# Finite group actions on dg categories and Hochschild homology

Ville Nordström

###### Abstract

We prove a decomposition of the Hochschild homology groups of the equivariant
dg category $\mathscr{C}^{G}$ associated to a small dg category $\mathscr{C}$
with direct sums on which a finite group $G$ acts. When the ground field is
$\mathbb{C}$ this decomposition is related to a categorical action of
$\text{Rep}(G)$ on $\mathscr{C}^{G}$ and the resulting action of the
representation ring $R_{\mathbb{C}}(G)$ on $HH_{\bullet}(\mathscr{C}^{G})$.

##  1 Introduction

Let $\mathscr{C}$ be a dg category over a field of characteristic zero. If $G$
is a finite group which acts on $\mathscr{C}$ via autoequivalences
$\rho_{g}:\mathscr{C}\to\mathscr{C}$ we can consider the dg category
$\mathscr{C}^{G}$ of equivariant objects in $\mathscr{C}$. When $\mathscr{C}$
is the module category of a dg algebra $A$ and the group action comes from an
action of $G$ on $A$ then it follows from [5] ([2] in the case of an ordinary
algebra) that

$\displaystyle
HH_{\bullet}(\mathscr{C}^{G})\cong\bigoplus_{[g]}HH_{\bullet}(\mathscr{C};\rho_{g})^{C(g)}$
(1)

where $[g]$ denotes the conjugacy class of $g$ and $C(g)$ is the centralizer
of $g$ in $G$. In this note we prove that this decomposition holds assuming
only that $\mathscr{C}$ is a small dg category that admits finite direct sums.
To our knowledge this has not appeared in the literature. Even in situations
where this decomposition was already known we believe proving it directly in
the dg category setting is illuminating. The projection and inclusion maps
associated to this decomposition arise very naturally in the dg category
setting; they are induced by two dg functors

$\mathscr{C}^{G}\overset{For}{\to}\mathscr{C},\quad\mathscr{C}\overset{Sym}{\to}\mathscr{C}^{G}$

and natural transformations

$Sym\circ\rho_{g}\cong Sym,\ \rho_{g}\circ For\cong For.$

Moreover, in the dg category setting this decomposition has another
interesting explanation when the ground field is $\mathbb{C}$. Namely, we
explain how the representation ring of $G$ over $\mathbb{C}$,
$R_{\mathbb{C}}(G)$ acts on $HH_{\bullet}(\mathscr{C}^{G})$. We then prove
that the decomposition above matches up with the decomposition of the
$R_{\mathbb{C}}(G)$-module

$HH_{\bullet}(\mathscr{C}^{G})=\bigoplus_{[g]}\varphi_{[g]}\cdot
HH_{\bullet}(\mathscr{C}^{G})$

where $\varphi_{[g]}\in R_{\mathbb{C}}(G)$ denotes the indicator function on
the conjugacy class of $g$.

Finally we note that as a consequence of the decomposition (1) above we have
that for any small dg category, $\text{Sym}^{n}(HH_{\bullet}(\mathscr{C}))$
appears naturally as a summand in $HH_{\bullet}(\text{Sym}^{n}(\mathscr{C}))$.
This is related to Conjecture 3.24 in [1].

###  1.1 Conventions

All vector spaces will be over a fixed ground field $k$ of characteristic
zero. Sometimes we will need $k=\mathbb{C}$ and will then make this clear.

All functors between dg categories will be dg functors. By a natural
transformation between two dg functors
$F,F^{\prime}:\mathscr{C}\to\mathscr{B}$ we will always mean a closed degree
zero natural transformations. An equivalence $F:\mathscr{C}\to\mathscr{B}$
between dg categories will mean a dg functor that admits an inverse
$F^{-1}:\mathscr{B}\to\mathscr{C}$ such that $F\circ
F^{-1}\cong\mathrm{id}_{\mathscr{B}}$ and $F^{-1}\circ
F\cong\mathrm{id}_{\mathscr{C}}$ and these natural isomorphisms are required
to be degree zero cycles.

Our convention on matrices is that they act on the left. So when we describe a
morphism in some category between objects with specified direct sum
decompositions by a matrix, the columns will be indexed by the summands of the
source and the rows by the summands of the target.

###  1.2 Acknowledgments

I am grateful to my advisor Alexander Polishchuk for bringing the
decomposition (1) to my attention.

##  2 Finite group actions on dg categories

When talking about an action of a finite group $G$ on a dg category
$\mathscr{C}$ there are different levels of strictness that one can require.
Here is the definition that we will work with.

######  Definition 1.

An action of $G$ on $\mathscr{C}$ is the data of an autoequivalence
$\rho_{g}:\mathscr{C}\overset{\sim}{\to}\mathscr{C}$ for every $g\in G$,
natural isomorphisms
$\theta_{g,g^{\prime}}:\rho_{g}\circ\rho_{g^{\prime}}\overset{\sim}{\implies}\rho_{g^{\prime}g}$
and $\eta:\rho_{e}\overset{\sim}{\implies}\mathrm{id}_{\mathscr{C}}$ and these
should satisfy the following two conditions.

i) For each $g\in G$ and $c\in\mathscr{C}$ we have

$(\theta_{e,g})_{c}=\eta_{\rho_{g}(c)}:\rho_{e}\rho_{g}(c)\to\rho_{g}(c)$

and

$(\theta_{g,e})_{c}=\rho_{g}(\eta_{c}):\rho_{g}\rho_{e}(c)\to\rho_{g}(c).$

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

${\rho_{g}\rho_{h}\rho_{k}(c)}$${\rho_{g}\rho_{kh}(c)}$${\rho_{hg}\rho_{k}(c)}$${\rho_{khg}(c)}$$\scriptstyle{\rho_{g}((\theta_{h,k})_{c})}$$\scriptstyle{(\theta_{g,h})_{\rho_{k}(c)}}$$\scriptstyle{(\theta_{g,kh})_{c}}$$\scriptstyle{(\theta_{hg,k})_{c}}$

######  Example 2.

Such group actions arise for example when a finite group $G$ acts on a smooth
variety $X$. We can take $\mathscr{C}=\text{perf}(X)$, the category of finite
complexes of locally free sheaves on $X$. Then $G$ acts on $\mathscr{C}$ in
the above sense by pulling back vector bundles.

Whenever we have a group action as above we can consider the category of
$G$-equivariant objects in $\mathscr{C}$.

