3.1 Some background on dg categories

Here, we give a brief summary of dg categories and their modules. We follow Section 2.1 of [Reference Polishchuk and Positselski7].

A right module over a dg category $\mathscr {C}$ is a dg functor $M:\mathscr {C}^{\mathrm{op}}\to \text {Com}(\mathbb {F}),$ where the later is the dg category whose objects are chain complexes of vector spaces and whose morphism complexes are defined in the usual way. Similarly, we define a left modules as functors $\mathscr {C}\to \text {Com}(k)$ . The collection of right (resp. left) dg modules themselves form a dg category in a natural way which we denote $\mathrm{mod}-\mathscr {C}$ (resp. $\mathscr {C}-\mathrm{mod}$ ). For any $c\in \mathsf {Ob}(\mathscr {C}),$ we denote by $h_c$ the right module represented by c. Modules of the form $\bigoplus _{i\in I}h_{c_i}\otimes V_i,$ where the $V_i$ ’s are chain complexes of vector spaces are called free modules and they are finitely generated free modules if I is finite and each $V_i$ is finite dimensional. A module M is semi-free if it admits a filtration

$$ \begin{align*}0=F_0\subset F_1\subset F_2\subset \dots M\end{align*} $$

such that $M=\cup _{n\geq 0}F_n$ and each sub quotient $F_{i+1}/F_i$ is free. The category ${Z^0(\mathrm{mod}-\mathscr {C})}$ has the structure of an exact category where a sequence of modules

$$ \begin{align*}M\to M"\to M\end{align*} $$

is defined to be exact if for any $c\in \mathsf {Ob}(\mathscr {C}),$ both

$$ \begin{align*}M(c)\to M"(c)\to M'(c)\end{align*} $$

is an exact sequence of complexes and

$$ \begin{align*}H(M(c))\to H(M"(c))\to H(M'(c))\end{align*} $$

is an exact sequence of graded vector spaces.

Proof It’s enough to consider a module of the form $V\otimes h^c$ , where V is a chain complex and $c\in \mathsf {Ob}(c)$ . Consider a diagram

The map $\varphi $ corresponds to a degree $0$ cycle

$$ \begin{align*}\hat{\varphi}\in \text{Hom}_{Z^0(\mathrm{mod}-\mathscr{C})}(h_c,\text{Hom}_k^\bullet(V,M(-))=\text{Hom}_{Ch(k)}(V,M(c)),\end{align*} $$

where $\text {Hom}_{Ch(k)}(A,B)$ denotes chain maps between complexes of vector spaces. Because $X(c)\to M(c)$ and $H(X(c))\to H(M(c))$ are both surjective, the induced map $\text {Hom}_{Ch(k)}(V,X(c))\to \text {Hom}_{Ch(k)}(V,M(c))$ is also surjective so we find a lift of $\hat {\varphi }$ , call it $\hat {\psi }$ . It corresponds to a map $\psi :V\otimes h_c\to X$ which makes the triangle above commute.

If M and N are $\mathscr {C}-$ modules, then a degree zero cycle $\varphi :M\to N$ gives rise to chain maps $\varphi _c:M(c)\to N(c)$ for every $c\in \mathsf {Ob}(\mathscr {C})$ . We say that $\varphi $ is a quasi isomorphism if $\varphi _c$ is a quasi isomorphism for all $c\in \mathsf {Ob}(\mathscr {C})$ . We define the cone of $\varphi $ to be the module $\mathrm{Cone}(\varphi ):c\mapsto \mathrm{Cone}(\varphi _c)$ . The homotopy category of the dg category of modules $H^0(mod-\mathscr {C})$ can be equipped with the structure of a triangulated category where the distinguished triangles are the ones isomorphic to triangles of the form

We define a perfect module to be one that lies in the smallest, thick triangulated subcategory of $H^0(\mathrm{mod}-\mathscr {C})$ which contains all the finitely generated free modules. We denote by $\text {perf}(\mathscr {C})$ the full sub category of $\mathrm{mod}-\mathscr {C}$ whose objects are perfect modules.

If $\mathscr {C}$ and $\mathscr {B}$ are two dg categories, we can form their tensor product $\mathscr {C}\otimes \mathscr {B}$ which by definition is the dg category whose objects are pairs $(c,b)$ with $c\in \mathsf {Ob}(\mathscr {C})$ and $b\in \mathsf {Ob}(\mathscr {B})$ and the morphism complexes are defined by

$$ \begin{align*}(\mathscr{C}\otimes \mathscr{B})((c,b),(c',b')):= \mathscr{C}(c,c')\otimes\mathscr{B}(b,b').\end{align*} $$

When defining composition of morphisms in $\mathscr {C}\otimes \mathscr {B}$ , we take the Koszul sign rule into account.

A right module over $\mathscr {C}\otimes \mathscr {B}^{\mathrm{op}}$ is also called a $\mathscr {B}-\mathscr {C}$ -bimodule. If M is a right $\mathscr {C}$ module and N is a left $\mathscr {C}$ , we define their tensor product $M\otimes _{\mathscr {C}}N$ to be the cokernel of the following map:

$$ \begin{align*}\bigoplus_{c,c'\in \mathscr{C}}M(c)\otimes \mathscr{C}(c',c)\otimes N(c)\to \bigoplus_{c\in \mathscr{C}}M(c)\otimes N(c), \ m\otimes f\otimes n\mapsto mf\otimes n-m\otimes fn.\end{align*} $$

Note that $M\otimes _{\mathscr {C}}N$ is just a chain complex. Similarly, if M is a $\mathscr {B}-\mathscr {C}$ -bimodule and N is a $\mathscr {C}-\mathscr {D}$ bimodule, we can define their tensor product $M\otimes _{\mathscr {C}}N$ which will be a $\mathscr {B}-\mathscr {D}$ -bimodule.

For any $\mathscr {C}$ , we have two diagonal bimodules. We denote by $\Delta ^r_{\mathscr {C}}$ the right $\mathscr {C}\otimes \mathscr {C}^{op}$ module defined by

$$ \begin{align*}(c_1,c_2)\mapsto \mathscr{C}(c_1,c_2).\end{align*} $$

We denote by $\Delta _{\mathscr {C}}^l$ the left $\mathscr {C}\otimes \mathscr {C}^{op}$ module defined by

$$ \begin{align*}(c_1,c_2)\mapsto \mathscr{C}(c_2,c_1).\end{align*} $$

There is a standard way to resolve $\Delta _{\mathscr {C}}^l$ by projective left $\mathscr {C}\otimes \mathscr {C}^{op}$ -modules, called the bar resolution which we will denote by $\text {Bar}^\bullet (\mathscr {C})$ . It is defined by

$$ \begin{align*}\text{Bar}^n(\mathscr{C}):=\Big((x,y)\mapsto\bigoplus_{c_0,\dots,c_{n}\in \mathscr{C}}\mathscr{C}(c_0,x)\otimes\mathscr{C}(c_1,c_0)\otimes\cdots\otimes \mathscr{C}(y,c_{n})\Big)\end{align*} $$

for $n\geq 1$ . For each $n\geq 1$ , we have the map of left $\mathscr {C}\otimes \mathscr {C}^{op}$ modules $d^n:\text {Bar}^n(\mathscr {C})\to \text {Bar}^{n-1}(\mathscr {C})$ defined by

