2.1. Hochschild and cyclic homology

Let $ \Lambda $ be the dg algebra generated by an indeterminate $ \epsilon $ of cohomological degree $ -1 $ with $ \epsilon ^{2}=0 $ and $ d\epsilon =0 $ . The underlying complex of $ \Lambda $ is

$$ \begin{align*}\cdots\rightarrow k\epsilon\rightarrow k\rightarrow0\rightarrow\cdots.\end{align*} $$

Then a mixed complex over $ k $ is a dg right $ \Lambda $ -module whose underlying dg $ k $ -module is $ (M,b) $ and where $ \epsilon $ acts by a closed endomorphism $ B $ . Suppose that $ M=(M,b,B) $ is a mixed complex. Then the shifted mixed complex $ \Sigma M $ is the mixed complex such that $ (\Sigma M)^{p}=M^{p-1} $ for all $ p $ , $ b_{\Sigma M}=-b, $ and $ B_{\Sigma M}=B $ . Let $ f:M\rightarrow M' $ be a morphism of mixed complexes. Then the mapping cone over $ f $ is the mixed complex

$$ \begin{align*}\bigg(M'\oplus \Sigma M,\begin{bmatrix} b_{M'} & f \\ 0 & -b_{M} \end{bmatrix},\begin{bmatrix} B_{M'} & 0 \\ 0 & B_{M} \end{bmatrix}\bigg).\end{align*} $$

We denote by $ {\mathcal M} ix $ the category of mixed complexes and by $ {\mathcal D}{\mathcal M} ix $ the derived category of the dg algebra $ \Lambda $ .

Let $ A $ be a dg $ k $ -algebra. We associate a precyclic chain complex $ C(A) $ (see [Reference Loday31, Definition 2.5.1]) with $ A $ as follows: For each $ n\in \mathbb {N} $ , its $ n $ th term is

$$ \begin{align*}C_{n}(A)=A\otimes_{k}A^{\otimes_{k}^{n}} .\end{align*} $$

The degeneracy maps are given by

$$ \begin{align*} d_{i}(a_{n},\ldots,a_{i},a_{i-1},\ldots,a_{0})=\left\{ \begin{aligned} (a_{n},\ldots,a_{i}a_{i-1},\ldots,a_{0})&&\text{if } i>0 ,\\ (-1)^{n+\sigma}(a_{0}a_{n},\ldots,a_{1})&&\text{if } i=0 , \end{aligned} \right. \end{align*} $$

where $ \sigma =(\mathrm {deg}a_{0})(\mathrm {deg}a_{1}+\cdots +\mathrm {deg}a_{n-1}) $ . The cyclic operator is given by

$$ \begin{align*}t(a_{n-1},\ldots,a_{0})=(-1)^{n+\sigma}(a_{0},a_{n-1},a_{n-2},\ldots,a_{1}) .\end{align*} $$

Then the corresponding total complex $ (H\!H(A),b) $ of $ (C(A),b=\sum _{i=0}^{n}(-1)^{i}d_{i}) $ is called the Hochschild complex of $ A $ and b is called the Hochschild differential of A. The Hochschild homology of $ A $ is defined to be the cohomology of this complex. By [Reference Van den Bergh37, Proposition B.1], the Hochschild complex is quasi-isomorphic to $ A\overset {\mathbf {L}}{\otimes }_{A^{e}}A $ in $ {\mathcal D}(k) $ .

We associate a mixed complex $ (M(A),b,B) $ with this precyclic chain complex as follows: Consider the total complex $ (H\!H(A),b') $ of $ (C(A),b'=\sum _{i=0}^{n-1}(-1)^{i}d_{i}) $ . The underlying dg module of $ M(A) $ is the mapping cone over $ (1-t) $ viewed as a morphism of complexes

$$ \begin{align*}1-t:(H\!H(A),b')\rightarrow(H\!H(A),b) ,\end{align*} $$

where $ b=\sum _{i=0}^{n}(-1)^{i}d_{i} $ and $ b'=\sum _{i=0}^{n-1}(-1)^{i}d_{i} $ . Its underlying module is $ H\!H(A)\oplus H\!H(A) $ ; it is endowed with the grading whose $ n $ th component is $ H\!H(A)_{n}\oplus H\!H(A)_{n-1} $ and the differential is

$$ \begin{align*}\begin{bmatrix} b & 1-t \\ 0 & -b' \end{bmatrix} .\end{align*} $$

The operator $ B\colon M\rightarrow M $ is

$$ \begin{align*}\begin{bmatrix} 0 & 0 \\ N & 0 \end{bmatrix} ,\end{align*} $$

where $ N=\sum _{i=0}^{n}t^{i} $ .

Let $ A' $ be an other dg $ k $ -algebra. Let $ f $ be a morphism from $ A' $ to $ A $ . Then $ f $ induces a canonical morphism between their Hochschild complexes

$$ \begin{align*}\gamma_{f}\colon H\!H(A')\rightarrow H\!H(A) .\end{align*} $$

Definition 2.5. The Hochschild homology $ H\!H_{\bullet }(f) $ of $ f $ is the cohomology of the relative Hochschild complex which is defined as follows:

$$ \begin{align*}H\!H(f)=\mathrm{Cone}(\gamma_{f}\colon HH(A')\rightarrow HH(A)) .\end{align*} $$

Definition 2.7. The cyclic homology $ H\!C_{\bullet }(A) $ of $ A $ is defined to be the cohomology of the cyclic chain complex of $ A $

$$ \begin{align*}H\!C(A)=M(A)\overset{\mathbf{L}}{\otimes}_{\Lambda}k .\end{align*} $$

The negative cyclic homology $ H\!N_{\bullet }(A) $ of $ A $ is defined to be the cohomology of the negative cyclic chain complex of $ A $

$$ \begin{align*}H\!N(A)=\mathrm{\mathbf{R}Hom}_{\Lambda}(k,M(A)).\end{align*} $$

The augmentation morphism $ \Lambda \to k $ induces natural morphisms in $ {\mathcal D}(k) $

$$ \begin{align*}H\!N(A)\rightarrow H\!H(A)\rightarrow H\!C(A).\end{align*} $$

The morphism $ f $ also induces a canonical morphism between their mixed complexes

$$ \begin{align*}\gamma_{f}\colon M(A')\rightarrow M(A) .\end{align*} $$

We denote by $ M(f) $ the mapping cone over $ \gamma _{f} $ .