######  Definition 3.

Let $(\rho,\theta,\eta)$ be an action of a finite group $G$ on a small dg
category $\mathscr{C}$. The dg category of equivariant objects, denoted
$\mathscr{C}^{G}$ has objects

$(c,(\alpha_{g}:c\overset{\sim}{\to}\rho_{g}(c))_{g\in G})$

where the $\alpha_{g}$ are isomorphisms that are required to satisfy the
following condition. For any $c\in\mathscr{C}$ and $g,h\in G$ the following
diagram

${c}$${\rho_{g}(c)}$${\rho_{hg(c)}}$${\rho_{g}\rho_{h}(c)}$$\scriptstyle{\alpha_{g}}$$\scriptstyle{\alpha_{hg}}$$\scriptstyle{\rho_{g}(\alpha_{h})}$$\scriptstyle{(\theta_{g,h}^{-1})_{c}}$

commutes.

A degree $n$ morphism between equivariant objects
$\varphi:(c,\alpha)\to(c^{\prime},\alpha^{\prime})$ is a degree $n$ morphism
in $\mathscr{C}$ such that

${c}$${c^{\prime}}$${\rho_{g}(c)}$${\rho_{g}(c^{\prime}).}$$\scriptstyle{\varphi}$$\scriptstyle{\alpha_{g}}$$\scriptstyle{\alpha_{g}^{\prime}}$$\scriptstyle{\rho_{g}(\varphi)}$

If the category $\mathscr{C}$ has finite direct sums we can define a tensor
product $\otimes:\text{Rep}(G)\times\mathscr{C}^{G}\to\mathscr{C}^{G}$.
Indeed, if $(c,\alpha)\in\mathscr{C}^{G}$ and $(V,\rho_{V})$ is a left
$G$-module we define their tensor product $(V,\rho_{V})\otimes(c,\alpha)$ as
follows. Let $\\{e_{1},...,e_{n}\\}$ be a basis for $V$. Then

$\bigoplus_{i=1}^{n}c\cdot e_{i}\in\mathscr{C}$

is the object underlying $(V,\rho_{V})\otimes(c,\alpha)$. The equivariant
structure is given in matrix form by
$((\rho_{V}\otimes\alpha)_{g})_{ij}:=\rho_{V}(g)_{ij}\cdot\alpha_{g}$. To
verify that this is indeed an equivariant object we have to check that
$(\alpha\otimes\rho_{V})_{g}$ is indeed an isomorphism and that the condition
from definition 3 is satisfied. To see that it is an isomorphism we simply
note that its inverse is given by the matrix
$((\rho_{g}^{-1})_{ij}\alpha_{g}^{-1})_{ij}$. To see that it is an equivariant
structure we note that

$\sum_{j}\rho_{V}(h)_{ij}\rho_{V}(g)_{jk}\rho_{g}(\alpha_{h})\alpha_{g}=\rho_{V}(hg)_{ik}\theta_{g,h}^{-1}\alpha_{hg}.$

There is a forgetful functor $For:\mathscr{C}^{G}\to\mathscr{C}$. If
$\mathscr{C}$ has finite direct sums then there is also a functor in the other
direction which we call the symmetrization functor and denote by $S$. It is
defined by

$S(c):=\Big{(}\bigoplus_{h\in
G}\rho_{h}(c),\alpha_{g}=(\delta_{hg=h^{\prime}}\theta_{g,h}^{-1})_{hh^{\prime}}\Big{)}.$

######  Proposition 4.

The symmetrization functor is left adjoint to the forgetful functor.

###### Proof.

Suppose $(c,\alpha)$ is an equivariant object and $c^{\prime}$ is just an
object. Define

${\mathscr{C}(c^{\prime},c)}$${\mathscr{C}^{G}(S(c^{\prime}),(c,\alpha))}$${\varphi}$${(\alpha_{g}^{-1}\circ\rho_{g}(\varphi))_{g}.}$

Now define

${\mathscr{C}^{G}(S(c^{\prime}),(c,\alpha))}$${\mathscr{C}(c^{\prime},c)}$${\psi}$${\eta_{c}^{-1}\circ\alpha_{e}\circ\psi_{e}\circ\eta_{c^{\prime}}.}$

These are inverse bijections which proves the proposition. ∎

######  Proposition 5.

We have $S\circ For\cong k[G]\otimes(-)$ and $For\circ
S\cong\bigoplus_{g}\rho_{g}$.

###### Proof.

Let $(c,\alpha)$ be an equivariant object. Consider the following diagram,
where the morphisms are written as matrices,

${k[G]\otimes c}$${S\circ For(c)}$${\rho_{h}(k[G]\otimes c)}$${\rho_{h}(S\circ
For(c)).}$$\scriptstyle{(\delta_{g=g^{\prime-1}}\alpha_{g})_{gg^{\prime}}}$$\scriptstyle{(\delta_{k^{\prime}=hg^{\prime}}\alpha_{h})_{k^{\prime}g^{\prime}}}$$\scriptstyle{(\delta_{kh=g}\theta_{h,k}^{-1})_{kg}}$$\scriptstyle{(\delta_{k=k^{\prime-1}}\rho_{h}(\alpha_{k}))_{kk^{\prime}}}$

The top horizontal arrow is an isomorphism in $\mathscr{C}$ so if the diagram
commutes it is also an isomorphism in $\mathscr{C}^{G}$. We check that it
commutes

$\displaystyle\sum_{g}\delta_{kh=g}\theta_{h,k}^{-1}\delta_{g=g^{\prime-1}}\alpha_{g}=\delta_{(g^{\prime})^{-1}=kh}\theta_{h,k}^{-1}\alpha_{kh}=\delta_{(g^{\prime})^{-1}=kh}\rho_{h}(\alpha_{k})\alpha_{h}=$
$\displaystyle\sum_{k^{\prime}}\delta_{k=k^{\prime-1}}\rho_{h}(\alpha_{k})\delta_{k^{\prime}=hg^{\prime}}\alpha_{h}.$

∎

##  3 Hochschild homology of dg categories

###  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 [4].

A right module over dg category $\mathscr{C}$ is a dg functor
$M:\mathscr{C}^{op}\to\underline{\text{Com}}(k)$ 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 module as
functors $\mathscr{C}\to\underline{\text{Com}}(k)$. The collection of right
(resp. left) dg modules themselves form a dg category in a natural way which
we denote $mod-\mathscr{C}$ (resp. $\mathscr{C}-mod$). For any
$c\in\mathsf{Ob}(\mathscr{C})$ we denote by $h_{c}$ the right module
represented by $c$. The category $Z^{0}(mod-\mathscr{C})$ has the structure of
an exact category where a sequence of modules

$M\to M^{\prime\prime}\to M$

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

$M(c)\to M^{\prime\prime}(c)\to M^{\prime}(c)$

is an exact sequence of complexes and

$H(M(c))\to H(M^{\prime\prime}(c))\to H(M^{\prime}(c))$

is an exact sequence of graded vector spaces.

######  Lemma 6.

If $V$ is any chain complex of vector spaces, and
$c\in\mathsf{Ob}(\mathscr{C})$ then the functor $V\otimes h_{c}$ is projective
in the exact structure defined above.