$$ \begin{align*}f_0|f_1|\cdots|f_{n+1}\mapsto \sum_{i=0}^{n+1}(-1)^{i}f_0|\cdots |f_if_{i+1}|\cdots |f_{n+1}\end{align*} $$

and we have the map $d^0:\text {Bar}^0(\mathscr {C})\to \Delta _{\mathscr {C}}^l$ defined by $f_0|f_1\mapsto f_0f_1$ . Note that each $\text {Bar}^n(\mathscr {C})$ is projective as a left $\mathscr {C}\otimes \mathscr {C}^{op}$ modules by Lemma 3.1. The fact that this is a resolution in the exact structure follows from the fact that for any pair $(x,y),$ we have a contracting homotopy (which is not natural in x and y however)

$$ \begin{align*}\text{Bar}^{n}(\mathscr{C})\to\text{Bar}^{n+1}(\mathscr{C}), \ f_0|\cdots |f_{n+1}\mapsto \mathrm{id}_x|f_0|\cdots |f_{n+1}.\end{align*} $$

We can also consider the total direct sum complex $\text {Tot}^{\oplus }(\text {Bar}^\bullet (\mathscr {C}))$ . This is a semi-free module with the filtration $F_n=\text {Bar}^{\leq n}(\mathscr {C})$ . The map $d^0:\text {Bar}^0(\mathscr {C})\to \Delta _{\mathscr {C}}^r$ gives rise to a quasi isomorphism $\text {Tot}^{\oplus }(\text {Bar}^\bullet (\mathscr {C}))\overset {\sim }{\to }\Delta _{\mathscr {C}}^r$ .

3.2 Hochschild homology

We first define the standard complex, which we will use to compute Hochschild homology.

Definition 3.1 Let $\mathscr {C}$ be a small dg category, and let M be a $\mathscr {C}-\mathscr {C}$ -bimodule. Set

$$ \begin{align*}C_n(\mathscr{C};M):=\bigoplus_{c_0,\dots,c_n\in \mathscr{C}}M(c_1,c_0)\otimes \mathscr{C}(c_2,c_1)\otimes\cdots\otimes \mathscr{C}(c_0,c_n).\end{align*} $$

Note that each summand above is a chain complex whose differential, denoted $d_1$ , is obtained from the differentials in $\mathscr {C}$ and the differential on the bimodule M. We will use the bar notation for elements in the standard complex; to shorten notation, we will denote a pure tensor

$$ \begin{align*}a_0\otimes a_1\otimes\cdots\otimes a_n\in C_n(\mathscr{C};F)\end{align*} $$

by $a_0[a_1|\cdots |a_n]$ . There is a second differential $d_{2}:C_{n}(\mathscr {C};F)\to C_{n-1}(\mathscr {C};F)$ defined by

$$ \begin{align*}d_2\big(m[a_1|\cdots |a_n]\big)&=ma_1[a_2|\cdots|a_n]\\ &\quad +\sum_{i=1}^{n-1}(-1)^{i}m[\cdots|a_ia_{i+1}|\cdots]\\ &\quad +(-1)^{n+|a_n|(|a_0|+\cdots+|a_{n-1}|)}a_nm[a_1|\cdots|a_{n-1}]. \end{align*} $$

The differentials $d_1$ and $d_2$ commute; and therefore, we get a total complex (cohomologically graded) by altering the sign of $d_1$ by $(-1)^n$ on $C_n(\mathscr {C};M)$ whose $k'$ th term for $k\in \mathbb {Z}$ is

$$ \begin{align*}\bigoplus_{n-m=k}C_m(\mathscr{C};M)^n.\end{align*} $$

The homology of this complex is the Hochschild homology of $\mathscr {C}$ with values in M and it is denoted $HH_\bullet (\mathscr {C};M)$ . We define $HH_\bullet (\mathscr {C}):=HH_\bullet (\mathscr {C};\Delta _{\mathscr {C}}^r)$ .

Now suppose we have two dg categories $\mathscr {C}$ , $\mathscr {C}'$ , two bimodules ${M\kern1.4pt{\in}\kern1.4pt \mathrm{bimod}\kern1.4pt{-}\kern1.4pt\mathscr {C}\kern1.4pt{-}\kern1.4pt\mathscr {C}}$ , $M'\in \mathrm{bimod}-\mathscr {C}'- \mathscr {C}'$ , a functor $\phi :\mathscr {C}\to \mathscr {C}'$ and a morphism of of bimodules $\epsilon :M\to \phi ^*(M'),$ where $\phi ^*(M')$ is the $\mathscr {C}-\mathscr {C}$ -bimodule defined by $\phi ^*(M')(c_1,c_2)=M'(\phi (c_1),\phi (c_2))$ . Then we get an induced morphism

$$ \begin{align*}(\phi,\epsilon)_*:HH_\bullet(\mathscr{C};M)\to HH_\bullet(\mathscr{C}';M')\end{align*} $$

defined as follows. First, if $m[f_1|\cdots |f_n]\in C_n(\mathscr {C};M),$ we map it to $\epsilon (m)[\phi (f_1)|\cdots |\phi (f_n)]\in C_n(\mathscr {C}';M')$ . This defines a chain map between standard complexes and $(\phi ,\epsilon )_*$ is the induced map at the level of homology. We will refer to the data of such a $\phi $ and $\epsilon $ as $(\phi ,\epsilon ):(\mathscr {C},M)\to (\mathscr {C}',M')$ .

Suppose, we have three dg categories with bimodules $(\mathscr {C},M), (\mathscr {C}',M')$ and $(\mathscr {C}",M")$ and $(\phi ,\eta ):(\mathscr {C},M)\to (\mathscr {C}',M')$ , $(\psi ,\epsilon ):(\mathscr {C}',M')\to (\mathscr {C}",M")$ . In this situation, we define

$$ \begin{align*}(\psi,\epsilon)\circ (\phi,\eta):=(\psi\circ \phi,\phi^*(\epsilon)\circ \eta:M\to \phi^*\psi^*(M")=(\psi\circ \phi)^*(M")).\end{align*} $$

As a consequence of the above lemma, Hochschild homology can be thought of as a functor on the category whose objects are pairs $(\mathscr {C},M)$ and whose morphisms are pairs $(\phi ,\epsilon )$ .

A special kind of a $\mathscr {C}-\mathscr {C}$ -bimodule arise when you have an endofunctor $F:\mathscr {C}\to \mathscr {C}$ . The associated bimodule, which we will denote by $\widetilde {F}$ , is defined by $\widetilde {F}(c,c') :=\mathscr {C}(c',F(c))$ . We will often denote $HH_\bullet (\mathscr {C};\tilde {F})$ simply by $HH_\bullet (\mathscr {C};F)$ . Now suppose we have a dg functor $\phi :\mathscr {C}\to \mathscr {C}'$ , and bimodules $M=\widetilde {F}$ and $M'=\widetilde {F'}$ which are both associated with some endofunctors $F:\mathscr {C}\to \mathscr {C}$ and $F':\mathscr {C}'\to \mathscr {C}'$ . Then any natural transformation $\epsilon :\phi \circ F\implies F'\circ \phi $ gives rise to a bimodule map $\widetilde {F}\to \phi ^*(\widetilde {F'})$ which is given pointwise by

and so we get an induced map $(\phi ,\epsilon )_*:HH_\bullet (\mathscr {C},F)\to HH_\bullet (\mathscr {C}';F')$ . At the chain level, the induced map is given by

$$ \begin{align*}(\phi,\epsilon)_*(f_0[f_1|\cdots |f_k])=\epsilon\circ \phi(f_0)[\phi(f_1)|\cdots |\phi(f_k)],\end{align*} $$

where $f_0\in \mathscr {C}(c_1,F(c_0))$ , $f_1\in \mathscr {C}(c_2,c_1)$ ,…, $f_k\in \mathscr {C}(c_0,c_k)$ .

