrix} a_2 & 1 \\ a_3 & \end{pmatrix} a_1 \varepsilon(b_{1,2}), 1 \rangle \\
&\quad - \langle a_3 f \begin{pmatrix} a_1 & b_{1,2} \\ a_2 & \end{pmatrix}, 1 \rangle + \langle f \begin{pmatrix} a_3 a_1 & b_{1,2} \\ a_2 & \end{pmatrix}, 1 \rangle - \langle f \begin{pmatrix} a_3 & 1 \\ a_1 a_2 \varepsilon(b_{1,2}) & \end{pmatrix}, 1 \rangle + \langle f \begin{pmatrix} a_3 & 1 \\ a_1 & \end{pmatrix} a_2 \varepsilon(b_{1,2}), 1 \rangle
\end{align*}$$

Now, using the properties of the inner product $\langle \cdot, \cdot \rangle$ on $A$ we see that the first term cancels with the eighth, the fourth term cancels with the ninth, the fifth term cancels with the twelfth. Thus, the above expression reduces to

$$\langle f \begin{pmatrix} a_1 & a_2 \varepsilon(b_{1,2}) & 1 \\ & a_3 & \end{pmatrix}, 1 \rangle - \langle f \begin{pmatrix} a_1 & b_{1,2} \\ a_2 a_3 & \end{pmatrix}, 1 \rangle + \langle f \begin{pmatrix} a_2 & a_3 & b_{1,2} \\ a_1 & & \end{pmatrix}, 1 \rangle - \langle f \begin{pmatrix} a_2 & b_{1,2} \\ a_3 a_1 & \end{pmatrix}, 1 \rangle \\
+ \langle f \begin{pmatrix} a_3 & a_1 & b_{1,2} \\ a_2 & & \end{pmatrix}, 1 \rangle - \langle f \begin{pmatrix} a_3 & 1 \\ a_1 a_2 \varepsilon(b_{1,2}) & \end{pmatrix}, 1 \rangle \quad (13)$$
On the other hand, we have

$$
\begin{align}
& \langle (\delta^1 \Delta^2 f) \begin{pmatrix} a_1 & b_{1,2} \\ a_2 & \end{pmatrix}, a_3 \rangle \nonumber \\
&= \langle a_1 \varepsilon(b_{1,2}) (\Delta^2 f)(a_2), a_3 \rangle - \langle (\Delta^2 f)(a_1 a_2 \varepsilon(b_{1,2})), a_3 \rangle + \langle (\Delta^2 f)(a_1) a_2 \varepsilon(b_{1,2}), a_3 \rangle \nonumber \\
&= \langle (\Delta^2 f)(a_2), a_3 a_1 \varepsilon(b_{1,2}) \rangle - \langle (\Delta^2 f)(a_1 a_2 \varepsilon(b_{1,2})), a_3 \rangle + \langle (\Delta^2 f)(a_1), a_2 a_3 \varepsilon(b_{1,2}) \rangle \nonumber \\
&= - \langle f \begin{pmatrix} a_2 & 1 \\ a_3 & a_1 \varepsilon(b_{1,2}) \end{pmatrix}, 1 \rangle + \langle f \begin{pmatrix} a_3 a_1 \varepsilon(b_{1,2}) & 1 \\ a_2 & \end{pmatrix}, 1 \rangle + \langle f \begin{pmatrix} a_1 a_2 \varepsilon(b_{1,2}) & 1 \\ a_3 & \end{pmatrix}, 1 \rangle \nonumber \\
&\quad - \langle f \begin{pmatrix} a_3 & 1 \\ a_1 & a_2 \varepsilon(b_{1,2}) \end{pmatrix}, 1 \rangle - \langle f \begin{pmatrix} a_1 & 1 \\ a_2 a_3 \varepsilon(b_{1,2}) \end{pmatrix}, 1 \rangle + \langle f \begin{pmatrix} a_2 a_3 \varepsilon(b_{1,2}) & 1 \\ a_1 & \end{pmatrix}, 1 \rangle \tag{14}
\end{align}
$$

From the expressions in (13) and (14), it is clear that the diagram (12) does not commute in general, even if we take A to be commutative and B = A.