Definition 2.8. The cyclic homology $ H\!C_{\bullet }(f) $ of $ f\colon A'\rightarrow A $ is defined to be the cohomology of the cyclic chain complex group of $ f $

$$ \begin{align*}H\!C(f)=M(f)\overset{\mathbf{L}}{\otimes}_{\Lambda}k.\end{align*} $$

The negative cyclic homology $ H\!N_{\bullet }(f) $ of $ f\colon B\rightarrow A $ is defined to be the cohomology of the negative cyclic chain complex of $ f $

$$ \begin{align*}H\!N(f)=\mathrm{\mathbf{R}Hom}_{\Lambda}(k,M(f)).\end{align*} $$

2.2. Relative deformed Calabi–Yau completions

Given a dg algebra B, let $(\mathrm {dga}_{k})_{B/}$ be the category of dg algebras under B. The forgetful functor $(\mathrm {dga}_{k})_{B/}\rightarrow {\mathcal C}(B^{e})$ , sending a dg morphism $B\rightarrow A$ to the B-bimodule A, has a left adjoint $T_{B}\colon {\mathcal C}(B^{e})\rightarrow (\mathrm {dga}_{k})_{B/}$ , that can be described as follows.

• Let M be a B-bimodule. The tensor algebra $T_{B}M$ is defined as

  $$ \begin{align*}T_{B}M=B\oplus M\oplus (M\otimes_{B}M)\oplus\cdots.\end{align*} $$

  The dg structure on $T_{B}M$ is given by the differentials of B and M and the multiplication is given by the concatenation product and we have a canonical morphism $B\rightarrow T_{B}M$ of dg algebras.

This is a Quillen adjunction and thus induces an adjunction between their homotopy categories. We will denote by $\mathbf {L} T_{B}$ the left derived functor.

Let $ f\colon B\rightarrow A $ be a morphism (not necessarily unital) between smooth dg algebras. Let $ [\xi ] $ be an element in $ H\!H_{n-2}(f) $ . In this section, we recall the construction of the deformed relative $ n $ -Calabi–Yau completion of $ f\colon B\rightarrow A $ with respect to the Hochschild homology class $ \xi \in H\!H_{n-2}(f) $ . For more details, we refer the reader to [Reference Yeung41].

The morphism $ f $ induces a morphism in $ {\mathcal D}(A^{e}) $

$$ \begin{align*}m_{f}\colon B\overset{\mathbf{L}}{\otimes}_{B^{e}}A^{e}\rightarrow A .\end{align*} $$

After taking the derived bimodule dual (Definition 2.3), using the smoothness of $ B $ , we get a morphism

$$ \begin{align*}m_{f}^{\vee}\colon A^{\vee}\rightarrow B^{\vee}\overset{\mathbf{L}}{\otimes}_{B^{e}}A^{e}.\end{align*} $$

Let $ \Xi $ be the cofiber of $ m_{f}^{\vee } $ . The dualizing bimodule $ \Theta _{f}=(\mathrm {cof}(B\overset {\mathbf {L}}{\otimes }_{B^{e}}A^{e}\rightarrow A))^{\vee } $ of $ f $ is quasi-isomorphic to $ \Sigma ^{-1}\Xi $ .

By the definition of Hochschild homology of $ f $ , we have the following long exact sequence:

(1) $$ \begin{align} \cdots\rightarrow H\!H_{n-2}(B)\rightarrow H\!H_{n-2}(A)\rightarrow H\!H_{n-2}(f)\rightarrow H\!H_{n-3}(B)\rightarrow\cdots. \end{align} $$

Thus, the Hochschild homology class $ \xi \in H\!H_{n-2}(f) $ induces an element $ \xi _{B} $ in $ H\!H_{n-3}(B) $ .

Notice that since $ B,A $ are smooth, we have the following isomorphisms: $\colon $

$$ \begin{align*} \mathrm{Hom}_{{\mathcal D}(B^{e})}(\Sigma^{n-2}B^{\vee},\Sigma B)\simeq & H^{3-n}(B\overset{\mathbf{L}}{\otimes}_{B^{e}}B)=HH_{n-3}(B),\\ \mathrm{Hom}_{{\mathcal D}(A^{e})}(\Sigma^{n-2}\Xi,\Sigma A)\simeq & H^{2-n}(\mathrm{Cone}(B\overset{\mathbf{L}}{\otimes}_{B^{e}}A\rightarrow A\overset{\mathbf{L}}{\otimes}_{A^{e}}A))\\ \leftarrow & H^{2-n}(\mathrm{Cone}(B\overset{\mathbf{L}}{\otimes}_{B^{e}}B\rightarrow A\overset{\mathbf{L}}{\otimes}_{A^{e}}A))\\ \simeq & HH_{n-2}(f). \end{align*} $$

Thus, the homology class $ [\xi ] $ induces a morphism in $ {\mathcal D}(A^{e}) $

$$ \begin{align*}\xi\colon\Sigma^{n-2}\Xi\rightarrow\Sigma A\end{align*} $$

and the homology class $ [\xi _{B}] $ induces a morphism in $ {\mathcal D}(B^{e}) $

$$ \begin{align*}\xi_{B}\colon\Sigma^{n-2}B^{\vee}\rightarrow\Sigma B .\end{align*} $$

Moreover, we have the following commutative diagram in $ {\mathcal D}(A^{e}): $

Therefore, the morphism $ \xi _{B} $ gives rise to a “deformation”

$$ \begin{align*}\boldsymbol{\Pi}_{n-1}(B,\xi_{B})\end{align*} $$

of $ \boldsymbol {\Pi }_{n-1}(B)=\mathbf {L} T_{B}(\Sigma ^{n-2}B^{\vee }) $ , obtained by adding $ \xi _{B} $ to the differential of $ \boldsymbol {\Pi }_{n-1}(B) $ ; the morphism $ \xi $ gives rise to a “deformation”

$$ \begin{align*}\boldsymbol{\Pi}_{n}(A,B,\xi)\end{align*} $$

of $ \boldsymbol {\Pi }_{n}(A,B)=\mathbf {L} T_{A}(\Sigma ^{n-2}\Xi ) $ , obtained by adding $ \xi $ to the differential of $ \mathbf {L} T_{A}(\Sigma ^{n-2}\Xi ) $ ; and the commutative diagram above gives rise to a morphism

$$ \begin{align*}\tilde{f}\colon\boldsymbol{\Pi}_{n-1}(B,\xi_{B})\rightarrow\boldsymbol{\Pi}_{n}(A,B,\xi) .\end{align*} $$

A standard argument shows that up to weak equivalence, the morphism $ \tilde {f} $ and the deformations $ \boldsymbol {\Pi }_{n-1}(B,\xi _{B}) $ , $ \boldsymbol {\Pi }_{n}(A,B,\xi ) $ only depend on the class $ \xi $ .