In Lemma 3.3, if the bimodules $M,M'$ and $M"$ all come from endofunctors $F,F'$ and $F"$ and the bimodule maps come from natural transformations $\eta : \phi \circ F\implies F'\circ \phi $ and $\epsilon :\psi \circ F'\implies F"\circ \psi $ as described above, then Lemma 3.3 has the following reformulation.

The following two lemmas will only be used in the setting of Hochschild homology with coefficients in an endofunctor so we don’t formulate them in the generality of bimodules.

Lemma 3.5 Let $\mathscr {C}, \ \mathscr {C}'$ be dg categories, and let $F\in \mathrm {End}(\mathscr {C}),F'\in \mathrm {End}(\mathscr {C}')$ be two endofunctors. For $i=1,\dots ,k$ , let $A_i:\mathcal {C}\to \mathcal {C}'$ two dg functors and for $1\leq i,j\leq k$ let $\eta _{ij}:A_j\circ F\implies F'\circ A_i$ natural transformations. Set

$$ \begin{align*}\eta:=(\eta_{ij}):\bigoplus_{j=1}^k A_j\circ F\to \bigoplus_{i=1}^kF'\circ A_i.\end{align*} $$

Then

$$ \begin{align*}\Big(\bigoplus_{i=1}^k A_i,\eta\Big)_*\simeq \sum_{i=1}^k(A_i,\eta_{ii})_*\end{align*} $$

as maps $C_\bullet (\mathscr {C};F)\to C_\bullet (\mathscr {C}';F')$ .

Proof of lemma

By induction, it is enough to consider the case when $k=2$ . Let $\pi _i$ and $\iota _i$ be the natural projections and inclusions associated with the direct sum $A:=A_1\oplus A_2$ . Note that for any morphism f in $\mathscr {C}$ , we have $A(f)=\iota _1A_1(f)\pi _1+\iota _2A_2(f)\pi _2$ . We have

(3.1) $$ \begin{align}(A,\eta)_*(f_0[f_1|\cdots f_k])=\nonumber\\ \eta \circ A(f_0)[A(f_1)|\cdots |A(f_k)]=\nonumber\\ (\iota_1\eta_{11}+\iota_2\eta_{21})A_1(f_0)\pi_1[\iota_1A_1(f_1)\pi_1|\cdots |\iota_1A_1(f_k)\pi_1]+\nonumber\\ (\iota_1\eta_{12}+\iota_2\eta_{22})A_2(f_0)\pi_2[\iota_2A_2(f_1)\pi_2|\cdots |\iota_2A_2(f_k)\pi_2]+\nonumber\\ \text{mixed terms}. \end{align} $$

For a term in (3.1), we say that the parity changes at position l (for some $0\leq l\leq k$ ) if the $l+1$ ’th factor ends with a $\iota _2$ and the $l'$ th factor begins with a $\pi _1$ or vice versa. When $l=k$ here then the $l+1$ ’th factor refers to the $0'th$ factor. So, for example,

$$ \begin{align*}\iota_1\eta_{11}A_1(f_0)\pi_1[\iota_2A_2(f_1)\pi_2|\iota_2A_2\pi_2]\end{align*} $$

changes parity at positions $0$ and $2$ and

$$ \begin{align*}\iota_1\eta_{12}A_2(f_0)\pi_2[\iota_2A_2(f_1)\pi_2|\iota_2A_2\pi_2]\end{align*} $$

changes parity only at position 2.

For $n\geq 0$ and $i=1,2,$ let $\phi ^{n}_i$ be the sum of all the terms in (3.1) which changes parity at n distinct positions and which has $A_i(f_0)$ in position zero. Then, each $\phi ^n_i$ is a chain map and $(A_1\oplus A_2,\eta )_*=\sum _{n,i}\phi ^n_i.$ We have

$$ \begin{align*} \phi_i^0(f_0[f_1|\cdots f_k])=\iota_i\eta_{ii}A_i(f_0)\pi_i[\iota_i A_i(f_1)\pi_i|\cdots |\iota_i A_i(f_k)\pi_i] \end{align*} $$

and this is homotopic to $(A_i,\eta _{ii})_*$ via the homotopy

$$ \begin{align*}H_i(f_0[f_1|\cdots |f_k]):=\\\sum_{j=0}^k (-1)^{j} \eta_{ii} A_i(f_0)\pi_i[\iota_iA_i(f_1)\pi_i|\cdots |\iota_iA_i(f_j)\pi_i|\iota_i| A_i(f_{j+1})|\cdots |A_i(f_k)].\end{align*} $$

Next, for $n\geq 1,$ we have $\phi ^n_1\simeq 0$ via the homotopy

$$ \begin{align*}H(f_0[f_1|\cdots |f_k]):=\\\sum (-1)^i\iota_\alpha\eta_{\alpha\beta}A_1(f_0)\pi_1[\iota_1A_1(f_1)\pi_1|\cdots|\iota_2| A_2(f_i)\pi_2|\cdots ]. \end{align*} $$

Here, the sum is over all the ways in which we can start with $A_1(f_0)$ and then change parity at precisely n positions and then we’ve inserted $\iota _2$ after the last position that has $A_1$ in it. Similarly, $\phi _2^n\simeq 0$ for $n\geq 1$ . This proves the lemma.

Now suppose we have $(\phi ,\eta ):(\mathscr {C},F)\to (\mathscr {C}',F')$ and $\alpha :\phi \overset {\sim }{\implies }\psi $ is an isomorphism of functors. Then by abuse of notation, we will denote by $\alpha \eta \alpha ^{-1}$ the natural transformation $\psi \circ F\implies F'\circ \psi $ defined as the composite

Proof The following homotopy proves the equality at the level of homology:

$$ \begin{align*}H(f_0[f_1|\cdots |f_k]):=\\\sum_{i=0}^{k} (-1)^{i}\eta\alpha^{-1}\psi(f_0)[\psi(f_1)|\cdots |\psi(f_i)|\alpha|\phi(f_{i+1})|\cdots|\phi(f_k)].\\[-40pt] \end{align*} $$

3.3 Künneth isomorphism

In this section, we recall the Künneth isomorphism for Hochschild homology and check that it is compatible with the $S_2$ action. For convenience, we include a proof. The proof given here is an adaption of the proof of Theorem 8.1 in Chapter 8 of [Reference Mac Lane5] to the setting of dg categories and dg modules.