4. Relation with extra degeneracy and norm operator

We continue with A being a finite dimensional k-algebra equipped with a symmetric, non-
degenerate and invariant bilinear form ⟨·,·⟩ : A × A → k and B a commutative k-algebra with
a morphism of k-algebras ε : B → A such that ε(B) ⊆ Z(A). In particular, the non-degenerate
pairing on A induces mutually inverse isomorphisms

$$
\phi : A^* \xrightarrow{\cong} A \qquad \phi^{-1} : A \xrightarrow{\cong} A^* \tag{15}
$$

We let $C^*(A, M)$ denote the ordinary Hochschild complex of $A$ with coefficients in an $A$-bimodule $M$ and its cohomology by $H^*(A, M)$. In particular, we may set $M = A^*$ equipped with the $A$-bimodule structure $(a'f a'')(a) := f(a''aa')$ for $f \in A^* = \operatorname{Hom}(A, k)$ and $a, a', a'' \in A$. In that case, the terms $\{C^n(A, A^*)\}_{n \ge 0}$ in the Hochschild complex $C^*(A, A^*)$ may also be written as $C^n(A, A^*) \cong \operatorname{Hom}(A^{\otimes n+1}, k)$. We denote by $\hat{C}^*(A)$ the corresponding complex defined by setting $\hat{C}^n(A) := \operatorname{Hom}(A^{\otimes (n+1)}, k)$ for $n \ge 0$.

From Tradler [11], we know that the operator $\Delta^* : C^*(A, A) \to C^{*-1}(A, A)$ on Hochschild cochains inducing the BV-structure on $H^*(A, A)$ fits into the following commutative diagram with the duals of Hochschild chains

$$
\begin{tikzcd}[column sep=2.8em, row sep=2.8em]
\hat{C}^{\bullet+1}(A) \arrow[r, "N_s"] & \hat{C}^{\bullet}(A) \arrow[d, "\cong"] \\
C^{\bullet}(A, A^{*}) & C^{\bullet-1}(A, A^{*}) \arrow[d, "\cong"] \\
\phi^{\bullet} \arrow[r, "\cong", "\Delta^{\bullet}"] \arrow[d, "\cong"] & \phi^{\bullet-1}
\arrow[r, "C^{\bullet-1}(A, A)"', "\cong"'] & C^{\bullet-1}(A, A)
\end{tikzcd}
\qquad (16)
$$

Here, each $\phi^* : C^*(A, A^*) \to C^*(A, A)$ is the isomorphism induced by $\phi : A^* \xrightarrow{\cong} A$, while $s$ and $N$
respectively are the usual extra degeneracy and norm operators given by

$$
\begin{align}
s : \hat{C}^{n+1}(A) &\longrightarrow \hat{C}^n(A) && (sf)(a_1, \dots, a_n) := f(a_1, \dots, a_n, 1) \nonumber \\
N := 1 + \lambda + \dots + \lambda^n : \hat{C}^n(A) &\longrightarrow \hat{C}^n(A) && (\lambda \cdot f)(a_0, \dots, a_n) := (-1)^n f(a_n, a_0, \dots, a_{n-1}) \tag{17}
\end{align}
$$

If we pass to the normalized Hochschild complex which is a quasi-isomorphic subcomplex of $C^*(A, A^*)$, then (16) induces the following commutative diagram

$$
\begin{tikzcd}[column sep=2.8em, row sep=2.8em]
H^*(A, A^*) \arrow[r, "B^*", "\cong"] & H^{\bullet-1}(A, A^*) \arrow[d, "\cong"] \\
\phi^* \arrow[r, "\cong", "\Delta^*"'] \arrow[d, "\cong"] & \phi^{\bullet-1} \arrow[d, "\cong"] \\
H^*(A, A) \arrow[r, "\cong", "\Delta^*"'] & H^{\bullet-1}(A, A)
\end{tikzcd}
\qquad (18)
$$
where $B^\bullet : H^*(A, A^*) \to H^{*-1}(A, A^*)$ is the standard Connes operator.

In the case of secondary Hochschild cohomology, we have shown in Section 3 that $\Delta^\bullet$ is not in general a morphism of complexes, i.e., it does not descend to cohomology. We will now show that the operator $\Delta^\bullet$ on secondary Hochschild cohomology $H^*(A, B, \varepsilon)$ fits into a commutative diagram similar to (18).

In [8], Laubacher, Staic and Stancu have introduced a co-simplicial module $\bar{C}^*(A, B, \varepsilon)$ which is used to compute the secondary Hochschild cohomology associated to the triple $(A, B, \varepsilon)$. The terms of this co-simplicial module are given by

$$ \bar{C}^*(A, B, \varepsilon) := \{ \operatorname{Hom} \left( A^{\otimes n} \otimes B^{\otimes \frac{n(n-1)}{2}}, \operatorname{Hom}(A \otimes B^n, k) \right) \}_{n \ge 0} \quad (19) $$

It is important to note (see [8, Remark 4.7]) that despite the similar names, the complex $\bar{C}^*(A, B, \varepsilon)$ cannot be expressed as the secondary Hochschild complex of $(A, B, \varepsilon)$ with coefficients in some $A$-bimodule. This is because the “coefficient module” $\operatorname{Hom}(A \otimes B^n, k)$ appearing in (19) varies with $n$.