Theorem 2.10 [Reference Bozec, Calaque and Scherotzke3, Theorem 5.36] and [Reference Yeung41, Theorem 3.23]

If $ \xi $ has a negative cyclic lift, then each choice of such a lift gives rise to a canonical left $ n $ -Calabi–Yau structure on the morphism

$$ \begin{align*}\tilde{f}\colon \boldsymbol{\Pi}_{n-1}(B,\xi_{B})\rightarrow\boldsymbol{\Pi}_{n}(A,B,\xi) .\end{align*} $$

3.1. Group actions on dg morphisms

Let $ G $ be a finite group. Denote by $ \mathbf {1} $ the neutral element of $ G $ and by $ k[G] $ the group algebra of $ G $ . Let $ A $ be a dg algebra over $ k $ .

Now suppose that $ G $ acts on $ A $ by dg automorphisms. For a dg A-module M, a compatible action of G on M by dg automorphisms is a morphism of complexes $k[G]\otimes _{k} M\rightarrow M $ such that, denoting by $ ^{g}m $ the image of $ g\otimes m $ , for all $ g\in G $ and $ m\in M $ , then

• • $^g(m+n) = \,^gm+\,^gn$ for all $ m,n\in M $ and $ g\in G. $

• • $ ^{\mathbf {1}}m=m $ and $ ^{gh}m=\,^{g}(^{h}m) $ for all $ m\in M $ and $ g,h\in G $ .

• • $^{g}(m\cdot a)=\,^{g}m\cdot \,^{g}a $ for all $g\in G$ , $m\in M$ and $a\in A$ .

The action of G on A induces an action of G on $ A^{e}$ which is defined as follows:

$$ \begin{align*}G\times A^{e}\rightarrow A^{e}\end{align*} $$

$$ \begin{align*}\qquad\qquad\qquad\quad (g,a\otimes b)\mapsto\,^{g}(a\otimes b):=\,^{g}a\otimes\,^{g}b. \end{align*} $$

The corresponding skew group dg algebra is denoted by $A^{e}*G$ . Moreover, the action of G on $A^{e}$ is compatible with the canonical right $A^{e}$ -module structure on $A^{e}$ .

Denote by $\Lambda _{A}$ the skew group dg algebra $A*G$ . We have a canonical morphism of dg algebras

$$ \begin{align*}\iota\colon A\rightarrow\Lambda_{A}\end{align*} $$

$$ \begin{align*}\hspace{0.8cm} a\mapsto a*\mathbf{1}.\end{align*} $$

3.2. The dg algebra $\triangle $ and its dg modules

Let $\triangle _{A}$ be the following dg subalgebra of $\Lambda _{A}^{e}$ :

$$ \begin{align*}\triangle_{A}:=\underset{g\in G}{\oplus}(A*g)\otimes(A*g^{-1})\subseteq\Lambda_A^{e}=(A*G)\otimes_{k}(A*G)^{op} .\end{align*} $$

Let M be a dg $\triangle _{A}$ -module. We can regard M as a dg $A^{e}$ -module with a compatible G-action as follows:

• • for all $a,b\in A$ , define $m\cdot (a\otimes b):= m\cdot ((a*\mathbf {1})\otimes (b*\mathbf {1}))$ ;

• • for all $g\in G$ , define $^{g}m:= m\cdot ((1*g^{-1})\otimes (1*g))$ .

Conversely, let N be a dg $A^{e}$ -module with a compatible G-action. The structure of dg $\triangle _{A}$ -module on M is given by $m\cdot ((a*g)\otimes (a*g^{-1})):= (\,^{g^{-1}}m)\cdot \,(^{g^{-1}}a\otimes b)$ .

For a dg $\triangle _{A}$ -module N, we associate a dg $\Lambda _{A}^{e}$ -module $M*G$ with underlying complex of vector spaces $M\otimes _{k}k[G]$ and with action of $\Lambda _{A}^{e}$ such that $(m*g)\cdot (a*h\otimes b*k):= (\,^{k}m\cdot (^{kg}a\otimes b))*kgh$ for all $m\in M$ , $a,b\in A,$ and $g,h,k\in G$ .

Notice that A can be endowed with the following dg $\triangle _{A}$ -module structure:

$$ \begin{align*}A\otimes\triangle_{A}\rightarrow A\end{align*} $$

$$ \begin{align*}\hspace{2.3cm}a\otimes(b*g\otimes c*g^{-1})\mapsto(-1)^{\deg(c)\cdot(\deg(b)+\deg(a))}c\,\,^{g^{-1}}\hspace{-0.3cm}a\,\,^{g^{-1}}\hspace{-0.07cm}b.\end{align*} $$

It is easy to see that the resulting dg $\Lambda _{A}^{e}$ -module $A*G$ is $\Lambda _{A}$ .

Proposition 3.6 [Reference Le Meur29, Lemmas 3.1.1 and 3.3.1 and Proposition 3.3.2]

We have an isomorphism of dg algebras

$$ \begin{align*}\triangle_{A}\rightarrow A^{e}*G\end{align*} $$

$$ \begin{align*}\hspace{-1.4cm} (a*g)\otimes(b*g^{-1})\mapsto(a\otimes\,^{g}b)*g .\end{align*} $$

Moreover, we have

1. (1) $\triangle _{A}\simeq (A^{e})^{\mathrm {Card}(G)}$ as dg $A^{e}$ -modules; $\Lambda _{A}^{e}\simeq \triangle _{A}^{\mathrm {Card}(G)}$ as dg $\triangle _{A}$ -modules.

2. (2) Let $M\in \mathrm {Mod}(\triangle _{A})$ , then

   1. (a) (a) the following mapping is an isomorphism of dg $\Lambda _{A}^{e}$ -modules

      $$ \begin{align*}M\otimes_{\triangle_{A}}\Lambda_{A}^{e}\rightarrow M*G\end{align*} $$
      $$ \begin{align*}\hspace{1.8cm}m\otimes(a*g\otimes b*h)\mapsto (m\cdot(a*h^{-1}\otimes b*h))*hg.\end{align*} $$

   2. (b) M is a direct summand of $M\otimes _{\triangle _{A}}\Lambda _{A}^{e}$ in $\mathrm {Mod}(\triangle _{A})$ . If $\mathrm {char}(k)\nmid \mathrm {Card}(G),$ then M is a direct summand of $M\otimes _{A^{e}}\triangle _{A}$ in $\mathrm {Mod}(\triangle _{A})$ .

3. (3) The restriction-of-scalars functor $\mathrm {Mod}(\triangle _{A})\rightarrow \mathrm {Mod}(A^{e})$ maps cofibrant objects to cofibrant objects. Moreover, for all cofibrant resolution $X\rightarrow A$ in $\mathrm {Mod}(\triangle _{A})$ , the composite morphism

   $$ \begin{align*}X*G\simeq X\otimes_{\triangle_{A}}\Lambda_{A}^{e}\rightarrow A\otimes_{\triangle_{A}}\Lambda_{A}^{e}\rightarrow A*G=\Lambda_{A}\end{align*} $$
   is a cofibrant resolution in $\mathrm {Mod}(\Lambda _{A}^{e})$ .