Given two dg categories $\mathscr {C}$ and $\mathscr {B}$ , we define

$$ \begin{align*}Sh:C_\bullet(\mathscr{C})\otimes C_\bullet(\mathscr{B})\to C_\bullet(\mathscr{C}\otimes \mathscr{B})\end{align*} $$

by the formula

$$ \begin{align*} f_0[f_1|\cdots |f_k]\otimes g_0[g_1|\cdots |g_l]\mapsto \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\\\sum_{\sigma\in (k,l)-shuffles}(-1)^{|\sigma|+\epsilon} f_0\otimes g_0[\cdots |f_{i}\otimes 1|\cdots|1\otimes g_j|\cdots|f_{i+1}\otimes 1|\cdots |1\otimes g_j|\cdots]. \end{align*} $$

The sign in the formula above is given by

$$ \begin{align*}\epsilon=\sum |g_t||f_s|,\end{align*} $$

where the sum is over all pairs $(f_s,g_t)$ so that $f_s$ appears after $g_t$ .

Proposition 3.7 The map $Sh$ defined above induces an isomorphism

$$ \begin{align*}HH_\bullet(\mathscr{C})\otimes HH_\bullet(\mathscr{B})\to HH_\bullet(\mathscr{C}\otimes \mathscr{B}).\end{align*} $$

When $\mathscr {C}=\mathscr {B}$ this isomorphism commutes with the $S_2$ action.

Before we prove the proposition, we recall some facts about Hochschild homology. Just like for algebras, Hochschild homology can be defined in a “basis independent” way as $\Delta _{\mathscr {C}}^r\overset {L}{\otimes }\Delta _{\mathscr {C}}^l$ . As we show in the lemma below, the standard complex is then obtained by computing this derived tensor product using the bar resolution of $\Delta _{\mathscr {C}}^l$ described in Section 3.1.

Lemma 3.8 For each n, we have an isomorphism of chain complexes

$$ \begin{align*}C_n(\mathscr{C})\cong \Delta_{\mathscr{C}}^r\otimes_{\mathscr{C}\otimes \mathscr{C}^{op}} \mathrm{Bar}^n(\mathscr{C}).\end{align*} $$

Moreover, this isomorphism is compatible with the Hochschild and Bar differentials and so it gives an isomorphism

$$ \begin{align*}C_\bullet(\mathscr{C})\cong Tot^{\oplus}(\Delta_{\mathscr{C}}^r\otimes_{\mathscr{C}\otimes\mathscr{C}^{op}}\mathrm{Bar}^\bullet(\mathscr{C})).\end{align*} $$

Proof The complex on the right is the cokernel of

$$ \begin{align*}\bigoplus_{(x,y),(x',y')}\mathscr{C}(x,y)\otimes(\mathscr{C}(y',y)\otimes \mathscr{C}(x,x'))\otimes \text{Bar}^n(\mathscr{C})(y',x') \to\bigoplus_{(x,y)}\mathscr{C}(x,y)\otimes \text{Bar}^n(y,x)\end{align*} $$

$$ \begin{align*}f\otimes (\alpha\otimes \beta)\otimes B\mapsto (-1)^{|\alpha||f|}\alpha f\beta\otimes B-(-1)^{(|B|+|\beta|)|\alpha|}f\otimes \beta B\alpha.\end{align*} $$

The isomorphism is given induced by

$$ \begin{align*}\bigoplus_{(x,y)}\mathscr{C}(x,y)\otimes \text{Bar}^n(y,x)\to C_n(\mathscr{C})\end{align*} $$

$$ \begin{align*}f\otimes b_0|b_1|\cdots|b_{n+1}\mapsto (-1)^{|b_{n+1}|(|b_0|+\dots+|b_n|+|f|)}b_{n+1}fb_0[b_1|\cdots b_n]\end{align*} $$

and its inverse sends $a_0[a_1|\cdots |a_n]$ to the image of $a_0\otimes 1|a_1|\dots |a_n|1$ in the cokernel.

If M is a right (or left) $\mathscr {C}$ module and N is a right (or left) $\mathscr {B}$ module, then we can form their tensor product over k which will be a right (or left) module over $\mathscr {C}\otimes \mathscr {B}$

$$ \begin{align*}(M\otimes_k N)(c,b):=M(c)\otimes N(b).\end{align*} $$

Note that tensor product of representable modules is again representable. More generally, if M and N are projective modules of the special type described in Lemma 3.1, then their tensor product is again of that form.

Lemma 3.9 Let $\mathscr {C}$ and $\mathscr {B}$ be two small dg categories. We have a homotopy equivalence of left $\mathscr {C}\otimes \mathscr {B}\otimes \mathscr {C}^{\mathrm{op}}\otimes \mathscr {B}^{\mathrm{op}}$ -modules

$$ \begin{align*}Sh':\mathrm{Bar}(\mathscr{C})\otimes \mathrm{Bar}(\mathscr{B})\to \mathrm{Bar}(\mathscr{C}\otimes \mathscr{B}).\end{align*} $$

Proof The map is defined by

$$ \begin{align*}\Big(\bigoplus_{c_1,\dots,c_{p+1}}\mathscr{C}(c_1,-)\mathscr{C}(c_2,c_1)\cdots \mathscr{C}(-,c_{p+1})\Big)\otimes \Big(\bigoplus_{b_1,\dots,b_{q+1}}\mathscr{B}(b_1,-)\mathscr{C}(b_2,b_1)\cdots \mathscr{B}(-,b_{q+1})\Big)\\\longrightarrow \bigoplus_{(c_1,b_1),\dots,(c_{p+q+1},b_{p+q+1})}\mathscr{C}\mathscr{B}((c_1,b_1),-)\mathscr{C}\mathscr{B}((c_2,b_2),(c_1,c_1))\cdots \mathscr{C}\mathscr{B}(-,(c_{p+q+1},b_{p+q+1}))\\f_0|f_1|\cdots| f_p\otimes g_0| g_1|\cdots | g_q\longmapsto \\(-1)^{\epsilon_1} \sum(f_0\otimes g_0 )| \cdots|(f_i\otimes1)|\cdots |(1\otimes g_j)|\cdots |(f_{i+1}\otimes 1)|\cdots |(1\otimes_{g_{j+1}})|\cdots| (f_p\otimes g_q), \end{align*} $$

where the sum is over all $(p-1,q-1)-suffles \sigma .$ The signs are given by

$$ \begin{align*}&\epsilon_1=|g_0|(|f_1|+\dots+|f_p|)+|f_p|(|g_1|+\dots+|g_q|)\\&\epsilon_2= |\sigma|+\sum |f_i||g_j|, \end{align*} $$

where the sum is over all pairs $(f_i,g_j)$ such that $g_j$ appears before $f_i$ . This is a chain map which fits into a diagram:

Because both $\text {Bar}(\mathscr {C})\otimes \text {Bar}(\mathscr {B})$ and $\text {Bar}(\mathscr {C}\otimes \mathscr {B})$ form resolutions of $\mathscr {C}\otimes \mathscr {B}$ by projective $\mathscr {C}\otimes \mathscr {B}\otimes \mathscr {C}^{\mathrm{op}}\otimes \mathscr {B}^{\mathrm{op}}$ -modules $Sh'$ is a homotopy equivalence.