In addition, the cosimplicial module $\bar{C}^*(A, B, \varepsilon)$ is equipped with a cyclic operator, which can be used to compute the secondary cyclic cohomology associated to the triple $(A, B, \varepsilon)$. Using the isomorphisms

$$ \Psi^n : \operatorname{Hom} \left( A^{\otimes n} \otimes B^{\otimes \frac{n(n-1)}{2}}, \operatorname{Hom}(A \otimes B^n, k) \right) \xrightarrow{\cong} \operatorname{Hom} \left( A^{\otimes (n+1)} \otimes B^{\otimes \frac{n(n+1)}{2}}, k \right) \quad (20) $$

given by

$$ (\Psi^n g) \otimes \left( \begin{pmatrix} a_0 & b_{0,1} & b_{0,2} & \dots & b_{0,n-1} & b_{0,n} \\ 1 & a_1 & b_{1,2} & \dots & b_{1,n-1} & b_{1,n} \\ 1 & 1 & a_2 & \dots & b_{2,n-1} & b_{2,n} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & 1 & \dots & a_{n-1} & b_{n-1,n} \\ 1 & 1 & 1 & \dots & 1 & a_n \end{pmatrix} \right) \\ = g \left( \otimes \begin{pmatrix} a_0 & b_{0,1} & \dots & b_{0,n-2} & b_{0,n-1} \\ 1 & a_1 & \dots & b_{1,n-2} & b_{1,n-1} \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \dots & a_{n-2} & b_{n-2,n-1} \\ 1 & 1 & \dots & 1 & a_{n-1} \end{pmatrix} \right) (a_n \otimes b_{0,n} \otimes b_{1,n} \otimes \dots \otimes b_{n-1,n}) \quad (21) $$

we first transfer the cyclic operator from [8] to a complex $\hat{C}^*(A, B, \varepsilon)$ whose terms are given by

$$ \{ \hat{C}^n(A, B, \varepsilon) := \operatorname{Hom}(A^{\otimes (n+1)} \otimes B^{\otimes \frac{n(n+1)}{2}}, k) \}_{n \ge 0} \quad (22) $$

**Lemma 8.** For each $n \ge 0$, there is an action of the cyclic group $\mathbb{Z}_{n+1} = \langle \lambda \rangle$ on the $k$-space $\hat{C}^n(A, B, \varepsilon) = \operatorname{Hom}(A^{\otimes (n+1)} \otimes B^{\otimes \frac{n(n+1)}{2}}, k)$ given by

$$ (\lambda \cdot f) \otimes \left( \begin{pmatrix} a_0 & b_{0,1} & b_{0,2} & \dots & b_{0,n-1} & b_{0,n} \\ 1 & a_1 & b_{1,2} & \dots & b_{1,n-1} & b_{1,n} \\ 1 & 1 & a_2 & \dots & b_{2,n-1} & b_{2,n} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & 1 & \dots & a_{n-1} & b_{n-1,n} \\ 1 & 1 & 1 & \dots & 1 & a_n \end{pmatrix} \right) = (-1)^n f \otimes \left( \begin{pmatrix} a_n & b_{0,n} & b_{1,n} & \dots & b_{n-2,n} & b_{n-1,n} \\ 1 & a_0 & b_{0,1} & \dots & b_{0,n-2} & b_{0,n-1} \\ 1 & 1 & a_1 & \dots & b_{1,n-2} & b_{1,n-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & 1 & \dots & a_{n-2} & b_{n-2,n-1} \\ 1 & 1 & 1 & \dots & 1 & a_{n-1} \end{pmatrix} \right) $$

for any $f \in \hat{C}^n(A, B, \varepsilon)$, $a_i \in A$ and $b_{i,j} \in B$.