4. (4) The functor $\mathrm {Hom}_{A^{e}}(-,A^{e})\colon \mathrm {Mod}(A^{e})\rightarrow \mathrm {Mod}(A^{e})$ induces a functor $\mathrm {Mod}(\triangle _{A})\rightarrow \mathrm {Mod}(\triangle _{A})$ also denoted by $\mathrm {Hom}_{A^{e}}(-,A^{e})$ and whose total derived functor $\mathrm {\mathbf {R}Hom}_{A^{e}}(-,A^{e})$ is such that

   $$ \begin{align*}\mathrm{\mathbf{R}Hom}_{A^{e}}(-,A^{e})\otimes_{\triangle_{A}}\Lambda_{A}^{e}\simeq\mathrm{\mathbf{R}Hom}_{\Lambda_{A}^{e}}(-\otimes_{\triangle_{A}}\Lambda_{A}^{e},\Lambda_{A}^{e}).\end{align*} $$
   In particular, we have
   $$ \begin{align*}A^{\vee}*G=\mathrm{\mathbf{R}Hom}_{A^{e}}(A,A^{e})\otimes_{\triangle_{A}}\Lambda_{A}^{e}\simeq\mathrm{\mathbf{R}Hom}_{\Lambda_{A}^{e}}(A\otimes_{\triangle_{A}}\Lambda_{A}^{e},\Lambda_{A}^{e})\simeq\mathrm{\mathbf{R}Hom}_{\Lambda_{A}^{e}}(\Lambda_{A},\Lambda_{A}^{e}).\end{align*} $$

3.3. Equivariant Cuntz–Quillen resolutions

Let $\Omega ^{1}(A)$ be the dg $A^{e}$ -module $\ker (A\otimes _{k}A\xrightarrow {m}A)$ , where $m\colon A\otimes _{k}A\rightarrow A$ is the multiplication map. Then $\Omega ^{1}(A)$ is generated by $D(f):= f\otimes 1-1\otimes f$ and it has a compatible G-action defined as $\,^{g}D(f):= D(^{g}f)$ . Hence, there is a short exact sequence of dg $\triangle _{A}$ -modules

$$ \begin{align*}0\rightarrow\Omega^{1}(A)\xrightarrow{\alpha}A\otimes_{k}A\xrightarrow{m}A\rightarrow0.\end{align*} $$

Denote by ${\mathcal S}(A)\in \mathrm {Mod}(\triangle _{A})$ the cone

$$ \begin{align*}{\mathcal S}(A):=\mathrm{Cone}(\alpha\colon\Omega^{1}(A)\rightarrow A\otimes_{k}A)\end{align*} $$

called the Cuntz–Quillen resolution. Clearly, there is a quasi-isomorphism in $\mathrm {Mod}(\triangle _{A})$

$$ \begin{align*}r_A\colon{\mathcal S}(A)\twoheadrightarrow A\end{align*} $$

induced by the multiplication map. The underlying graded space of ${\mathcal S}(A)$ is $\Sigma \Omega ^{1}(A)\oplus (A\otimes _{k}A)$ . An element in $\Sigma \Omega ^{1}(A)$ has the form $sD(f)$ .

If A is finitely cellular, that is, there exists a finite graded quiver Q such that $A=T_{kQ_0}(kQ_1)$ , then ${\mathcal S}(A)$ is cellular of finite rank, with basis $\{sD(f)\,|\,f\in Q_1\}\cup \{E_x\,|\,x\in Q_0\},$ where $E_x=e_x\otimes e_x$ are the basis elements in $A\otimes _k A$ .

In particular, it is cofibrant and perfect [Reference Yeung41, Corollary 2.19]. By Proposition 3.6, the canonical morphism ${\mathcal S}(A)*G\rightarrow A*G$ is a cofibrant replacement of $A*G$ .

Moreover, the $ A $ -bimodule $ {\mathcal S}(A)^{\vee } $ is also cellular of finite rank, with basis $ \{g^{\vee }|g\in Q_{1}\} \cup \{c_{y}|y\in Q_{0}\}, $ where the arrow $ g^{\vee } $ has degree $ |g^{\vee }| =1-|g|$ , and points in the opposite direction to $ g $ ; the loop $ c_{y} $ has degree $ |c_{y}|=0 $ and is based at $ y $ . In particular, it is cofibrant and perfect.

3.4. Equivariant relative Calabi–Yau completions

Let G be a finite group. Let $ f\colon B\rightarrow A $ be a dg morphism (not necessarily unital) between finitely cellular dg algebras such that f is G-equivariant. By [Reference Yeung40, Remark 24.2.8], we can assume that $ f\colon B \to A$ is a semi-free extension, that is, there is a finite graded quiver $ Q $ and a subquiver $ F\subseteq Q $ such that the underlying graded $ k $ -categories of $ B $ and $ A $ are isomorphic to $ T_{kF_{0}}(kF_{1}) $ and $ T_{kQ_{0}}(kQ_{1}) $ , respectively. In particular, B and A are smooth [Reference Yeung41, Corollary 2.22].

Let $r_{B}\colon {\mathcal S}(B)\rightarrow B$ (resp. $r_{A}\colon {\mathcal S}(A) \rightarrow A$ ) be the Cuntz–Quillen resolution of B (resp. A) in $\mathrm {Mod}(\triangle _{B})$ (resp. $\mathrm {Mod}(\triangle _{A})$ ). Then there exists a morphism $r_{f}\colon {\mathcal S}(B)\rightarrow {\mathcal S}(A)$ of $\triangle _{B}$ -modules such that the following square commutes in $\mathrm {Mod}(\triangle _{B }):$

(2)

Notice that ${\mathcal S}(B)$ and ${\mathcal S}(A)$ are also cofibrant in $\mathrm {Mod}(B^{e})$ and $\mathrm {Mod}(A^{e}),$ respectively (see [Reference Le Meur29, Lemma 3.1.1]). By the above diagram (2), we obtain a morphism of Hochschild chain complexes

$$ \begin{align*}\Theta_{f}\colon {\mathcal S}(B)\otimes_{B^{e}}B\rightarrow {\mathcal S}(A)\otimes_{A^{e}}A.\end{align*} $$

Moreover, it is G-equivariant provided that ${\mathcal S}(B)\otimes _{B^{e}}B$ and $ {\mathcal S}(A)\otimes _{A^{e}}A$ are endowed with the following actions of $G:$

• • for all $g\in G$ , $x\in {\mathcal S}(B),$ and $b\in B$ , let $^{g}(x\otimes b)=\,^{g}x\otimes \,^{g}b$ ;

• • similar action for ${\mathcal S}(A)\otimes _{A^{e}}A$ .