3.4 Morita invariance

A quasi functor from $\mathscr {C}$ to $\mathscr {C}'$ , denoted by a dashed arrow $\mathscr {C}\dashrightarrow \mathscr {C}'$ , is a functor $\mathscr {C}\to \text {perf}(\mathscr {C}')$ .

Quasi functors can alternatively be as certain $\mathscr {C}-\mathscr {C}'$ -bimodules P with the property that $P(-,c)$ is a perfect $\mathscr {C}'$ module for all $c\in \mathsf {Ob}(\mathscr {C})$ . Indeed, given a quasi functor F, we can associate with it a $\mathscr {C}-\mathscr {C}'$ -bimodule defined by $(c,c')\mapsto F(c)(c')$ . This bimodule clearly has the property described above. Conversely, any $\mathscr {C}-\mathscr {C}'$ -bimodule P, gives rise to a functor $T_P:\mathrm{mod}-\mathscr {C}\to \mathrm{mod}-\mathscr {C}'$ defined by $M\mapsto M\otimes _{\mathscr {C}}P$ . If $P(-,c)$ is a perfect $\mathscr {C}'$ -module for all $c\in \mathsf {Ob}(\mathscr {C}),$ then $T_P$ restricts to a functor $\text {perf}(\mathscr {C})\to \text {perf}(\mathscr {C}')$ which we can precompose with the Yoneda embedding of $\mathscr {C}$ if we like to get a functor $\mathscr {C}\to \text {perf}(\mathscr {C}')$ .

One special quasi functor $\mathscr {C}\dashrightarrow \mathscr {C}$ is given by the Yoneda embedding $Y:\mathscr {C}\to \text {perf}(\mathscr {C})$ . As a bimodule, this corresponds to the diagonal $\mathscr {C}-\mathscr {C}$ -bimodule $\Delta _{\mathscr {C}}^r$ introduced in Section 3.1. Tensoring with $\Delta _{\mathscr {C}}^r$ on the left or on the right is naturally isomorphic to the identity functor. We say that a quasi functor P is invertible if there is another quasi functor Q such that $P\otimes _{\mathscr {C}} Q\cong Q\otimes P\cong \Delta _{\mathscr {C}}^r$ .

Now if P is a quasi functor, which we think of as a $\mathscr {C}-\mathscr {C}$ -bimodule, then $T_P$ is an endofunctor $\text {perf}(\mathscr {C})\to \text {perf}(\mathscr {C})$ . If we denote by $\widetilde {T_P}$ the associated $\text {perf}(\mathscr {C})-\text {perf}(\mathscr {C})$ -bimodule, we have an isomorphism of bimodules $\epsilon : P\to Y^*(\widetilde {T_P})$ and as is explained in Section 3.2, we get an induced map $(Y,\epsilon )_*:HH_\bullet (\mathscr {C};P)\to HH_\bullet (\text {perf}(\mathscr {C});T_P)$ .

Proof This is [Reference Keller3] Theorem 2.4(b) but with coefficients. We include a proof for convenience. First, let us define

$$ \begin{align*}\mathscr{C}':=\text{perf}(\mathscr{C}), \ \text{Bar}(\mathscr{C}):=\text{Tot}^{\oplus}(\text{Bar}^\bullet(\mathscr{C})), \ \text{Bar}(\mathscr{C}'):=\text{Tot}^{\oplus}(\text{Bar}^\bullet(\mathscr{C})).\end{align*} $$

Let us also define the following $\mathscr {C}'-\mathscr {C}'$ -bimodules:

$$ \begin{align*}\text{Bar}^n(\mathscr{C}',\mathscr{C},\mathscr{C}'):=\Big((x',y')\mapsto\bigoplus_{c_0,\dots,c_{n}\in \mathscr{C}}\mathscr{C}'(c_0,x')\otimes\mathscr{C}(c_1,c_0)\otimes\dots\otimes \mathscr{C}'(y',c_{n})\Big).\end{align*} $$

Again, we have a differential $\text {Bar}^{n}(\mathscr {C}',\mathscr {C},\mathscr {C}')\to \text {Bar}^{n-1}(\mathscr {C}',\mathscr {C},\mathscr {C}')$ given by the formula

$$ \begin{align*}f_0|f_1|\cdots|f_{n+1}\mapsto \sum_{i=0}^{n+1}(-1)^{i}f_0|\cdots |f_if_{i+1}|\cdots |f_{n+1}\end{align*} $$

and we define $\text {Bar}(\mathscr {C}',\mathscr {C},\mathscr {C}'):=\text {Tot}^{\oplus } (\text {Bar}^\bullet (\mathscr {C}',\mathscr {C},\mathscr {C}'))$ .

Now, the first claim is that the natural morphism of bimodules $p:\text {Bar}(\mathscr {C}',\mathscr {C},\mathscr {C}')\to \Delta _{\mathscr {C}}^r$ is a quasi isomorphism. To see this, let $x\in \mathscr {C}$ and note that $y'\mapsto \mathrm{Cone}(p_{(x,y')})$ defines an exact functor $H^0(\mathscr {C}')\to H^0(\text {Com}(\mathbb {F}))$ which vanishes on $\mathscr {C}$ and must therefore vanish on all of $\mathscr {C}'$ . Now, let $y'\in \mathsf {Ob}(\mathscr {C}')$ and note that the functor $x'\mapsto \mathrm{Cone}(p_{(x',y')})$ defines an exact functor $H^0(\mathscr {C}')\to H^0(\text {Com}(\mathbb {F}))$ . By the previous argument, it vanishes on $\mathscr {C}$ and must therefore vanish on all of $\mathscr {C}'$ . This proves the claim.

By the previous claim, $\text {Bar}(\mathscr {C}',\mathscr {C},\mathscr {C}')$ is a semi-free replacement of $\Delta _{\mathscr {C}'}^r$ so we can use it to compute Hochschild homology with coefficients:

$$ \begin{align*}HH_\bullet(\mathscr{C}';T_P)\simeq \widetilde{T_P}\otimes_{\mathscr{C}'\otimes \mathscr{C}'^{\mathrm{op}}}\text{Bar}(\mathscr{C}',\mathscr{C},\mathscr{C}'). \end{align*} $$

Now there is a canonical map

$$ \begin{align*}Y^*(\widetilde{T_P})\otimes_{\mathscr{C}\otimes \mathscr{C}^{op}}Y^*(\text{Bar}(\mathscr{C}',\mathscr{C},\mathscr{C}'))\to \widetilde{T_P}\otimes_{\mathscr{C}'\otimes \mathscr{C}'^{\mathrm{op}}}\text{Bar}(\mathscr{C}',\mathscr{C},\mathscr{C}') \end{align*} $$

and it is an isomorphism (not just quasi isomorphism) by [Reference Polishchuk and Positselski7], Lemma 1.5B. It remains only to note that $Y^*(\widetilde {T_P})\cong P$ and $Y^*(\text {Bar}(\mathscr {C}',\mathscr {C},\mathscr {C}'))=\text {Bar}(\mathscr {C})$ .