###### Proof.

Consider a diagram

${X}$${V\otimes h_{c}}$${M.}$$\scriptstyle{\pi}$$\scriptstyle{\varphi}$

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

$\hat{\varphi}\in\text{Hom}_{Z^{0}(mod-\mathscr{C})}(h_{c},\text{Hom}_{k}^{\bullet}(V,M(-))=\text{Hom}_{Ch(k)}(V,M(c))$

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 $\mathscr{C}$ 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 momorphism complexes are defined by

$(\mathscr{C}\otimes\mathscr{B})((c,b),(c^{\prime},b^{\prime})):=\mathscr{C}(c,c^{\prime})\otimes\mathscr{B}(b,b^{\prime}).$

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

A right module over $\mathscr{C}\otimes\mathscr{B}^{op}$ 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

$\bigoplus_{c,c^{\prime}\in\mathscr{C}}M(c)\otimes\mathscr{C}(c^{\prime},c)\otimes
N(c)\to\bigoplus_{c\in\mathscr{C}}M(c)\otimes N(c),\ m\otimes f\otimes
n\mapsto mf\otimes n-m\otimes fn.$

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

$(c_{1},c_{2})\mapsto\mathscr{C}(c_{1},c_{2}).$

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

$(c_{1},c_{2})\mapsto\mathscr{C}(c_{2},c_{1}).$

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

$\text{Bar}^{n}(\mathscr{C}):=\Big{(}(x,y)\mapsto\bigoplus_{c_{0},...,c_{n}\in\mathscr{C}}\mathscr{C}(c_{0},x)\otimes\mathscr{C}(c_{1},c_{0})\otimes...\otimes\mathscr{C}(y,c_{n})\Big{)}$

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

$f_{0}|f_{1}|\cdots|f_{n+1}\mapsto\sum_{i=0}^{n+1}(-1)^{i}f_{0}|\cdots|f_{i}f_{i+1}|\cdots|f_{n+1}$

and we have the map
$d^{0}:\text{Bar}^{0}(\mathscr{C})\to\Delta_{\mathscr{C}}^{l}$ defined by
$f_{0}|f_{1}\mapsto f_{0}f_{1}$. Note that each $\text{Bar}^{n}(\mathscr{C})$
is projective as a left $\mathscr{C}\otimes\mathscr{C}^{op}$ modules by lemma
6. 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)

$\text{Bar}^{n}(\mathscr{C})\to\text{Bar}^{n+1}(\mathscr{C}),\
f_{0}|\cdots|f_{n+1}\mapsto\mathrm{id}_{x}|f_{0}|\cdots|f_{n+1}.$

###  3.2 Hochschild homology

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

######  Definition 7.

Let $\mathscr{C}$ be a small dg category and let $F:\mathscr{C}\to\mathscr{C}$
be an endofunctor. Set

$C_{n}(\mathscr{C};F):=\bigoplus_{c_{0},...,c_{n}\in\mathscr{C}}\mathscr{C}(c_{1},F(c_{0}))\otimes\mathscr{C}(c_{2},c_{1})\otimes...\otimes\mathscr{C}(c_{0},c_{n}).$

There is a differential $d_{2}:C_{n}(\mathscr{C};F)\to C_{n-1}(\mathscr{C};F)$
defined by

$\displaystyle d_{2}\big{(}a_{0}[a_{1}|\cdots|a_{n}]\big{)}$
$\displaystyle=a_{0}a_{1}[a_{2}|\cdots|a_{n}]$
$\displaystyle+\sum_{i=1}^{n-1}(-1)^{i}a_{0}[\cdots|a_{i}a_{i+1}|\cdots]$
$\displaystyle+(-1)^{n+|a_{n}|(|a_{0}|+...+|a_{n-1}|)}F(a_{n})a_{0}[a_{1}|\cdots|a_{n-1}].$

Each $C_{n}(\mathscr{C};F)$ is itself a cohomologically graded chain complex
$(C_{n}(\mathscr{C};F)^{\bullet},d_{1})$. 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 $C_{n}(\mathscr{C};F)$ whose
$k^{\prime}$th term for $k\in\mathbb{Z}$ is

$\bigoplus_{n-m=k}C_{m}(\mathscr{C};F)^{n}.$

The homology of this complex is the Hochschild homology of $\mathscr{C}$ with
values in $F$ and it is denoted $HH_{\bullet}(\mathscr{C};F)$.

######  Remark 8.

Different sign conventions appear for the Hochschild differential but they all
give rise to isomorphic chain complexes and so we use this one as it is the
least complicated. This sign convention appears for example in [3].

If we have two dg categories $\mathscr{C}$, $\mathscr{C}^{\prime}$ and two
endofunctors $F:\mathscr{C}\to\mathscr{C}$,
$F^{\prime}:\mathscr{C}^{\prime}\to\mathscr{C}^{\prime}$ and
$\phi:\mathscr{C}\to\mathscr{C}^{\prime}$ is a functor and $\epsilon:\phi\circ
F\implies F^{\prime}\circ\phi$ is a natural transformation, then we get an
induced map on Hochschild homology

$(\phi,\epsilon)_{*}:HH_{\bullet}(\mathscr{C};F)\to
HH_{\bullet}(\mathscr{C}^{\prime};F^{\prime})$

defined at the chain level as

${\mathscr{C}(c_{1},F(c_{0}))\otimes...\otimes\mathscr{C}(c_{0},c_{n})}$${\mathscr{C}^{\prime}(\phi(c_{1}),F^{\prime}\circ\phi(c_{0}))\otimes...\otimes\mathscr{C}^{\prime}(\phi(c_{0}),\phi(c_{n}))}$${f_{0}[f_{1}|\cdots|f_{n}]}$${\epsilon_{c_{0}}\phi(f_{0})\big{[}\phi(f_{1})|\cdots|\phi(f_{n})\big{]}.}$

We will refer to the data of such a $\phi$ and $\epsilon$ as
$(\phi,\epsilon):(\mathscr{C},F)\to(\mathscr{C}^{\prime},F^{\prime})$.