Then Hochschild homology $ H\!H_{\bullet }(f) $ of $ f $ is computed by $\mathrm {Cone}(\Theta _{f})$ and G also acts on $H\!H_{\bullet }(f)$ . For each $k\in \mathbb {Z}$ , denote by $H\!H_{k}(B)^{G}$ , $H\!H_{k}(A)^{G}$ , $H\!H_{k}(f)^{G}$ the corresponding G-invariant subspace.

By the definition of $H\!H_{\bullet }(f)$ , we have the following long exact sequence:

$$ \begin{align*} \cdots\rightarrow H\!H_{n-2}(B)\rightarrow H\!H_{n-2}(A)\rightarrow H\!H_{n-2}(f)\xrightarrow{\delta} H\!H_{n-3}(B)\rightarrow\cdots. \end{align*} $$

And $\delta $ induces a map between the G-invariant subspaces

(3) $$ \begin{align} \delta^{G}\colon H\!H_{n-2}(f)^{G}\rightarrow H\!H_{n-3}(B)^{G}. \end{align} $$

The morphism $r_{f}\colon {\mathcal S}(B)\rightarrow {\mathcal S}(A)$ of dg $\triangle _{B}$ -modules in (2) induces a morphism $m_{f}\colon {\mathcal S}(B)\otimes _{\triangle _{B}}\triangle _{A}\rightarrow {\mathcal S}(A)$ of dg $\triangle _{A}$ -modules. After taking the bimodule dual, we get a morphism in $\mathrm {Mod}(\triangle _{A})$

$$ \begin{align*}m_{f}^{\vee}\colon {\mathcal S}(A)^{\vee}\rightarrow {\mathcal S}(B)^{\vee}\otimes_{\triangle_{B}}\triangle_{A}.\end{align*} $$

Equivalently, we get a morphism of dg $A^{e}$ -modules

$$ \begin{align*}s_{f}^{\vee}\colon{\mathcal S}(A)^{\vee} \rightarrow{\mathcal S}(B)^{\vee}\otimes_{B^{e}}A^{e},\end{align*} $$

which is compatible with G-actions on ${\mathcal S}(A)^{\vee }$ and ${\mathcal S}(B)^{\vee }\otimes _{B^{e}}A^{e}$ (see Lemma 3.5).

Notice that ${\mathcal S}(A)^{\vee }$ and ${\mathcal S}(B)^{\vee }\otimes _{B^{e}}A^{e}$ are also cofibrant. Let $ \Xi $ be the cone of $ s_{f}^{\vee } $ . Then $\Xi $ is a dg $A^{e}$ -module with a compatible G-action, that is, $\Xi $ is a $\triangle _{A}$ -module.

Remark 3.7. The $ B $ -bimodule $ {\mathcal S}(B)^{\vee } $ is cellular of finite rank, with basis $ \{f^{\vee }|f\in F_{1}\} \cup \{c_{x}|x\in F_{0}\}, $ where the arrow $ f^{\vee } $ has degree $ |f^{\vee }| =1-|f|$ , and points in the opposite direction to $ f $ ; the loop $ c_{x} $ has degree $ |c_{x}|=0 $ and is based at $ x $ . Similarly, the $ A $ -bimodule $ {\mathcal S}(A)^{\vee } $ is also cellular of finite rank, with basis $ \{g^{\vee }|g\in Q_{1}\} \cup \{c_{y}|y\in Q_{0}\}, $ where the arrow $ g^{\vee } $ has degree $ |g^{\vee }| =1-|g|$ , and points in the opposite direction to $ g $ ; the loop $ c_{y} $ has degree $ |c_{y}|=0 $ and is based at $ y $ . The map

$$ \begin{align*}s_{f}^{\vee}\colon{\mathcal S}(A)^{\vee} \rightarrow{\mathcal S}(B)^{\vee}\otimes_{B^{e}}A^{e}\end{align*} $$

is surjective. Let ${\mathcal K}$ be the kernel of $s_{f}^{\vee }$ . Then $\Xi $ is quasi-isomorphic to $\Sigma {\mathcal K}$ [Reference Wu39, Section 3.6].

Let $\xi $ be an element of $H\!H_{n-2}(f)^{G}\subseteq H\!H_{n-2}(f)$ and let $ \xi _{B}=\delta ^{G}(\xi )\in H\!H_{n-3}(B)^{G}\subseteq H\!H_{n-3}(B) $ , where $\delta ^{G}$ is the map (3).

The homology class $ \xi \in H\!H_{n-2}(f)^{G}\subseteq H\!H_{n-2}(f) $ induces a morphism in $ {\mathcal D}(A^{e}) $

$$ \begin{align*}\xi\colon\Sigma^{n-2}\Xi\rightarrow\Sigma A\end{align*} $$

which is compatible with the G-actions, and the homology class $ \xi _{B} $ induces a morphism in $ {\mathcal D}(B^{e}) $

$$ \begin{align*}\xi_{B}\colon\Sigma^{n-2}{\mathcal S}(B)^{\vee}\rightarrow\Sigma B,\end{align*} $$

which is also compatible with the G-actions. Moreover, we have the following commutative diagram in $ {\mathcal D}(A^{e}): $

(4)

Therefore, the morphism $ \xi _{B} $ gives rise to a “deformation”

$$ \begin{align*}\boldsymbol{\Pi}_{n-1}(B,\xi_{B})\end{align*} $$

of $ \boldsymbol {\Pi }_{n-1}(B)=T_{B}(\Sigma ^{n-2}{\mathcal S}(B)^{\vee }) $ , obtained by adding $ \xi _{B} $ to the differential of $ T_{B}(\Sigma ^{n-2}{\mathcal S}(B)^{\vee }) $ ; the morphism $ \xi $ gives rise to a “deformation”

$$ \begin{align*}\boldsymbol{\Pi}_{n}(A,B,\xi)\end{align*} $$

of $ \boldsymbol {\Pi }_{n}(A,B)=T_{A}(\Sigma ^{n-2}\Xi ) $ , obtained by adding $ \xi $ to the differential of $ T_{A}(\Sigma ^{n-2}\Xi ) $ ; and the commutative diagram above gives rise to a morphism

$$ \begin{align*}\tilde{f}\colon\boldsymbol{\Pi}_{n-1}(B,\xi_{B})\rightarrow\boldsymbol{\Pi}_{n}(A,B,\xi) .\end{align*} $$