4.1 Hochschild homology of equivariant category

Let us again consider an action of a finite group G on a small dg category $\mathscr {C}$ . Let $G^{gr}$ denote the groupoid whose objects are $\mathsf {Ob}(G^{\mathrm{gr}})=G$ and $G^{\mathrm{gr}}(g,g')=\{h\in G|h^{-1}gh=g'\}$ . Composition is given by the opposite of the product in G. Note that for any $g\in G,$ we have $G^{\mathrm{gr}}(g,g)=C(g)^{\mathrm{op}}$ where the latter denotes the centralizer of g in G. We will define a functor $A:G^{gr}\to \text {Vect}$ which sends $g\in G=\mathsf {Ob}(G^{\mathrm{gr}})$ to $HH_\bullet (\mathscr {C};\rho _g)$ . Let $h\in G^{\mathrm{gr}}(g,g')$ and define $A(h)=(\rho _h,C_{h,g,g'})_*:HH_\bullet (\mathscr {C};\rho _g)\to HH_\bullet (\mathscr {C},\rho _{g'})$ where $C_{h,g,g'}=\theta _{g',h}^{-1}\circ \theta _{h,g}$ .

Proof We have

$$ \begin{align*}(\rho_h,C_{h,g',g})\circ(\rho_{h'},C_{h',g",g'})=(\rho_h\circ\rho_{h'},C_{h,g',g}\star C_{h',g",g'}). \end{align*} $$

We also have $\theta _{h,h'}(C_{h,g}\star C_{h',g})\theta _{h,h'}^{-1}=C_{h'h,g",g}$ . It then follows from Lemmas 3.4 and 3.6 that $(\rho _h,C_{h,g})_*\circ (\rho _{h'},C_{h',g})_*=(\rho _{h'h},C_{h'h,g})_*$ .

Whenever we have a functor $A:G^{\mathrm{gr}}\to \text {Vect}$ , we get a right group action of G on $\bigoplus _{g\in G}V_g$ , where $V_g:=A(g)$ . Indeed, $g\in G$ acts on $V_{h}$ via $A(g:h\to g^{-1}hg):V_h\overset {\sim }{\to } V_{g^{-1}hg}$ . Moreover, for each $g\in G,$ this group action restricts to an action of $C(g)$ on $V_g$ .

Lemma 4.2 We have an isomorphism of vector spaces

$$ \begin{align*}\Big(\!\bigoplus_{g\in G}V_g\Big)^G\cong \bigoplus_{[g]\in G/\sim}V_g^{C(g)}.\end{align*} $$

The tensor product functor $\text {Rep}(G)\times \mathscr {C}^G\to \mathscr {C}^G$ makes $HH_\bullet (\mathscr {C}^G)$ into a module over the ring $R_{\mathbb {F}}(G):=R(G)\otimes _{\mathbb {Z}} \mathbb {F}$ , where $R(G)$ is the representation ring of G. When $\mathbb {F}$ is algebraically closed, it follows from character theory that this ring is nothing but the ring of functions on the set of conjugacy classes of G. In this setting, we denote by $\varphi _{[g]}\in R_{\mathbb {F}}(G)$ the indicator function on the conjugacy class of g. Note that the $\varphi _{[g]}$ form a collection of pairwise orthogonal idempotents which sum to the identity. Therefore, any $R_{\mathbb {F}}(G)$ -module M decomposes as $\bigoplus _{[g]\in G/\sim }\varphi _{[g]}\cdot M$ .

Now we are ready to state and prove our main theorem.

Theorem 4.3 Let G be a finite group which acts on a small dg category $\mathscr {C}$ and assume that $\mathscr {C}$ has finite direct sums. Then we have

$$ \begin{align*}HH_\bullet(\mathscr{C}^G)\cong\bigoplus_{[g]\in G/\sim}HH_\bullet(\mathscr{C};\rho_g)^{^{C(g)}},\end{align*} $$

where $\sim $ refers to conjugacy. Moreover, when the ground field is algebraically closed, this isomorphism identifies $HH_\bullet (\mathscr {C};\rho _g)^{C(g)}$ with the submodule $\varphi _{[g]}\cdot HH_\bullet (\mathscr {C}^G)\subset HH_\bullet (\mathscr {C}^G)$ .

Proof We will first define inclusions

$$ \begin{align*}\iota_g:HH_\bullet(\mathscr{C};\rho_g)^{C(g)}\to HH_\bullet(\mathscr{C}^G)\end{align*} $$

and projections

$$ \begin{align*}\pi_g:HH_\bullet(\mathscr{C}^G)\to HH_\bullet(\mathscr{C};\rho_g)^{C(g)}\end{align*} $$

and show that they give rise to a decomposition $HH_\bullet (\mathscr {C}^G)\cong \bigoplus _{[g]\in G/\sim } HH_\bullet (\mathscr {C},\rho _g)^{C(g)}$ . Then we check that the image of $\iota _g\circ \pi _g$ is precisely the submodule $\varphi _{[g]}HH_\bullet (\mathscr {C}^G)\subset HH_\bullet (\mathscr {C}^G)$ .

Let $U:\mathscr {C}^G\to \mathscr {C}$ and $S:\mathscr {C}\to \mathscr {C}^G$ be the functors introduced in Section 2. We define $\pi _g=(U,\alpha _g)_*:HH_\bullet (\mathscr {C}^G)\to HH_\bullet (\mathscr {C};\rho _g),$ where (by abuse of notation) $\alpha _g$ denotes the natural isomorphism $U\overset {\sim }{\implies }\rho _g\circ U$ whose value at an equivariant object $(c,\alpha )$ is $\alpha _g:c\to \rho _g(c)$ .

Claim 1 The map $\pi _g$ lands in $HH_\bullet (\mathscr {C};\rho _g)^{C(g)}$ .

Proof of claim 1

We consider $(\rho _h,C_{h,g})\circ (U,\alpha _g)=(\rho _h\circ U,C_{h,g}\star \alpha _g)$ and note that $\alpha _h:U\overset {\sim }{\implies }\rho _h\circ U$ and $C_{h,g}\star \alpha _g=\alpha _h \alpha _g\alpha _h^{-1}$ . The claim then follows from Lemmas 3.4 and 3.6.

Now define $\iota _g:HH_\bullet (\mathscr {C};\rho _g)^{C(g)}\to HH_\bullet (\mathscr {C}^G)$ by

$$ \begin{align*}HH_\bullet(\mathscr{C};\rho_g)^{C(g)}\to HH_\bullet(\mathscr{C};\rho_g)\overset{S_g}{\to}HH_\bullet(\mathscr{C}^G),\end{align*} $$

where $S_g=(S,\phi _{g})_*$ and $\phi _g:S\circ \rho _g\to S$ is the natural isomorphism

$$ \begin{align*}(\phi_g)_{hh'}=\delta_{gh'=h}\theta_{h',g}.\end{align*} $$

Claim 2 $\pi _g\circ \iota _g\simeq |C(g)|\cdot \mathrm {id}$ .