######  Lemma 9.

Let $\mathscr{C},\ \mathscr{C}^{\prime}$ be dg categories,
$F\in\mathrm{End}(\mathscr{C}),F^{\prime}\in\mathrm{End}(\mathscr{C}^{\prime})$be
two endofunctors, $A_{1},A_{2}:\mathcal{C}\to\mathcal{C}^{\prime}$ two dg
functors and $\eta_{ij}:A_{j}\circ F\implies F^{\prime}\circ A_{i}$ natural
transformations. Set

$\eta:=\begin{pmatrix}\eta_{11}&\eta_{12}\\\
\eta_{21}&\eta_{22}\end{pmatrix}:\begin{matrix}A_{1}\circ F\\\ \oplus\\\
A_{2}\circ F\end{matrix}\to\begin{matrix}F\circ A_{1}\\\ \oplus\\\ F\circ
A_{2}\end{matrix}.$

Then

$(A_{1}\oplus
A_{2},\eta)_{*}\simeq(A_{1},\eta_{11})_{*}+(A_{2},\eta_{22})_{*}$

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

###### Proof of lemma.

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
$A(f)=\iota_{1}A_{1}(f)\pi_{1}+\iota_{2}A_{2}(f)\pi_{2}$. We have

$\displaystyle(A,\eta)_{*}(f_{0}[f_{1}|\cdots f_{k}])=$
$\displaystyle\eta\circ A(f_{0})[A(f_{1})|\cdots|A(f_{k})]=$
$\displaystyle(\iota_{1}\eta_{11}+\iota_{2}\eta_{21})A_{1}(f_{0})\pi_{1}[\iota_{1}A_{1}(f_{1})\pi_{1}|\cdots|\iota_{1}A_{1}(f_{k})\pi_{1}]+$
$\displaystyle(\iota_{1}\eta_{12}+\iota_{2}\eta_{22})A_{2}(f_{0})\pi_{2}[\iota_{2}A_{2}(f_{1})\pi_{2}|\cdots|\iota_{2}A_{2}(f_{k})\pi_{2}]+$
$\displaystyle\text{mixed terms}.$

For a term in (LABEL:terms) 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 $l^{\prime}$th factor begins with a $\pi_{1}$ or vice versa. When $l=k$
here then the $l+1$’th factor refers to the $0^{\prime}th$ factor. So for
example

$\iota_{1}\eta_{11}A_{1}(f_{0})\pi_{1}[\iota_{2}A_{2}(f_{1})\pi_{2}|\iota_{2}A_{2}\pi_{2}]$

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

$\iota_{1}\eta_{12}A_{2}(f_{0})\pi_{2}[\iota_{2}A_{2}(f_{1})\pi_{2}|\iota_{2}A_{2}\pi_{2}]$

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
$(\ref{terms})$ 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

$\displaystyle\phi_{i}^{0}(f_{0}[f_{1}|\cdots
f_{k}])=\iota_{i}\eta_{ii}A_{i}(f_{0})\pi_{i}[\iota_{i}A_{i}(f_{1})\pi_{i}|\cdots|\iota_{i}A_{i}(f_{k})\pi_{i}]$

and this is homotopic to $(A_{i},\eta_{ii})_{*}$ via the homotopy

$\displaystyle H_{i}(f_{0}[f_{1}|\cdots|f_{k}]):=$
$\displaystyle\sum_{j=0}^{k}(-1)^{j}\eta_{ii}A_{i}(f_{0})\pi_{i}[\iota_{i}A_{i}(f_{1})\pi_{i}|\cdots|\iota_{i}A_{i}(f_{j})\pi_{i}|\iota_{i}|A_{i}(f_{j+1})|\cdots|A_{i}(f_{k})].$

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

$\displaystyle H(f_{0}[f_{1}|\cdots|f_{k}]):=$
$\displaystyle\sum(-1)^{i}\iota_{\alpha}\eta_{\alpha\beta}A_{1}(f_{0})\pi_{1}[\iota_{1}A_{1}(f_{1})\pi_{1}|\cdots|\iota_{2}|A_{2}(f_{i})\pi_{2}|\cdots].$

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.

∎

Suppose we have three dg categories with endofunctors
$(\mathscr{C},F),(\mathscr{C}^{\prime},F^{\prime})$ and
$(\mathscr{C}^{\prime\prime},F^{\prime\prime})$ and
$(\phi,\eta):(\mathscr{C},F)\to(\mathscr{C}^{\prime},F^{\prime})$,
$(\psi,\epsilon):(\mathscr{C}^{\prime},F^{\prime})\to(\mathscr{C}^{\prime\prime},F^{\prime\prime})$.
In this situation we define

$(\psi,\epsilon)\circ(\phi,\eta):=(\psi\circ\phi,\epsilon\star\eta)$

where $\epsilon\star\eta$ is the composite

${\psi\circ\phi\circ F}$${\psi\circ
F^{\prime}\circ\phi}$${F^{\prime\prime}\circ\psi\circ\phi.}$$\scriptstyle{\psi(\eta)}$$\scriptstyle{\epsilon_{\phi}}$

######  Lemma 10.

We have
$\big{(}(\psi,\epsilon)\circ(\phi,\eta)\big{)}_{*}=(\psi,\epsilon)_{*}\circ(\phi,\eta)_{*}$.

###### Proof.

They are just equal already at the chain level. ∎

Now suppose we have
$(\phi,\eta):(\mathscr{C},F)\to(\mathscr{C}^{\prime},F^{\prime})$ 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 $\psi\circ F\implies F^{\prime}\circ\psi$ defined as the
composite

${\psi F}$${\phi\circ
F}$${F^{\prime}\circ\phi}$${F^{\prime}\circ\psi.}$$\scriptstyle{\alpha_{F}^{-1}}$$\scriptstyle{\eta}$$\scriptstyle{F^{\prime}(\alpha)}$

######  Lemma 11.

We have $(\phi,\eta)_{*}=(\psi,\alpha\eta\alpha^{-1})_{*}$.

###### Proof.

The following homotopy proves the equality at the level of homology

$\displaystyle H(f_{0}[f_{1}|\cdots|f_{k}]):=$
$\displaystyle\sum_{i=0}^{k}(-1)^{i}\eta\alpha^{-1}\psi(f_{0})[\psi(f_{1})|\cdots|\psi(f_{i})|\alpha|\phi(f_{i+1})|\cdots|\phi(f_{k})].$

∎

###  3.3 Künneth isomorphism

In this section we give a proof of the Künneth isomorphism for Hochschild
homology and check that it is compatible with the $S_{2}$ action. Given two dg
categories $\mathscr{C}$ and $\mathscr{B}$ we define

$Sh:C_{\bullet}(\mathscr{C})\otimes C_{\bullet}(\mathscr{B})\to
C_{\bullet}(\mathscr{C}\otimes\mathscr{B})$

by the formula

$\displaystyle f_{0}[f_{1}|\cdots|f_{k}]\otimes
g_{0}[g_{1}|\cdots|g_{l}]\mapsto\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$
$\displaystyle\sum_{\sigma\in(k,l)-shuffles}(-1)^{|\sigma|+\epsilon}f_{0}\otimes
g_{0}[\cdots|f_{i}\otimes 1|\cdots|1\otimes g_{j}|\cdots|f_{i+1}\otimes
1|\cdots|1\otimes g_{j}|\cdots].$

The sign in the formula above is given by

$\epsilon=\sum|g_{t}||f_{s}|$

where the sum is over all pairs $(f_{s},g_{t})$ so that $f_{s}$ appears after
$g_{t}$.