**Proof.** This is clear from the definition in [8, §4.2] and the isomorphisms in (20). $\square$

<figure><img src="image_4.png" /></figure>
The norm operator $N: \hat{C}^n(A, B, \varepsilon) \to \hat{C}^n(A, B, \varepsilon)$ is then defined as $N = 1 + \lambda + \dots + \lambda^n$. The extra degeneracy $s: \hat{C}^n(A, B, \varepsilon) \to \hat{C}^{n-1}(A, B, \varepsilon)$ is given by

$$ (sf) \left( \otimes \left( \begin{pmatrix} a_0 & b_{0,1} & b_{0,2} & \dots & b_{0,n-2} & b_{0,n-1} \\ 1 & a_1 & b_{1,2} & \dots & b_{1,n-2} & b_{1,n-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & 1 & \dots & a_{n-2} & b_{n-2,n-1} \\ 1 & 1 & 1 & \dots & 1 & a_{n-1} \end{pmatrix} \right) \right) = f \left( \otimes \left( \begin{pmatrix} a_0 & b_{0,1} & b_{0,2} & \dots & b_{0,n-1} & 1 \\ 1 & a_1 & b_{1,2} & \dots & b_{1,n-1} & 1 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & 1 & \dots & a_{n-1} & 1 \\ 1 & 1 & 1 & \dots & 1 & 1 \end{pmatrix} \right) \right) $$

for any $f \in \hat{C}^n(A, B, \varepsilon)$, $a_i \in A$ and $b_{i,j} \in B$.

For $n \ge 0$, let $\alpha^n : (A \otimes B^{\otimes n})^* \to A^*$ be the map defined by $p \mapsto \bar{p} := \alpha^n(p)$, where $\bar{p}(a) := p(a \otimes 1_B \otimes \dots \otimes 1_B)$. We denote by $\alpha^* : \bar{C}^*(A, B, \varepsilon) \to C^*((A, B, \varepsilon); A^*)$ the induced map.

We also let $\Phi^* : C^*((A, B, \varepsilon); A^*) \to C^*((A, B, \varepsilon); A)$ be the map induced by the isomorphism $\phi : A^* \to A$ and $\Phi'^* : C^*((A, B, \varepsilon); A) \to C^*((A, B, \varepsilon); A^*)$ be the map induced by $\phi^{-1}$. It may also be verified that the inverse $\Psi'_n : \text{Hom}(A^{\otimes(n+1)} \otimes B^{\otimes \frac{n(n+1)}{2}}, k) \to \text{Hom}(A^{\otimes n} \otimes B^{\otimes \frac{n(n-1)}{2}}, \text{Hom}(A \otimes B^n, k))$ of the map in (21) is given by

$$ (\Psi'^n f) \left( \otimes \left( \begin{pmatrix} a_1 & b_{1,2} & \dots & b_{1,n-1} & b_{1,n} \\ 1 & a_2 & \dots & b_{2,n-1} & b_{2,n} \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \dots & a_{n-1} & b_{n-1,n} \\ 1 & 1 & \dots & 1 & a_n \end{pmatrix} \right) \right) \left( \begin{pmatrix} a_{n+1} \\ b_{1,n+1} \\ b_{2,n+1} \\ \vdots \\ b_{n-1,n+1} \\ b_{n,n+1} \end{pmatrix} \right) = f \left( \otimes \left( \begin{pmatrix} a_1 & b_{1,2} & \dots & b_{1,n-1} & b_{1,n} & b_{1,n+1} \\ 1 & a_2 & \dots & b_{2,n-1} & b_{2,n} & b_{2,n+1} \\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ 1 & 1 & \dots & a_{n-1} & b_{n-1,n} & b_{n-1,n+1} \\ 1 & 1 & \dots & 1 & a_n & b_{n,n+1} \\ 1 & 1 & \dots & 1 & 1 & a_{n+1} \end{pmatrix} \right) \right) \quad (23) $$

**Proposition 9.** Let $A$ be a finite dimensional $k$-algebra equipped with a symmetric, non-degenerate and invariant bilinear form $\langle \cdot, \cdot \rangle : A \times A \to k$ and $B$ be a commutative $k$-algebra with a morphism of $k$-algebras $\varepsilon : B \to A$ such that $\varepsilon(B) \subseteq Z(A)$. Then, the following diagram commutes:

$$ \begin{tikzcd} \bar{C}^*(A, B, \varepsilon) \arrow[r, "N_s"] & \bar{C}^{*-1}(A, B, \varepsilon) \arrow[d, "\Phi^{*-1}\circ\alpha^{*-1}"] \\ C^*((A, B, \varepsilon); A) \arrow[r, "\Delta^*"'] & C^{*-1}((A, B, \varepsilon); A) \arrow[d, "\Phi^{*-1}\circ\alpha^{*-1}"] \end{tikzcd} \quad (24) $$