Proof of claim 2

Let us denote by $\hat {S}:=U\circ S$ and recall from Proposition 2.5 that it is isomorphic (or in fact equal) to $\bigoplus _{g\in G}\rho _g$ . By Lemma 3.4, we have $\pi _g\circ S_g=(U\circ S,\alpha _g\star \phi _g)_*$ . The natural transformation

$$ \begin{align*}\alpha_g\star\phi_g:\hat{S}\circ\rho_g=\bigoplus_{h\in G}\rho_{h}\rho_g\to \bigoplus_{h\in G}\rho_g\rho_h=\rho_g\circ\hat{S}\end{align*} $$

is given in matrix form by $(\alpha _g\star \phi _g)_{hh'}=\delta _{hg=gh'}\theta _{g,h}^{-1}\circ \theta _{h',g}$ . Then we get

$$ \begin{align*}\pi_g\circ S_g=&\Big(\!\bigoplus_{h\in G}\rho_h,\alpha_g\star\phi_g\Big)_*\\ =&\sum_{h\in G}(\rho_h,(\alpha_g\star \phi_g)_{hh})_*\\ =&\sum_{h\in C(g)}(\rho_h,C_{h,g})_*,\end{align*} $$

where we used Lemma 3.5 for the second equality and we used the description of ${(\alpha _g\star \phi _g)_{hh'}}$ for the last equality. Restrict this to $HH_\bullet (\mathscr {C},\rho _g)^{C(g)}$ gives precisely ${|C(g)|\cdot \mathrm {id}}$ .

Claim 3 $\pi _g\circ \iota _{g'}\simeq 0$ if $g\nsim g'$ .

Proof of claim 3

Note that $\pi _g\circ S_{g'}=(U\circ S,\alpha _g\star \phi _{g'})_*,$ where $\alpha _g\star \phi _{g'}$ is given in matrix notation by $(\alpha _g\star \phi _{g'})_{hh'}=\delta _{hg=g'h'}\theta _{g,h}^{-1}\circ \theta _{h',g'}$ . If $g\nsim g'$ we see that ${(\alpha _g\star \phi _{g'})_{hh}=0}$ and as in the previous claim we get

$$ \begin{align*}\pi_g\circ S_{g'}=&\Big(\!\bigoplus_{h\in G}\rho_h,\alpha_g\star\phi_{g'}\Big)_*\\ =&\sum_{h\in G}(\rho_h,(\alpha_g\star \phi_{g'})_{hh})_*\\ =&0,\end{align*} $$

where we used Lemma 3.5 for the second equality.

Claim 4 If g and $g'$ are conjugate, then $\iota _g\pi _g\simeq \iota _{g'}\pi _{g'}$ .

Proof of claim 4

Suppose $g'=h^{-1}gh. $ Consider the diagram

We have a natural isomorphism $\alpha _h:F\implies \rho _h\circ F$ and

$$ \begin{align*}(\rho_h,\theta_{g',h}^{-1}\circ \theta_{h,g})\circ (F,\alpha_g)=(\rho_h\circ F,\rho_{g'}(\alpha_h)\circ \alpha_{g'}\circ \alpha_h^{-1}).\end{align*} $$

Also, we have a natural isomorphism $\phi _h:S\circ \rho _h\implies S$ and

$$ \begin{align*}(S,\phi_{g'})\circ (\rho_h,\theta_{g',h}^{-1}\circ \theta_{h,g})=(S\circ \rho_h, \phi_h^{-1}\circ \phi_g\circ (\phi_h)_{\rho_g}).\end{align*} $$

Now using Lemmas 3.4 and 3.6, we get

$$ \begin{align*}\iota_g\pi_g &=(S,\phi_g)_*\circ (F,\alpha_g)_*\\&=(S,\phi_{g'})_*\circ (\rho_h,\theta_{g',h}^{-1}\circ \theta_{h,g})_*\circ (F,\alpha_g)_*\\ &=(S,\phi_{g'})_*\circ (F,\alpha_{g'})_*=\iota_{g'}\circ \pi_{g'}.\\[-34pt] \end{align*} $$

Claim 5

$$ \begin{align*}\sum_{[g]\in G/\sim}\frac{\iota_g\circ\pi_g}{|C(g)|}=\mathrm{id}_{HH_\bullet(\mathscr{C}^G).}\end{align*} $$

Proof of claim 5

By the previous claim, we have

$$ \begin{align*} \sum_{[g]\in G/\sim}\frac{\iota_g\circ\pi_g}{|C(g)|}&=\sum_{[g]\in G/\sim}|conj(g)|\frac{\iota_g\circ\pi_g}{|G|}\\ &=\frac{1}{|G|}\sum_{g\in G}\iota_g\circ \pi_g. \end{align*} $$

Next, we note that

$$ \begin{align*} \iota_g\circ \pi_g=(S,\phi_g)_*\circ(U,\alpha_g)_*=(S\circ U,\phi_g\star \alpha_g)_* \end{align*} $$

and therefore

$$ \begin{align*} \sum_{g\in G}\iota_g\circ \pi_g=(S\circ U,\sum_{g\in G}\phi_g\star\alpha_g)_*. \end{align*} $$

Moreover,

$$ \begin{align*}\sum_{g\in G}\phi_g\star\alpha_g:S\circ U\implies S\circ U\end{align*} $$

is given in matrix form by

$$ \begin{align*}\Big(\sum_{g\in G}\phi_g\star\alpha_g\Big)_{h'h}=\alpha_{h'}\circ \alpha_h^{-1}=(I\circ P)_{h'h},\end{align*} $$

where

$$ \begin{align*}I:\mathrm{id}_{\mathscr{C}^G}\implies S\circ U, \ P:S\circ U\implies \mathrm{id}_{\mathscr{C}^G}\end{align*} $$

are the inclusion- and projection-natural transformation associated with the identity component of (see Proposition 2.5). We may also point out that P is the counit of the adjunction in Proposition 2.4. Let $S'=S\circ U$ . Now the following formula defines a chain homotopy $(S',I\circ P)_*\simeq \mathrm {id}_{C_\bullet (\mathscr {C}^G)}$ :

$$ \begin{align*}H(f_0[f_1|\cdots |f_k)=\\ \sum_{i=0}^k (-1)^{i}P\circ S'(f_0)[S'(f_0)|\cdots |S'(f_i)|I|f_{i+1}|\cdots |f_k]. \end{align*} $$

Indeed, H as defined above commutes with $d_1$ and $d_2H+Hd_2=\mathrm {id}_{C_\bullet (\mathscr {C}^G)}-(S',I\circ P)_*.$

It remains to prove the second part of the theorem, that when the ground field is algebraically closed, we have $\iota _g\circ \pi _g(HH_\bullet (\mathscr {C}^G))=\varphi _{[g]}\cdot HH_\bullet (\mathscr {C}^G)$ . Since we know now that $HH_\bullet (\mathscr {C}^G)=\bigoplus _{[g]\in G/\sim }\iota _g\circ \pi _g(HH_\bullet (\mathscr {C}^G))$ and $HH_\bullet (\mathscr {C}^G)=\bigoplus _{[g]\in G/\sim }\varphi _{[g]}\cdot HH_\bullet (\mathscr {C}^G))$ it suffices to prove that $\iota _g\circ \pi _g(HH_\bullet (\mathscr {C}^G))\subset \varphi _{[g]}\cdot HH_\bullet (\mathscr {C}^G)$ for each conjugacy class $[g]$ . Note that for any $R_{\mathbb {F}}(G)$ module M then we have that

$$ \begin{align*}\varphi_{[g]}\cdot M=\{m\in M|[V]\cdot m=\chi_V(g)\cdot m\text{ for any representation }V\}.\end{align*} $$