Hence, by constructions, the dg algebras $\boldsymbol {\Pi }_{n-1}(B,\xi _{B})$ and $\boldsymbol {\Pi }_{n}(A,B,\xi )$ have natural G-actions. Moreover, the dg morphism $\tilde {f}$ is compatible with G-actions, that is, G acts on $\tilde {f}$ . Therefore, we obtain a dg morphism between the corresponding skew group dg algebras

$$ \begin{align*}\tilde{f}*G\colon\boldsymbol{\Pi}_{n-1}(B,\xi_{B})*G\rightarrow\boldsymbol{\Pi}_{n}(A,B,\xi)*G\end{align*} $$

given by

• • for each $g\in G$ and $b\in B$ , we have $(\tilde {f}*G)(b*g)=f(b)*g$ ;

• • for each $g\in G$ and $\alpha _1\otimes \alpha _2\otimes \cdots \otimes \alpha _n$ in $(\Sigma ^{n-1}{\mathcal S}(B)^{\vee })^{\otimes ^{n}}$ , we have

  $$ \begin{align*}(\tilde{f}*G)((\alpha_1\otimes\alpha_2\otimes\cdots\otimes\alpha_n)*g)=(l(\alpha_1)\otimes l(\alpha_2)\cdots\otimes l(\alpha_n))*g,\end{align*} $$

  where l is the map in (4).

On the other hand, the action of $\alpha $ on f also induces a dg morphism between the corresponding skew group dg algebras

$$ \begin{align*}f*G\colon B*G\rightarrow A*G\end{align*} $$

given by

$$ \begin{align*}\hspace{1.5cm}b*h\mapsto f(b)*h.\end{align*} $$

If $\mathrm {char}(k)\nmid \mathrm {Card}(G)$ , by [Reference Le Meur29, Proposition 3.2.1], the skew group dg algebras $B*G$ and $A*G$ are smooth.

By the functoriality of the Hochschild homology, we have the following commutative diagram:

3.5. Application to Ginzburg dg functors

Let $Q=(Q_{0},Q_{1})$ be a finite graded k-quiver. Recall that the space of potentials on Q is the graded vector space

$$ \begin{align*}\frac{kQ\otimes_{kQ^{e}_{0}}kQ_{0}}{\langle uv\otimes e_{x}-(-1)^{\deg(u)\deg(v)}vu\otimes e_{y}\,|\,u\in e_{x}kQe_{y}, v\in e_{y}kQe_{x}\,\text{homogeneous}, x,y\in Q_{0}\rangle}.\end{align*} $$

Therefore, potentials are expressed as linear combinations of oriented cycles in Q, with each cycle being considered up to cyclic permutation with Koszul-type signs. Moreover, a homogeneous potential of degree $-d$ on Q can be viewed as an element of $H\!H_d(kQ)$ . Let $F=(F_{0},F_{1})$ be a graded subquiver of Q (which may not be full). Let n be a positive integer. Let W be a homogeneous potential on Q of degree $3-n$ .

Definition 3.9. Let $ \widetilde {Q} $ be the graded quiver with the same vertices as $ Q $ and whose arrows are

• • the arrows of $ Q $ ;

• • an arrow $ a^{\vee }:j\to i $ of degree $ 2-n-|a| $ for each unfrozen arrow $ a\colon i\rightarrow j $ ;

• • a loop $ t_{i}:i\to i $ of degree $ 1-n $ for each unfrozen vertex $ i $ .

Define the n-dimensional relative Ginzburg dg algebra $ \boldsymbol {\Gamma }_{n}(Q,F,W) $ as the dg algebra whose underlying graded space is the completed graded path algebra $k\widetilde {Q} $ . Its differential is the unique $ k $ -linear continuous endomorphism of degree 1 which satisfies the Leibniz rule

for all homogeneous $ u $ of degree $ p $ and all $ v $ and takes the following values on the arrows of $ \widetilde {Q} $ :

• • $ d(a)=0 $ for each arrow $ a $ of $ Q $ ;

• • $d(a^{\vee })=\partial _{a}W$ for each unfrozen arrow $ a $ ;

• • $ d(t_{i})=e_{i}(\sum _{a\in Q_{1}}[a,a^{\vee }])e_{i} $ for each unfrozen vertex $ i $ , where $ e_{i} $ is the lazy path corresponding to the vertex $ i $ and $[\,,\,]$ denotes the supercommutator.

Definition 3.10. Let $ \widetilde {F} $ be the graded quiver with the same vertices as $ F $ and whose arrows are

• • the arrows of $ F $ ;

• • an arrow $ \tilde {a}:j\to i $ of degree $ 3-n-|a|$ for each arrow $ a $ of $ F $ ;

• • a loop $ r_{i}:i\to i $ of degree $ 2-n $ for each vertex $ i $ of $ F $ .

Define derived preprojective algebra $ \boldsymbol {\Pi }_{n-1}(F) $ as the dg algebra whose underlying graded space is the completed graded path algebra $ k\widetilde {F} $ . Its differential is the unique $ k $ -linear continuous endomorphism of degree 1 which satisfies the Leibniz rule

for all homogeneous $ u $ of degree $ p $ and all $ v $ , and takes the following values on the arrows of $ \widetilde {F} $ :

• • $ d(a)=0 $ for each arrow $ a $ of $ F $ ;

• • $d(\tilde {a})=0 $ for each arrow $ a $ in $ F $ ;

• • $ d(r_{i})=e_{i}(\sum _{a\in F_{1}}[a,\tilde {a}])e_{i} $ for each vertex $ i $ of $ F $ , where $ e_{i} $ is the lazy path corresponding to the vertex $ i $ .

Let c be the image of W under Connes’ boundary in Hochschild homology $H\!H_{n-3} (kQ) \rightarrow H\!H_{n-2}(kQ)$ (see [Reference Keller26, Section 6.1]). Let $ \pi \colon kF\hookrightarrow kQ $ be the canonical dg inclusion. Then $ \xi =(0,c) $ is an element of $ H\!H_{n-2}(G) $ . Via the deformed relative n-Calabi–Yau completion of $ G $ with respect to the class $ \xi $ , we get a dg functor

$$ \begin{align*}\boldsymbol{\tilde{\pi}}\colon\boldsymbol{\Pi}_{n-1}(F)\rightarrow\boldsymbol{\Pi}_{n}(kQ,kF,\xi) ,\end{align*} $$

which has a canonical left $ n $ -Calabi–Yau structure (see [Reference Bozec, Calaque and Scherotzke3, Theorem 5.36] and [Reference Yeung41, Theorem 3.23]).