######  Proposition 12.

The map $Sh$ defined above induces an isomorphism

$HH_{\bullet}(\mathscr{C})\otimes HH_{\bullet}(\mathscr{B})\to
HH_{\bullet}(\mathscr{C}\otimes\mathscr{B}).$

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 13.

For each n we have an isomorphism of chain complexes

$C_{n}(\mathscr{C})\cong\Delta_{\mathscr{C}}^{r}\otimes_{\mathscr{C}\otimes\mathscr{C}^{op}}\text{Bar}^{n}(\mathscr{C}).$

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

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

###### Proof.

The complex on the right is the cokernel of

$\bigoplus_{(x,y),(x^{\prime},y^{\prime})}\mathscr{C}(x,y)\otimes(\mathscr{C}(y^{\prime},y)\otimes\mathscr{C}(x,x^{\prime}))\otimes\text{Bar}^{n}(\mathscr{C})(y^{\prime},x^{\prime})\to\bigoplus_{(x,y)}\mathscr{C}(x,y)\otimes\text{Bar}^{n}(y,x)$
$f\otimes(\alpha\otimes\beta)\otimes B\mapsto(-1)^{|\alpha||f|}\alpha
f\beta\otimes B-(-1)^{(|B|+|\beta|)|\alpha|}f\otimes\beta B\alpha.$

The isomorphism is given induced by

$\bigoplus_{(x,y)}\mathscr{C}(x,y)\otimes\text{Bar}^{n}(y,x)\to
C_{n}(\mathscr{C})$ $f\otimes
b_{0}|b_{1}|\cdots|b_{n+1}\mapsto(-1)^{|b_{n+1}|(|b_{0}|+...+|b_{n}|+|f|)}b_{n+1}fb_{0}[b_{1}|\cdots
b_{n}]$

and its inverse sends $a_{0}[a_{1}|\cdots|a_{n}]$ to the image of
$a_{0}\otimes 1|a_{1}|...|a_{n}|1$ 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 $\mathscr{C}\otimes\mathscr{B}$

$(M\otimes_{k}N)(c,b):=M(c)\otimes N(b).$

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 6 then their tensor product is again of that form.

######  Lemma 14.

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

$Sh^{\prime}:\text{Bar}(\mathscr{C})\otimes\text{Bar}(\mathscr{B})\to\text{Bar}(\mathscr{C}\otimes\mathscr{B}).$

###### Proof.

The map is defined by

$\displaystyle\Big{(}\bigoplus_{c_{1},...,c_{p+1}}\mathscr{C}(c_{1},-)\mathscr{C}(c_{2},c_{1})\cdots\mathscr{C}(-,c_{p+1})\Big{)}\otimes\Big{(}\bigoplus_{b_{1},...,b_{q+1}}\mathscr{B}(b_{1},-)\mathscr{C}(b_{2},b_{1})\cdots\mathscr{B}(-,b_{q+1})\Big{)}$
$\displaystyle\longrightarrow\bigoplus_{(c_{1},b_{1}),...,(c_{p+q+1},b_{p+q+1})}\mathscr{C}\mathscr{B}((c_{1},b_{1}),-)\mathscr{C}\mathscr{B}((c_{2},b_{2}),(c_{1},c_{1}))\cdots\mathscr{C}\mathscr{B}(-,(c_{p+q+1},b_{p+q+1}))$
$\displaystyle f_{0}|f_{1}|\cdots|f_{p}\otimes
g_{0}|g_{1}|\cdots|g_{q}\longmapsto$
$\displaystyle(-1)^{\epsilon_{1}}\sum(f_{0}\otimes g_{0})|\cdots|(f_{i}\otimes
1)|\cdots|(1\otimes g_{j})|\cdots|(f_{i+1}\otimes
1)|\cdots|(1\otimes_{g_{j+1}})|\cdots|(f_{p}\otimes g_{q})$

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

$\displaystyle\epsilon_{1}=|g_{0}|(|f_{1}|+...+|f_{p}|)+|f_{p}|(|g_{1}|+...+|g_{q}|)$
$\displaystyle\epsilon_{2}=|\sigma|+\sum|f_{i}||g_{j}|$

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

${\text{Bar}(\mathscr{C})\otimes\text{Bar}(\mathscr{B})}$${\text{Bar}(\mathscr{C}\otimes\mathscr{B})}$${\mathscr{C}\otimes\mathscr{B}}$${\mathscr{C}\otimes\mathscr{B}}$$\scriptstyle{\sim}$$\scriptstyle{Sh^{\prime}}$$\scriptstyle{\sim}$$\scriptstyle{=}$

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
$\mathscr{C}\otimes\mathscr{B}\otimes\mathscr{C}^{op}\otimes\mathscr{B}^{op}$-modules
$Sh^{\prime}$ is a homotopy equivalence. ∎

###### Proof of proposition 12.

The isomorphism is obtained by tensoring the homotopy equivalence
$Sh^{\prime}$ from the previous lemma on the left with
$\Delta_{\mathscr{C}\otimes\mathscr{B}}^{r}$. ∎

##  4 Hochschild homology of equivariant category

Now we again consider an action of a finite group $G$ on a small dg category
$\mathscr{C}$. Let $g\in G$ and denote by $C(g)$ the subgroup of elements that
commute with $g$. Then, for any $h\in C(g)$ we have a natural isomorphism

$C_{h,g}:=\theta_{g,h}^{-1}\circ\theta_{h,g}:\rho_{h}\circ\rho_{g}\implies\rho_{g}\circ\rho_{h}$

and they induce isomorphisms

$(\rho_{h},C_{h,g})_{*}:HH_{\bullet}(\mathscr{C},\rho_{g})\overset{\sim}{\to}HH_{\bullet}(\mathscr{C},\rho_{g}).$

######  Lemma 15.

The assignment $h\mapsto(\rho_{h},C_{h,g})_{*}$ defines a right action of
$C(g)$ on $HH_{\bullet}(\mathscr{C},\rho_{g}).$

###### Proof.

We have

$\displaystyle(\rho_{h},C_{h,g})\circ(\rho_{h^{\prime}},C_{h^{\prime},g})=(\rho_{h}\circ\rho_{h^{\prime}},(C_{h,g})_{\rho_{h^{\prime}}}\circ\rho_{h}(C_{h^{\prime},g})).$

and
$\theta_{h,h^{\prime}}:\rho_{h}\circ\rho_{h^{\prime}}\overset{\sim}{\implies}\rho_{h^{\prime}h}$
and $\theta_{h,h^{\prime}}(C_{h,g}\star
C_{h^{\prime},g})\theta_{h,h^{\prime}}^{-1}=C_{h^{\prime}h,g}$. It then
follows from lemmas 10 and 11 that
$(\rho_{h},C_{h,g})_{*}\circ(\rho_{h^{\prime}},C_{h^{\prime},g})_{*}=(\rho_{h^{\prime}h},C_{h^{\prime}h,g})_{*}$.