**Proof.** We will show that for any $n \ge 0$, the following diagram is commutative:

$$ \begin{tikzcd} \operatorname{Hom}(A^{\otimes(n+1)} \otimes B^{\otimes \frac{n(n+1)}{2}}, k) \arrow[r, "N_s"] & \operatorname{Hom}(A^{\otimes n} \otimes B^{\otimes \frac{n(n-1)}{2}}, k) \arrow[d, "\Psi'^n"] \\ \operatorname{Hom}(A^{\otimes n} \otimes B^{\otimes \frac{n(n-1)}{2}}, (A \otimes B^{\otimes n})^*) & \operatorname{Hom}(A^{\otimes(n-1)} \otimes B^{\otimes \frac{(n-1)(n-2)}{2}}, (A \otimes B^{\otimes(n-1)})^*) \arrow[d, "\Psi'^{n-1}"] \\ \operatorname{Hom}(A^{\otimes n} \otimes B^{\otimes \frac{n(n-1)}{2}}, A^*) & \operatorname{Hom}(A^{\otimes(n-1)} \otimes B^{\otimes \frac{(n-1)(n-2)}{2}}, A^*) \arrow[d, "\Phi^{n-1}"] \\ \operatorname{Hom}(A^{\otimes n} \otimes B^{\otimes \frac{n(n-1)}{2}}, A) \arrow[r, "\Delta"] & \operatorname{Hom}(A^{\otimes(n-1)} \otimes B^{\otimes \frac{(n-1)(n-2)}{2}}, A) \arrow[d, "\Phi^{n-1}"] \end{tikzcd} \quad (25) $$
Since $\Phi'^{n-1}$ is an isomorphism, it suffices to check that this diagram is commutative when composed with $\Phi'^{n-1}: \operatorname{Hom}(A^{\otimes(n-1)} \otimes B^{\otimes\frac{(n-1)(n-2)}{2}}, A) \to \operatorname{Hom}(A^{\otimes(n-1)} \otimes B^{\otimes\frac{(n-1)(n-2)}{2}}, A^*)$. Let $f \in \operatorname{Hom}(A^{\otimes(n+1)} \otimes B^{\otimes\frac{n(n+1)}{2}}, k)$. Then, for $a_i \in A$ and $b_{i,j} \in B$, we have

$$ \left\langle (\Delta \circ \Phi^n \circ \alpha^n \circ \Psi'^n(f)) \otimes \left( \begin{pmatrix} a_1 & b_{1,2} & b_{1,3} & \dots & b_{1,n-2} & b_{1,n-1} \\ 1 & a_2 & b_{2,3} & \dots & b_{2,n-2} & b_{2,n-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & 1 & \dots & a_{n-2} & b_{n-2,n-1} \\ 1 & 1 & 1 & \dots & 1 & a_{n-1} \end{pmatrix} \right) \right\rangle, a_n $$

$$ = \sum_{i=1}^{n} (-1)^{(n-1)i} \left\langle \Phi^n \alpha^n \Psi'^n(f) \otimes \left( \begin{pmatrix} a_i & b_{i,i+1} & b_{i,i+2} & \dots & b_{i,n-1} & 1 & b_{1,i} & b_{2,i} & \dots & b_{i-1,i} \\ 1 & a_{i+1} & b_{i+1,i+2} & \dots & b_{i+1,n-1} & 1 & b_{1,i+1} & b_{2,i+1} & \dots & b_{i-1,i+1} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \dots & a_{n-1} & 1 & b_{1,n-1} & b_{2,n-1} & \dots & b_{i-1,n-1} \\ 1 & 1 & \dots & 1 & a_n & 1 & \dots & 1 & \dots & 1 \\ 1 & 1 & \dots & 1 & 1 & a_1 & b_{1,2} & \dots & b_{1,i-1} & \dots \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \dots & \dots & \dots & \dots & \dots & \dots & a_{i-1} & \dots \end{pmatrix} \right) \right\rangle, 1 $$

$$ = \sum_{i=1}^{n} (-1)^{(n-1)i} \phi^{-1}(\phi \alpha^n \Psi'^n(f)) \otimes \left( \begin{pmatrix} a_i & b_{i,i+1} & b_{i,i+2} & \dots & b_{i,n-1} & 1 & b_{1,i} & b_{2,i} & \dots & b_{i-1,i} \\ 1 & a_{i+1} & b_{i+1,i+2} & \dots & b_{i+1,n-1} & 1 & b_{1,i+1} & b_{2,i+1} & \dots & b_{i-1,i+1} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \dots & a_{n-1} & 1 & b_{1,n-1} & b_{2,n-1} & \dots & b_{i-1,n-1} \\ 1 & 1 & \dots & 1 & a_n & 1 & \dots & 1 & \dots & 1 \\ 1 & 1 & \dots & 1 & 1 & a_1 & b_{1,2} & \dots & b_{1,i-1} & \dots \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \dots & \dots & \dots & \dots & \dots & \dots & a_{i-1} & \dots \end{pmatrix} \right) \quad (1) $$