Proposition 3.11 [Reference Wu39, Proposition 3.18]

We have the following commutative diagram (up to homotopy) in the category of pseudocompact dg algebras:

where $ i $ is a quasi-equivalent dg inclusion and $ \boldsymbol {G}_{rel} $ is given explicitly as follows:

• • $ \boldsymbol {G}_{rel}(i)=i $ for each frozen vertex $ i\in F_{0} $ ;

• • $ \boldsymbol {G}_{rel}(a)=a $ for each arrow $ a\in F_{1} $ ;

• • $ \boldsymbol {G}_{rel}(\tilde {a})=-\partial _{a}W $ for each arrow $ a\in F_{1} $ ;

• • $ \boldsymbol {G}_{rel}(r_{i})=e_{i}(\sum _{a\in Q_{1}\setminus F_{1}}[a,a^{\vee }])e_{i} $ for each frozen vertex $ i\in F_{0} $ .

Thus, the functor $ \boldsymbol {G}_{rel}: \boldsymbol {\Pi }_{n-1}(F)\rightarrow \boldsymbol {\Gamma }_{n}(Q,F,W) $ has a canonical left $ n $ -Calabi–Yau structure.

We call $ \boldsymbol {G}_{rel}\colon \boldsymbol {\Pi }_{n-1}(F)\rightarrow \boldsymbol {\Gamma }_{n}(Q,F,W) $ the n-dimensional Ginzburg functor associated with $ (Q,F,W) $ .

Let G be a finite group such that $\mathrm {char}(k) \nmid \mathrm {Card}(G)$ . We make the following assumption.

Definition 3.13 [Reference Demonet11] and [Reference Le Meur29, Definition 4.4.1]

Let $[G\backslash Q_0]$ be a complete set of representatives of the G-orbits of vertices of Q. For each $i \in Q_0$ , denote by $G_i$ the stabilizer of i and let $[G/G_i]$ be a complete set of representatives of the cosets of G modulo $G_i$ . Finally, for all $i \in Q_0$ , let $\text {irr}(G_i)$ be a complete set of representatives of the isomorphism classes of the irreducible representations of $G_i$ ; it is convenient to assume that $\rho = kG_i e_\rho $ for some primitive idempotent $e_\rho $ of $kG_i$ , for all $\rho \in \text {irr}(G_i)$ .

1. (1) Let $\epsilon $ be the following idempotent of $kQ*G:$

   $$\begin{align*}\epsilon = \sum_{i \in [G\backslash Q_0], \rho \in \text{irr}(G_i)} e_i * e_\rho\,,\end{align*}$$
   where $e_i$ ranges over a complete set of representatives of the G-orbits of vertices of Q and $\rho $ ranges over a complete set of representatives of the isomorphism classes of the irreducible representations of $G_i$ .

2. (2) Let $Q_{G}$ be the quiver defined as follows:

   • ⋆ Its vertices are the pairs $(i, \rho ),$ where $i \in [G\backslash Q_0]$ and $\rho \in \text {irr}(G_i)$ .

   • ⋆ For all vertices $(i, \rho )$ and $(j, \tau )$ , denote by $M(i, j; \tau )$ the following vector subspace of $kQ*G:$

     $$\begin{align*}M(i, j; \tau) = \bigoplus_{y\in[G/G_{j}]} (e_i (kQ_1) e_{y\cdot j}) * (y (kG_j) e_\tau). \end{align*}$$

     Note that $kQ*G$ is a representation of G through left multiplication and, by restriction, it is a representation of $G_i$ ; then $M(i, j; \tau )$ is a sub-representation of this left $G_i$ -module.

   • ⋆ The vector space spanned by the set of arrows in $Q_{G}$ from $(i, \rho )$ to $(j, \tau )$ is $\text {Hom}_{G_i}(\rho , M(i, j; \tau ))$ .

By [Reference Le Meur29, Section 4.4], there exists an isomorphism of algebras

$$\begin{align*}\gamma_{Q}\colon kQ_{G} \xrightarrow{\sim} \epsilon (kQ*G) \epsilon \end{align*}$$

given as follows:

• • For all vertices $(i,\rho )$ of $Q_{G}$ , the corresponding idempotent of $kQ_{G}$ is mapped onto $e_i * e_\rho $ .

• • For all vertices $(i, \rho )$ and $(j,\tau )$ of $Q_{G}$ , and for all $f \in \text {Hom}_{G_i}(\rho , M(i, j; \tau ))$ , then f is mapped onto $f(e_\rho )$ . The arrow $f : (i, \rho ) \rightarrow (j,\tau )$ of $Q_{G}$ can be identified with the corresponding element $f(e_\rho )$ of $(e_i * e_\rho ) \cdot (kQ*G) \cdot (e_j * e_\tau )$ .

Similarly, let $\theta $ be the following idempotent of $kF*G$ :

$$\begin{align*}\theta = \sum_{i \in [G\backslash F_0], \rho \in \text{irr}(G_i)} e_i * e_\rho.\end{align*}$$

Let $F_{G}$ be the quiver defined as follows:

• ⋆ Its vertices are the pairs $(i, \rho ),$ where $i \in [G\backslash F_0]$ and $\rho \in \text {irr}(G_i)$ .

• ⋆ For all vertices $(i, \rho )$ and $(j, \tau )$ , denote by $N(i, j; \tau )$ the following vector subspace of $kF*G$ :

  $$\begin{align*}N(i, j; \tau) = \bigoplus_{y\in[G/G_{j}]} (e_i (kF_1) e_{y\cdot j})*(y (kG_j) e_\tau). \end{align*}$$

• ⋆ The vector space spanned by the set of arrows in $Q_{G}$ from $(i, \rho )$ to $(j, \tau )$ is $\text {Hom}_{G_i}(\rho , N(i, j; \tau ))$ .

By [Reference Le Meur29, Section 4.4], we also have an isomorphism of algebras

(5) $$ \begin{align} \gamma_{F}\colon kF_{G} \xrightarrow{\sim} \theta (kF*G) \theta \end{align} $$

given as follows:

• • For all vertices $(i,\rho )$ of $F_{G}$ , the corresponding idempotent of $kF_{G}$ is mapped onto $e_i * e_\rho $ .

• • For all vertices $(i, \rho )$ and $(j,\tau )$ of $F_{G}$ , and for all $f \in \text {Hom}_{G_i}(\rho , N(i, j; \tau ))$ , then f is mapped onto $f(e_\rho )$ . The arrow $f\colon (i, \rho ) \rightarrow (j,\tau )$ of $F_{G}$ can be identified with the corresponding element $f(e_\rho )$ of $(e_i * e_\rho ) \cdot (kQ*F) \cdot (e_j * e_\tau )$ .