∎

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
$R_{k}(G):=R(G)\otimes_{\mathbb{Z}}k$ where $R(G)$ is the representation ring
of $G$. When $k=\mathbb{C}$ it follows from character theory that this ring is
nothing but the ring of complex valued functions on the set of conjugacy
classes of $G$. Denote by $\varphi_{[g]}\in R_{\mathbb{C}}(G)$ 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 $R_{\mathbb{C}}(G)$-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 16.

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

$HH_{\bullet}(\mathscr{C}^{G})\cong\bigoplus_{[g]\in
G/\sim}HH_{\bullet}(\mathscr{C};\rho_{g})^{{}^{C(g)}}$

where $\sim$ refers to conjugacy. Moreover, when the ground field is
$\mathbb{C}$, 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 and projections $\iota_{g},\pi_{g}$ 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 $For:\mathscr{C}^{G}\to\mathscr{C}$ and $S:\mathscr{C}\to\mathscr{C}^{G}$
be the functors introduced in section 2. We define
$\pi_{g}=(For,\alpha_{g})_{*}:HH_{\bullet}(\mathscr{C}^{G})\to
HH_{\bullet}(\mathscr{C};\rho_{g})$ where (by abuse of notation) $\alpha_{g}$
denotes the natural isomorphism $For\overset{\sim}{\implies}\rho_{g}\circ For$
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
$(\rho_{h},C_{h,g})\circ(For,\alpha_{g})=(\rho_{h}\circ
For,C_{h,g}\star\alpha_{g})$ and note that
$\alpha_{h}:For\overset{\sim}{\implies}\rho_{h}\circ For$ and
$C_{h,g}\star\alpha_{g}=\alpha_{h}\alpha_{g}\alpha_{h}^{-1}$. The claim then
follows from lemmas 10 and 11.

Let us denote by $\hat{S}:=For\circ S$ and recall from proposition 5 that it
is isomorphic (or in fact equal) to $\bigoplus_{g\in G}\rho_{g}$. Now define
$\iota_{g}:HH_{\bullet}(\mathscr{C};\rho_{g})^{C(g)}\to
HH_{\bullet}(\mathscr{C}^{G})$ by

$HH_{\bullet}(\mathscr{C};\rho_{g})^{C(g)}\to
HH_{\bullet}(\mathscr{C};\rho_{g})\overset{S_{g}}{\to}HH_{\bullet}(\mathscr{C}^{G})$

where $S_{g}=(S,\phi_{g})_{*}$ and $\phi_{g}:S(\rho_{g}(x_{0}))\to S(x_{0})$
is the natural isomorphism

$(\phi_{g})_{hh^{\prime}}=\delta_{gh^{\prime}=h}\theta_{h^{\prime},g}.$

Claim 2: $\pi_{g}\circ\iota_{g}\simeq|C(g)|\cdot\mathrm{id}$.
Proof of claim 2: Note that $\pi_{g}\circ S_{g}=(For\circ
S,\alpha_{g}\star\phi_{g})$ and
$\alpha_{g}\star\phi_{g}:\hat{S}(\rho_{g}(x_{0}))\to\rho_{g}(\hat{S}(x_{0}))$
is give by
$(\alpha_{g}\star\phi_{g})_{hh^{\prime}}=\delta_{hg=gh^{\prime}}\theta_{g,h}^{-1}\circ\theta_{h^{\prime},g}$.
By lemma 9 we see that $\pi_{g}\circ\tilde{S}=\sum_{h\in
C(g)}(\rho_{h},C_{h,g})_{*}$ and from this the claim follows.

Claim 3: $\pi_{g}\circ\iota_{g^{\prime}}\simeq 0$ if $g\nsim g^{\prime}$.
This also follows from lemma 9.

Claim 4: If $g$ and $g^{\prime}$ are conjugate then
$\iota_{g}\pi_{g}\simeq\iota_{g^{\prime}}\pi_{g^{\prime}}$.
Proof of claim 4: Suppose $g^{\prime}=h^{-1}gh.$Consider the diagram

${(\mathscr{C},\rho_{g})}$${(\mathscr{C}^{G},\mathrm{id})}$${(\mathscr{C}^{G},\mathrm{id})}$${(\mathscr{C},\rho_{g^{\prime}}).}$$\scriptstyle{(\rho_{h},\theta_{g^{\prime},h}^{-1}\circ\theta_{h,g})}$$\scriptstyle{(S,\phi_{g})}$$\scriptstyle{(F,\alpha_{g})}$$\scriptstyle{(F,\alpha_{g^{\prime}})}$$\scriptstyle{(S,\phi_{g^{\prime}})}$

We have a natural isomorphism $\alpha_{h}:F\implies\rho_{h}\circ F$ and

$(\rho_{h},\theta_{g^{\prime},h}^{-1}\circ\theta_{h,g})\circ(F,\alpha_{g})=(\rho_{h}\circ
F,\rho_{g^{\prime}}(\alpha_{h})\circ\alpha_{g^{\prime}}\circ\alpha_{h}^{-1}).$

Also, we have a natural isomorphism $\phi_{h}:S\circ\rho_{h}\implies S$ and

$(S,\phi_{g^{\prime}})\circ(\rho_{h},\theta_{g^{\prime},h}^{-1}\circ\theta_{h,g})=(S\circ\rho_{h},\phi_{h}^{-1}\circ\phi_{g}\circ(\phi_{h})_{\rho_{g}}).$

Now using lemmas 10 and 11 we get

$\displaystyle\iota_{g}\pi_{g}$
$\displaystyle=(S,\phi_{g})_{*}\circ(F,\alpha_{g})_{*}$
$\displaystyle=(S,\phi_{g^{\prime}})_{*}\circ(\rho_{h},\theta_{g^{\prime},h}^{-1}\circ\theta_{h,g})_{*}\circ(F,\alpha_{g})_{*}$
$\displaystyle=(S,\phi_{g^{\prime}})_{*}\circ(F,\alpha_{g^{\prime}})_{*}=\iota_{g^{\prime}}\circ\pi_{g^{\prime}}.$

Claim 5:

$\sum_{[g]\in
G/\sim}\frac{\iota_{g}\circ\pi_{g}}{|C(g)|}=\mathrm{id}_{HH_{\bullet}(\mathscr{C}^{G})}$

Proof of claim 5: By the previous claim we have

$\displaystyle\sum_{[g]\in G/\sim}\frac{\iota_{g}\circ\pi_{g}}{|C(g)|}$
$\displaystyle=\sum_{[g]\in G/\sim}|conj(g)|\frac{\iota_{g}\circ\pi_{g}}{|G|}$
$\displaystyle=\frac{1}{|G|}\sum_{g\in G}\iota_{g}\circ\pi_{g}.$