Let $(V,\rho _V)$ be a representation with basis $\{e_1,\dots ,e_n\}$ . Let $T_V:=V\otimes (-):\mathscr {C}^G\to \mathscr {C}^G$ . Then, multiplication by $[V]$ on Hochschild homology is the same as $(T_V,1)_*$ . But note that using Proposition 2.5, we have the following isomorphism of functors:

(4.1) $$ \begin{align}T_V\circ S\circ U\cong T_V\circ T_{k[G]}\cong T_{k[G]}\circ T_V=S\circ U\circ T_V\cong \oplus_{i=1}^nS\circ U. \end{align} $$

Call the composite of these natural isomorphisms $\tau $ . It is given in matrix form by

$$ \begin{align*}\tau_{(i,k),(j,h)}=\delta_{h=k}\rho_V(k)_{ij},\end{align*} $$

where $1\leq i,j\leq n$ and $h,k\in G$ . Let $I_j:S\circ U\implies T_V\circ S\circ U$ and $P_j:T_V\circ S\circ U\implies S\circ U$ be the inclusion and projection-natural transformations associated with the decomposition (4.1) above. We have

$$ \begin{align*}P_j\circ T_V(\phi_g\star \alpha_g)\circ I_j=\rho_V(g)_{jj}\cdot(\phi_g\star\alpha_g).\end{align*} $$

We then have

$$ \begin{align*} (T_V,1)_*\circ \iota_g\circ \pi_g=&(T_V,1)_*\circ (S,\phi_g)_*\circ (U,\alpha_g)_*\\ =& (T_V\circ S\circ U,T_V(\phi_g\star \alpha_g))_*\\ =&\sum_{j=1}^n\rho_V(g)_{jj}\cdot (S\circ U,\phi_g\star\alpha_g)_*\\ =&\chi_V(g)( \iota_g\circ \pi_g), \end{align*} $$

where the second equality is Lemma 3.4 and the third equality combines the decomposition (4.1) above and Lemma 3.5.

Remark 4.4 In view of Lemma 4.2, the decomposition in Theorem 4.3 could be rewritten as

$$ \begin{align*}HH_\bullet(\mathscr{C}^G)\cong \Big(\bigoplus_{g\in G}HH_\bullet(\mathscr{C};\rho_g)\Big)^G.\end{align*} $$

4.2 Group actions via bimodules

Here, we consider a slightly more general notion of group actions.

Definition 4.1 Let G be a finite group, and let $\mathscr {C}$ be a dg category. A bimodule group action of G on $\mathscr {C}$ is the data of an invertible quasi functor $P_g$ for every ${g\in G}$ , bimodule isomorphisms $\theta _{g,g'}:P_{g'}\otimes _{\mathscr {C}}P_{g}\overset {\sim }{\to }P_{g'g}$ , $\eta :P_e\overset {\sim }{\to }\Delta _{\mathscr {C}}^r$ and these should satisfy the following two conditions.

i) For each $g\in G$ , we have

$$ \begin{align*}\theta_{g,e}=C\circ (\eta\otimes_{\mathscr{C}} \mathrm{id}_{P_g}):P_e\otimes_{\mathscr{C}} P_g \to P_g\end{align*} $$

and

$$ \begin{align*}\theta_{e,g}=C\circ (\mathrm{id}_{P_g}\otimes_{\mathscr{C}} \eta):P_g\otimes P_e\to P_g,\end{align*} $$

where C denotes either of the natural isomorphisms $\Delta _{\mathscr {C}}^r \otimes _{\mathscr {C}}(-)\cong \mathrm {id}_{\mathrm{bimod}-\mathscr {C}-\mathscr {C}}$ or $(-)\otimes _{\mathscr {C}} \Delta _{\mathscr {C}}^r \cong \mathrm {id}_{\mathrm{bimod}-\mathscr {C}-\mathscr {C}}$ .

ii) For all $g,h,k\in G$ , we have the following commutative diagrams:

Now, we observe that a bimodule action of G on $\mathscr {C}$ gives rise to a group action in the sense of Definition 2.1 of G on $\text {perf}(\mathscr {C}),$ where we define $\rho _g:=(-)\otimes _{\mathscr {C}} P_g$ . Theorem 4.3 gives us a decomposition

$$ \begin{align*}HH_\bullet(\text{perf}(\mathscr{C})^G)\cong \bigoplus_{g\in G/\sim}HH_\bullet(\text{perf}(\mathscr{C});\rho_g)^{C(g)}\end{align*} $$

and by Proposition 3.10 the right-hand side can be rewritten to obtain the following corollary.

Corollary 4.5 Suppose we are given a bimodule group action of G on $\mathscr {C}$ where $g\in G$ acts via the bimodule $P_g$ . Then we have a decomposition

$$ \begin{align*}HH_\bullet(\text{perf}(\mathscr{C})^G)\cong\bigoplus_{g\in G/\sim}HH_\bullet(\mathscr{C};P_g)^{C(g)}.\end{align*} $$

4.3 The case of symmetric powers

Now, for any dg category $\mathscr {C}$ , we can define its tensor powers. Recall that by definition $\mathscr {C}^{\otimes n}$ is the dg category with objects $x_\bullet =(x_1,x_2,\dots ,x_n)$ with each $x_i\in \mathsf {Ob}(\mathscr {C})$ . The homomorphism spaces are given by $\mathrm{Hom}_{\mathscr {C}^{\otimes n}}(x_\bullet ,y_\bullet ):=\mathscr {C}(x_1,y_1)\otimes \mathscr {C}(x_2,y_2)\otimes \dots \otimes \mathscr {C}(x_n,y_n)$ . There is a right action of $S^n$ on $\mathscr {C}$ defined on objects by $\sigma (x_1,\dots ,x_n)=(x_{\sigma (1)},\dots ,x_{\sigma (n)}).$ This action is strict in the strongest possible sense. All the higher categorical data are trivial, $\theta _{\sigma ,\tau }=\mathrm {id}$ , $\eta =\mathrm {id}$ . Let $\widehat {\mathscr {C}^{\otimes n}}$ be the pre triangulated hull of $\mathscr {C}^{\otimes n}$ and note that $S_n$ still acts on $\widehat {\mathscr {C}^{\otimes n}}$ . If we set $\text {Sym}^n(\mathscr {C})=(\widehat {\mathscr {C}^{\otimes n}})^{S_n}$ , then the theorem tells us

$$ \begin{align*}HH_\bullet(\text{Sym}^n(\mathscr{C}))=\bigoplus_{[\sigma]\in S_n/\sim } HH_\bullet\big(\widehat{\mathscr{C}^{\otimes n}},\rho_\sigma\big)^{C(\sigma)}.\end{align*} $$

In particular, it contains one summand (the one corresponding to the identity permutation)

$$ \begin{align*}HH_\bullet\big(\widehat{\mathscr{C}^{\otimes n}}\big)^{S_n}\cong HH_\bullet(\mathscr{C}^{\otimes n})^{S_n}\cong (HH_\bullet(\mathscr{C})^{\otimes n})^{S_n}=\text{Sym}^n(HH_\bullet(\mathscr{C})),\end{align*} $$

where the second isomorphism is the Künneth isomorphism (which is an isomorphism of $S_n$ -modules). We get the following corollary.