$$ = \sum_{i=1}^{n} (-1)^{(n-1)i} (\alpha^n \Psi'^n(f)) \otimes \left( \begin{pmatrix} a_i & b_{i,i+1} & b_{i,i+2} & \dots & b_{i,n-1} & 1 & b_{1,i} & b_{2,i} & \dots & b_{i-1,i} \\ 1 & a_{i+1} & b_{i+1,i+2} & \dots & b_{i+1,n-1} & 1 & b_{1,i+1} & b_{2,i+1} & \dots & b_{i-1,i+1} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \dots & a_{n-1} & 1 & b_{1,n-1} & b_{2,n-1} & \dots & b_{i-1,n-1} \\ 1 & 1 & \dots & 1 & a_n & 1 & \dots & 1 & \dots & 1 \\ 1 & 1 & \dots & 1 & 1 & a_1 & b_{1,2} & \dots & b_{1,i-1} & \dots \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \dots & \dots & \dots & \dots & \dots & \dots & a_{i-1} & \dots \end{pmatrix} \right) \quad (1) $$

$$ = \sum_{i=1}^{n} (-1)^{(n-1)i} (\alpha^n \Psi'^n(f)) \otimes \left( \begin{pmatrix} a_i & b_{i,i+1} & b_{i,i+2} & \dots & b_{i,n-1} & 1 & b_{1,i} & b_{2,i} & \dots & b_{i-1,i} \\ 1 & a_{i+1} & b_{i+1,i+2} & \dots & b_{i+1,n-1} & 1 & b_{1,i+1} & b_{2,i+1} & \dots & b_{i-1,i+1} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \dots & a_{n-1} & 1 & b_{1,n-1} & b_{2,n-1} & \dots & b_{i-1,n-1} \\ 1 & 1 & \dots & 1 & a_n & 1 & \dots & 1 & \dots & 1 \\ 1 & 1 & \dots & 1 & 1 & a_1 & b_{1,2} & \dots & b_{1,i-1} & \dots \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \dots & \dots & \dots & \dots & \dots & \dots & a_{i-1} & \dots \end{pmatrix} \right) \quad (1) $$
$$ = \sum_{i=1}^{n} (-1)^{(n-1)i} (\Psi'^n(f)) \otimes \left( \begin{pmatrix} a_i & b_{i,i+1} & b_{i,i+2} & \dots & b_{i,n-1} & 1 & b_{1,i} & b_{2,i} & \dots & b_{i-1,i} \\ 1 & a_{i+1} & b_{i+1,i+2} & \dots & b_{i+1,n-1} & 1 & b_{1,i+1} & b_{2,i+1} & \dots & b_{i-1,i+1} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & \dots & a_{n-1} & 1 & b_{1,n-1} & b_{2,n-1} & \dots & b_{i-1,n-1} & 1 \\ 1 & 1 & \dots & 1 & a_n & 1 & \dots & 1 & \dots & 1 \\ 1 & 1 & \dots & 1 & 1 & a_1 & b_{1,2} & \dots & b_{1,i-1} & 1 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \dots & \dots & \dots & \dots & \dots & \dots & a_{i-1} & 1 \end{pmatrix} \right) (1 \otimes 1 \otimes \dots \otimes 1) $$

$$ = \sum_{i=1}^{n} (-1)^{(n-1)i} f \otimes \left( \begin{pmatrix} a_i & b_{i,i+1} & b_{i,i+2} & \dots & b_{i,n-1} & 1 & b_{1,i} & b_{2,i} & \dots & b_{i-1,i} \\ 1 & a_{i+1} & b_{i+1,i+2} & \dots & b_{i+1,n-1} & 1 & b_{1,i+1} & b_{2,i+1} & \dots & b_{i-1,i+1} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \dots & a_{n-1} & 1 & b_{1,n-1} & b_{2,n-1} & \dots & b_{i-1,n-1} & 1 \\ 1 & 1 & \dots & 1 & a_n & 1 & \dots & 1 & \dots & 1 \\ 1 & 1 & \dots & 1 & 1 & a_1 & b_{1,2} & \dots & b_{1,i-1} & 1 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \dots & \dots & \dots & \dots & \dots & \dots & a_{i-1} & 1 \\ 1 & 1 & \dots & \dots & \dots & \dots & \dots & \dots & 1 & 1 \end{pmatrix} \right) $$