Next we note that

$\displaystyle\iota_{g}\circ\pi_{g}=(S,\phi_{g})_{*}\circ(For,\alpha_{g})_{*}=(S\circ
For,\phi_{g}\star\alpha_{g})_{*}$

and therefore

$\displaystyle\sum_{g\in G}\iota_{g}\circ\pi_{g}=(S\circ For,\sum_{g\in
G}\phi_{g}\star\alpha_{g}).$

Moreover

$\sum_{g\in G}\phi_{g}\star\alpha_{g}:S\circ For\implies S\circ For$

is given in matrix form by

$\Big{(}\sum_{g\in
G}\phi_{g}\star\alpha_{g}\Big{)}_{h^{\prime}h}=\alpha_{h^{\prime}}\circ\alpha_{h}^{-1}=(I\circ
P)_{h^{\prime}h}$

where

$I:\mathrm{id}_{\mathscr{C}^{G}}\implies S\circ For,\ P:S\circ
For\implies\mathrm{id}_{\mathscr{C}^{G}}$

are the inclusion- and projection-natural transformation associated with the
identity component of $\mathbbm{1}\otimes(-)\subset k[G]\otimes(-)$ (see
proposition 5). Let $S^{\prime}=S\circ For$. Now the following homotopy
completes the proof of the claim.

$\displaystyle H(f_{0}[f_{1}|\cdots|f_{k})=$
$\displaystyle\sum_{i=0}^{k}(-1)^{i}P\circ
S^{\prime}(f_{0})[S^{\prime}(f_{0})|\cdots|S^{\prime}(f_{i})|I|f_{i+1}|\cdots
f_{k}].$

It remains to prove the second part of the theorem, that when the ground field
is $\mathbb{C}$ 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 $R_{\mathbb{C}}(G)$ module $M$ the we have that

$\varphi_{[g]}\cdot M=\\{m\in M|[V]\cdot m=\chi_{V}(g)\cdot m\text{ for any
representation }V\\}.$

Let $(V,\rho_{V})$ be a representation with basis $\\{e_{1},...,e_{n}\\}$. 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 5 we have the following isomorphism of functors

$\displaystyle T_{V}\circ S\circ For\cong T_{V}\circ T_{k[G]}\cong
T_{k[G]}\circ T_{V}=S\circ For\circ T_{V}\cong\oplus_{i=1}^{n}S\circ For.$ (3)

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

$\tau_{(i,k),(j,h)}=\delta_{h=k}\rho_{V}(k)_{ij}$

where $1\leq i,j\leq n$ and $h,k\in G$. Let $I_{j}:S\circ For\implies
T_{V}\circ S\circ For$ and $P_{j}:T_{V}\circ S\circ For\implies S\circ For$ be
the inclusion and projection-natural transformations associated with the
decomposition (3) above. We have

$P_{j}\circ T_{V}(\phi_{g}\star\alpha_{g})\circ
I_{j}=\rho_{V}(g)_{jj}\cdot(\phi_{g}\star\alpha_{g}).$

We then have

$\displaystyle(T_{V},1)_{*}\circ\iota_{g}\circ\pi_{g}=$
$\displaystyle(T_{V},1)_{*}\circ(S,\phi_{g})_{*}\circ(For,\alpha_{g})_{*}$
$\displaystyle=$ $\displaystyle(T_{V}\circ S\circ
For,T_{V}(\phi_{g}\star\alpha_{g}))_{*}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\rho_{V}(g)_{jj}\cdot(S\circ
For,\phi_{g}\star\alpha_{g})_{*}$ $\displaystyle=$
$\displaystyle\chi_{V}(g)(\iota_{g}\circ\pi_{g})$

where the second equality is lemma 10 and the third equality combines the
decomposition (3) above and lemma 9.

∎

###  4.1 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
$x_{\bullet}=(x_{1},x_{2},...,x_{n})$ with each
$x_{i}\in\mathsf{Ob}(\mathscr{C})$. The homomorphism spaces are given by
$Hom_{\mathscr{C}^{\otimes
n}}(x_{\bullet},y_{\bullet}):=\mathscr{C}(x_{1},y_{1})\otimes\mathscr{C}(x_{2},y_{2})\otimes...\otimes\mathscr{C}(x_{n},y_{n})$.
There is a right action of $S^{n}$ on $\mathscr{C}$ defined on objects by
$\sigma(x_{1},...,x_{n})=(x_{\sigma(1)},...,x_{\sigma(n)}).$ This action is
strict in the strongest possible sense. All the higher categorical data is
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

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

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

$HH_{\bullet}\big{(}\widehat{\mathscr{C}^{\otimes n}}\big{)}^{S_{n}}\cong
HH_{\bullet}(\mathscr{C}^{\otimes
n})^{S_{n}}\cong(HH_{\bullet}(\mathscr{C})^{\otimes
n})^{S_{n}}=\text{Sym}^{n}(HH_{\bullet}(\mathscr{C}))$

where the second isomorphism is the Künneth isomorphism (which is an
isomorphism of $S_{n}$-modules). We get the following corollary.

######  Corollary 17.

If $\mathscr{C}$ is a small dg category then
$HH_{\bullet}(\text{Sym}^{n}(\mathscr{C}))$ contains naturally
$\text{Sym}^{n}(HH_{\bullet}(\mathscr{C}))$ as a summand.

## References

  * [1] P. Belmans, L. Fu, A. Krug, Hochschild cohomology of Hilbert schemes of points on surfaces. arXiv:2309.06244 [math.AG]
  * [2] E. Getzler, J.D. Jones, The cyclic homology of crossed product algebras. J. Reine Angew. Math.445(1993), 161–174.
  * [3] J-L. LodayCyclic Homology. 2nd ed., Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin, 1998.
  * [4] A. Polishchuk, L. Positselski Hochschild (co)homology of the second kind I. Trans. Amer. Math. Soc.364(2012), no.10, 5311–5368.
  * [5] S. Quddus Equivariant Cartan homotopy formulae for the crossed product of DG algebra. Rocky Mountain J. Math.51(2021), no.4, 1399–1406.


# Dichalcogenides and difulfides nanostructures for hydrogen storage

Flavio Bento de Oliveira  Andréia Luisa da Rosa<EMAIL_ADDRESS>Institute
of Physica, Federal University of Goiás, Campus Samambaia, 74690-900, Goiânia,
Goiás, Brazil.

###### Abstract

Large-surface-area, two-dimensional (2D) layered materials are expected to
have an advantage in hydrogen storage applications. Among a large number of 2D
materials, monolayer dichacogenides have emerged as promising candidate for
hydrogen clean energy. In this work we investigate MoS2, porous MoS2 and XS2
as alternative materials for hydrogen storage. In this work we have
investigated porous MoS2 and XS2 (X = Pd, Ni and Pt) as alternative materials
for hydrogen storage. Nickel selenide appears as a viable candidate for
hydrogen storage, boasting a gravimetrics of 6.87%. We also show that MoS2
nanopores could enhance the gravimetry for hydrogen storage. This enhancement
occurs through the adsorption of H2 molecules within the pore. Finally, AIMD
calculations suggested that the structures identified in this work could be
stable even at room temperature.