It is not hard to see that $F_{G}$ is a graded sub-quiver of $Q_{G}$ . And the inclusion of algebras

$$ \begin{align*}\iota\colon kF_{G}\hookrightarrow kQ_{G}\end{align*} $$

induces the following commutative diagram:

(6)

Corollary 3.19. The equivalence $ {\mathcal D}(\boldsymbol {\Gamma }_{n}(Q_{G},F_{G},W_G))\xrightarrow {\sim }{\mathcal D}(\boldsymbol {\Gamma }_{n}(Q,F,W)*G)$ restricts to equivalences

$$ \begin{align*}\mathrm{per}(\boldsymbol{\Gamma}_{n}(Q_{G},F_{G},W_G))\xrightarrow{\sim}\mathrm{per}(\boldsymbol{\Gamma}_{n}(Q,F,W)*G)\end{align*} $$

and

$$ \begin{align*}\mathrm{pvd}(\boldsymbol{\Gamma}_{n}(Q_{G},F_{G},W_G))\xrightarrow{\sim}\mathrm{pvd}(\boldsymbol{\Gamma}_{n}(Q,F,W)*G).\end{align*} $$

Example 3.20. Let $ (Q,F,W) $ be the following ice quiver with potential:

By [Reference Wu39, Corollary 8.7], the relative Ginzburg algebra $\boldsymbol {\Gamma }_{3}(Q,F,W)$ is quasi-isomorphic to $H^{0}(\boldsymbol {\Gamma }_{3}(Q,F,W))$ . We define a $ \mathbb {Z}/2\mathbb {Z}=\langle \sigma \rangle $ -action on $ (Q,F) $ as follows:

• • $\sigma (1)=1$ , $\sigma (2)=3$ , $\sigma (4)=4,$ and $\sigma (5)=6$ .

• • $\sigma (a)=-b$ , $\sigma (c)=c$ , $\sigma (e)=d$ , $\sigma (s)=r,$ and $\sigma (f)=-g$ .

It is clear that W is invariant up to cyclic permutation. We compute the following data to define $Q_{G} $ :

• • $[G\setminus Q_{0}]=\{1,2,4,5\}$ , $G_{1}=G=G_{4}$ , and $G_{2}=\{\mathbf {1}\}=G_{5}$ .

• • $[G/G_{1}]=\{\mathbf {1}\}=[G/G_{4}]$ and $[G/G_{2}]=G=[G/G_{5}]$ .

• • The irreducible representations of $G_1$ and $G_4$ consist of the trivial representation $\rho _+$ given by $kG \cdot (\mathbf {1} +\sigma )$ and the non-trivial one $\rho _-$ given by $kG \cdot (\mathbf {1}-\sigma )$ .

• • The irreducible representations of $G_2$ and $G_{5}$ consist of the trivial representation $\mathbb {K}$ of $\{\mathbf {1}\}$ .

Then we have $\epsilon =\epsilon _{1}^{+}+\epsilon _{1}^{-}+\epsilon _{4}^{+}+\epsilon _{4}^{-}+\epsilon _{2}+\epsilon _{5}$ , where $\epsilon ^{+}_{1} = \frac {1}{2} (e_1 \ast \mathbf {1} + e_1 \ast \sigma )=\epsilon ^{+}_{4}$ , $\epsilon ^{-}_{1} = \frac {1}{2} (e_1 \ast \mathbf {1} - e_1 \ast \sigma )=\epsilon ^{-}_{4}$ , and $\epsilon _2 = e_2 \ast \mathbf {1}=\epsilon _5$ . Hence,

$$ \begin{align*}M(1,2;k)=\mathrm{span}(a*\mathbf{1},b*\sigma)\simeq\mathrm{span}(\epsilon_{1}^{+}a\epsilon_{2})\oplus\mathrm{span}(\epsilon_{1}^{-}a\epsilon_{2})\simeq\rho_+\oplus\rho_-,\end{align*} $$

$$ \begin{align*}M(4,5;k)=\mathrm{span}(f*\mathbf{1},g*\sigma)\simeq\rho_+\oplus\rho_-,\end{align*} $$

$$ \begin{align*}M(4,1;\rho_+)=\mathrm{span}(c\epsilon^{+}_{1})\simeq\rho_+,\end{align*} $$

$$ \begin{align*}M(4,1;\rho_-)=\mathrm{span}(c\epsilon^{-}_{1})\simeq\rho_-,\end{align*} $$

$$ \begin{align*}M(2,4;\rho_+)=\mathrm{span}(e\epsilon^{+}_{1})\simeq\mathbb{K},\end{align*} $$

$$ \begin{align*}M(2,4;\rho_-)=\mathrm{span}(e\epsilon^{+}_{1})\simeq\mathbb{K},\end{align*} $$

$$ \begin{align*}M(5,2;\mathbb{K})=\mathrm{span}(s*\mathbf{1})\simeq\mathbb{K}.\end{align*} $$

Then the quiver $Q_{G}$ is given as follows:

The potential $W_G$ on $Q_{G}$ is given by $x_{12}^{+}x_{41}^{+}x_{24}-x_{52}x_{45}^{+}x_{24}+x_{45}^{-}x_{24}x_{52}-x_{24}x_{12}^{-}x_{41}^{-}$ .

Similarly, the quiver $F_{G}$ is given by the following full sub-quiver of $Q_{G}$ :

We thus obtain an ice quiver with potential $(Q_{G},F_{G},W_G)$ and $\boldsymbol {\Gamma }_{3}(Q,F,W)*G$ is derived Morita equivalent to $ \boldsymbol {\Gamma }_{3}(Q_{G},F_{G},W_G)$ .

Let $\overline {Q}$ be the quiver obtained from Q by deleting all vertices in F and all arrows incident with vertices in F. Let $\overline {W}$ be the potential on Q obtaining by deleting all cycles passing through vertices of F in W. We define $(\overline {Q_{G}},\overline {W_{G}})$ similarly.

Corollary 3.21 [Reference Le Meur29, Corollary 4.4.3]

There is a non-unital dg algebra homomorphism

$$ \begin{align*}\boldsymbol{\Gamma}_{n}(\overline{Q_{G}},\overline{W_G})\rightarrow\boldsymbol{\Gamma}_{n}(\overline{Q},\overline{W})*G\end{align*} $$

and the restriction-of-scalars functor induces the following equivalences of triangulated categories:

$$\begin{align*}{\mathcal D}(\boldsymbol{\Gamma}_{n}(\overline{Q},\overline{W})*G)\simeq{\mathcal D}(\boldsymbol{\Gamma}_{n}(\overline{Q_{G}},\overline{W_G})),\end{align*}$$

$$\begin{align*}\mathrm{per}(\boldsymbol{\Gamma}_{n}(\overline{Q},\overline{W})*G)\simeq\mathrm{per}(\boldsymbol{\Gamma}_{n}(\overline{Q_{G}},\overline{W_G})),\end{align*}$$

$$\begin{align*}\mathrm{pvd}(\boldsymbol{\Gamma}_{n}(\overline{Q},\overline{W})*G)\simeq\mathrm{pvd}(\boldsymbol{\Gamma}_{n}(\overline{Q_{G}},\overline{W_G})).\end{align*}$$