On the other hand, let $g := (Ns)(f)$. Then, we have

$$ (\alpha^{n-1} \circ \Psi'^{n-1}(g)) \otimes \left( \begin{pmatrix} a_1 & b_{1,2} & b_{1,3} & \dots & b_{1,n-2} & b_{1,n-1} \\ 1 & a_2 & b_{2,3} & \dots & b_{2,n-2} & b_{2,n-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & 1 & \dots & a_{n-2} & b_{n-2,n-1} \\ 1 & 1 & 1 & \dots & 1 & a_{n-1} \end{pmatrix} \right) (a_n) $$

$$ = (\Psi'^{n-1}(g)) \otimes \left( \begin{pmatrix} a_1 & b_{1,2} & b_{1,3} & \dots & b_{1,n-2} & b_{1,n-1} \\ 1 & a_2 & b_{2,3} & \dots & b_{2,n-2} & b_{2,n-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & 1 & \dots & a_{n-2} & b_{n-2,n-1} \\ 1 & 1 & 1 & \dots & 1 & a_{n-1} \end{pmatrix} \right) (a_n \otimes 1 \otimes \dots \otimes 1) $$

$$ = (Ns)(f) \otimes \left( \begin{pmatrix} a_1 & b_{1,2} & b_{1,3} & \dots & b_{1,n-2} & b_{1,n-1} & 1 \\ 1 & a_2 & b_{2,3} & \dots & b_{2,n-2} & b_{2,n-1} & 1 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ 1 & 1 & 1 & \dots & a_{n-2} & b_{n-2,n-1} & 1 \\ 1 & 1 & 1 & \dots & 1 & a_{n-1} & 1 \\ 1 & 1 & 1 & \dots & 1 & 1 & a_n \end{pmatrix} \right) $$

$$ = \sum_{i=1}^{n} (-1)^{(n-1)i} f \otimes \left( \begin{pmatrix} a_i & b_{i,i+1} & b_{i,i+2} & \dots & b_{i,n-1} & 1 & b_{1,i} & b_{2,i} & \dots & b_{i-1,i} \\ 1 & a_{i+1} & b_{i+1,i+2} & \dots & b_{i+1,n-1} & 1 & b_{1,i+1} & b_{2,i+1} & \dots & b_{i-1,i+1} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \dots & a_{n-1} & 1 & b_{1,n-1} & b_{2,n-1} & \dots & b_{i-1,n-1} & 1 \\ 1 & 1 & \dots & 1 & a_n & 1 & \dots & 1 & \dots & 1 \\ 1 & 1 & \dots & 1 & 1 & a_1 & b_{1,2} & \dots & b_{1,i-1} & 1 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \dots & \dots & \dots & \dots & \dots & \dots & a_{i-1} & 1 \\ 1 & 1 & \dots & \dots & \dots & \dots & \dots & \dots & 1 & 1 \end{pmatrix} \right) $$

This proves the result.

<figure><img src="image_5.png" /></figure>
We now let $\overline{\overline{\mathcal{C}}}^*(A, B, \varepsilon)$ denote the normalized complex associated to the cosimplicial module $\overline{\mathcal{C}}^*(A, B, \varepsilon)$ given in (19). Again, using the isomorphisms in (20), the complex $\overline{\overline{\mathcal{C}}}^*(A, B, \varepsilon)$ becomes isomorphic to the normalized complex $\overline{\mathcal{C}}^*(A, B, \varepsilon)$ whose terms are given by

$$ \overline{\mathcal{C}}^n(A, B, \varepsilon) := \mathrm{Ker} \left( \overline{\mathcal{C}}^n(A, B, \varepsilon) \xrightarrow{s_j} \bigoplus_{j=0}^{n-1} \overline{\mathcal{C}}^{n-1}(A, B, \varepsilon) \right) $$

where $s_j : \overline{\mathcal{C}}^n(A, B, \varepsilon) \to \overline{\mathcal{C}}^{n-1}(A, B, \varepsilon)$ for $0 \le j \le n-1$ is the degeneracy

$$ (s_j f) \left( \bigotimes \begin{pmatrix} a_1 & b_{1,2} & \dots & b_{1,n-1} & b_{1,n} \\ 1 & a_2 & \dots & b_{2,n-1} & b_{2,n} \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \dots & a_{n-1} & b_{n-1,n} \\ 1 & 1 & \dots & 1 & a_n \end{pmatrix} \right) = f \left( \bigotimes \begin{pmatrix} a_1 & b_{1,2} & \dots & b_{1,j} & 1 & b_{1,j+1} & \dots & b_{1,n-1} & b_{1,n} \\ 1 & a_2 & \dots & b_{2,j} & 1 & b_{2,j+1} & \dots & b_{2,n-1} & b_{2,n} \\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \dots & 1 & \dots & \dots & \dots & a_{n-1} & b_{n-1,n} \\ 1 & 1 & \dots & 1 & \dots & \dots & \dots & 1 & a_n \end{pmatrix} \right) $$

**Theorem 10.** Let A be a finite dimensional k-algebra equipped with a symmetric, non-degenerate and invariant bilinear form $\langle \cdot, \cdot \rangle : A \times A \to k$ and B be a commutative k-algebra with a morphism of k-algebras $\varepsilon : B \to A$ such that $\varepsilon(B) \subseteq Z(A)$. Then, the following diagram commutes:

$$ \begin{tikzcd} \overline{\mathcal{C}}^*(A, B, \varepsilon) \arrow[r, "B"] & \overline{\mathcal{C}}^{\bullet-1}(A, B, \varepsilon) \arrow[d, "\overline{\mathcal{C}}^*(A, B, \varepsilon)"'] \\ \overline{\mathcal{C}}^*(A, B, \varepsilon) \arrow[r, "Ns"] & \overline{\mathcal{C}}^{\bullet-1}(A, B, \varepsilon) \arrow[d, "\Phi^{(\bullet-1)\circ\alpha^{(\bullet-1)}}"'] \\ C^*((A, B, \varepsilon); A) \arrow[r, "\Delta^*"'] & C^{\bullet-1}((A, B, \varepsilon); A) \arrow[d, "\Phi^{(\bullet-1)\circ\alpha^{(\bullet-1)}}"'] \end{tikzcd} \quad (26) $$

where B is Connes' operator.

**Proof.** The commutativity of the lower square has already been shown in Proposition 9. By definition, Connes' operator on the complex $\overline{\mathcal{C}}^*(A, B, \varepsilon)$ is given by $B = Ns(1 - \lambda)$ which reduces to $Ns$ on the normalized complex $\overline{\overline{\mathcal{C}}}^*(A, B, \varepsilon)$. Hence, it may be directly verified that the upper diagram commutes. $\square$

## Acknowledgements

The authors wish to thank the referee for a very detailed report and for making suggestions that helped significantly improve the paper.

## References

[1] M. Balodi, A. Banerjee, A. Naolekar, "Weak comp algebras and cup products in secondary Hochschild cohomology of entwining structures", https://arxiv.org/abs/1909.05476v3, 2020.

[2] B. R. Corrigan-Salter, M. D. Staic, "Higher-order and secondary Hochschild cohomology", C. R. Math. Acad. Sci. Paris 354 (2016), no. 11, p. 1049-1054.
[3] M. Gerstenhaber, "The cohomology structure of an associative ring", Ann. Math. **78** (1963), p. 267-288.

[4] M. Gerstenhaber, A. A. Voronov, "Higher-order operations on the Hochschild complex", Funkts. Anal. Prilozh. **29** (1995), no. 1, p. 1-6, translation in *Funct. Anal. Appl.* **29** (1995), no. 1, p. 1-5.

[5] ——, "Homotopy $G$-algebras and moduli space operad", Int. Math. Res. Not. **1995** (1995), no. 3, p. 141-153.

[6] Y. Kosmann-Schwarzbach, "Quasi, twisted, and all that... in Poisson geometry and Lie algebroid theory", in *The breadth of symplectic and Poisson geometry*, Progress in Mathematics, vol. 232, Birkhäuser, 2005, p. 363-389.

[7] J.-L. Koszul, "Crochet de Schouten-Nijenhuis et cohomologie", in *The mathematical heritage of Élie Cartan (Lyon, 1984)*, Astérisque, Société Mathématique de France, 1985, p. 257-271.

[8] J. Laubacher, M. D. Staic, A. Stancu, "Bar simplicial modules and secondary cyclic (co)homology", J. Noncommut. Geom. **12** (2018), no. 3, p. 865-887.

[9] M. D. Staic, "Secondary Hochschild cohomology", Algebr. Represent. Theory **19** (2016), no. 1, p. 47-56.

[10] M. D. Staic, A. Stancu, "Operations on the secondary Hochschild cohomology", Homology Homotopy Appl. **17** (2015), no. 1, p. 129-146.

[11] T. Tradler, "The Batalin-Vilkovisky algebra on Hochschild cohomology induced by infinity inner products", Ann. Inst. Fourier **58** (2008), no. 7, p. 2351-2379.