##  I Introduction

The demand for energy development has continually increased, and therefore
energy consumption is constantly increasing, causing global warming through
the emission of pollutants into the atmosphere, leading to a probable future
shortage of fossil fuels. With this there is a growing demand for renewable
energy, with a growth rate per year of 14.7% Sen (2023). Hydrogen is a
potential energy carrier since its origin may come from clean sources.
Nevertheless, storing hydrogen at densities greater than its natural density
poses a challenge. There are four main methods for hydrogen storage:
compression, liquefaction, metal hydrides and physisorption. The primary
challenge lies in developing lightweight hydrogen storage materials at
reasonable costs. Hydrogen boasts three times more energy value compared to
fossil fuels Sen (2023). Nevertheless, storing 1 kg of hydrogen requires
approximately 11 m3 at ambient temperature, necessitating the storage of
hydrogen fuel in cylinders with pressures ranging from 20 MPa to 80 MPa
Schlapbach and Zuttel (2001); Züttel (2004). This method is referred to as
compression. Another approach is liquefaction, which involves cooling hydrogen
down to 33.2 K. While hydrogen can be stored in open systems using this
method, the evaporation and consumption of hydrogen are high and limited. The
fuel is consumed in a very short time, limiting its application to specific
cases, such as aircrafts Zhou (2005); Tarhan and Çil (2021). The currently
used hydrogen storage in the industry is either metallic or in form of complex
hydrides. However these hydrides cannot resolve the high pressure condition,
and contain a large volume to be stored. Besides storage, another problem when
using these hydrides is the fact that after a few cycles the metals used end
up degrading or becoming brittle. Another problem often associated is the
thermodynamics of metal hydrides reactions, the large amount of heat generated
during refuelling, thus making it difficult to store hydrogen.

It is stimates that storing hydrogen requires a weight (wt) between 4 wt% to 7
wt% of hydrogen Graetz (2009); Lynch (2011); Cao (2021). This has lead to the
search for new technologies. In the search for new materials, two-dimensional
nanostructures have gained interest following the discovery of graphene, whose
properties differed significantly from those of its three-dimensional
counterpart, graphite Geim and Novoselov (2007). These materials exhibit
electrical properties highly appealing for technological applications and hold
promise for reducing the dimensions of currently produced electronic devices.
This has led to a growing interest in exploration of other two-dimensional
materials, in particular for hydrogen storage Züttel (2004); Chen and Kuriyama
(2001); Chen J (2001). It is expected that large-surface-area, two-dimensional
(2D) layered materials have an advantage in hydrogen storage applications.
Therefore, finding materials that can effectively contain hydrogen is crucial
for the application of hydrogen energy.

MX2 (M = Mo,W and X=S,Se,Te) have several polytypes the 2H (2H-MoS2) has
semiconductor behavior and 1T(1T-MoS2) or 1T’(1T’-MoS2) both metallic phase.
It has been suggested that the the performance of the layered material has a
dependence with the structural phase Chen  _et al._ (2020). For example, MoS2
nanotubes showed 1.2 % wt hydrogen storage capacity. The authors attributed
these results to the high specific surface area of MoS2 nanotubes Chen and
Kuriyama (2001). Another usual approach for improving hydrogen storage
capacity is doping. Hydrogen storage performances of pure MoS2 monolayer and
the one decorated with 3d and 4d atoms were studied via the first-principles
DFT calculationsYang  _et al._ (2021a, b). Storage capacity of around 6 wt%.
Yang  _et al._ (2021a) has been achieved. Although doping is a possible route
to enhance adsorption energies on MoS2, doping has proven to be difficult in
low-dimensional materials Pham and Yeom (2016); Altuntepe (2024). Previous
investigations on hydrogen intercalation in MoS2 bilayers Zhu  _et al._ (2016)
has suggested that the insertion energies of hydrogen molecules as a function
of hydrogen concentration in MoS2 can be accommodated in MoS2 bilayers without
a major change in the interlayer spacing.

Previous work on hydrogen intercalation in MoS2 bilayersZhu  _et al._ (2016).
The insertion energies of hydrogen molecules as a function of hydrogen
concentration in MoS2 have shown that at a resonable number of H2 molecules
can be accommodated in MoS2 bilayers without a major change in the interlayer
spacing. Hydrogen can be liquified at ambient pressure at a temperature of
-253C, which increases its density to about 70 kg /m3, facilitating
transportation of smaller volumes. Recently, bulk and exfoliated MoS2 were
used to understand hydrogen storage potential of the bulk and exfoliated MoS2
was addressed under 1-9 bar hydrogen pressure. After the absorption process.
The bulk and exfoliated MoS2 stored 220 and 450 sccm hydrogen, respectively.
Besides, th e weight percent hydrogen storage was determined to be 1.2 and 2.4
wt%, respectivelyAltuntepe  _et al._ (2024).

On the other hand similar layered structures of formed by Pd, Ni and Pt TMDs
remained largely unexplored with few publications focusing on their
electronic, mechanical, optical and thermoelectric properties of monolayer
PdS2 and PdSe2 Moujaes and Diery (2021); Deng  _et al._ (2018); Zhao (2020).
These structures show a different reactivity than their MoS2 counterparts and
therefore could lead to other weight percent hydrogen.

In this work we investigate MoS2, porous MoS2 and XS2 as alternative materials
for hydrogen storage. In this work we have investigated porous MoS2 and XS2 (X
= Pd, Ni and Pt) as alternative materials for hydrogen storage. Nickel
selenide appears as a viable candidate for hydrogen storage, boasting a
gravimetrics of 6.87%. We also show that MoS2 nanopores could enhance the
gravimetry for hydrogen storage. This enhancement occurs through the
adsorption of H2 molecules within the pore. Our AIMD calculations suggests
that the structures identified in this work could be stable even at room
temperature.

##  II Computational Details

In this work, we compute electronic and atomic properties using VASP (Vienna
Ab-initio Simulation Package) computational code Kresse and Joubert (1999);
Kresse and Furthmüller (1996). The calculations presented here have 400 eV. We
use the PAWBlöchl (1994) method together with the exchange-correlation
functional in the PBE to describe the electronic interactions Perdew (1996).
The criteria used for the convergence of energy was $10^{-4}\textrm{eV}$. AIMD
calculations have been performed using the Andersen thermostat at 300K. Time
steps of for a total simulation time of 5 ps. The NEB method is used to
determine the path which a atom on the crystal energy surface would take when
connected by two states, which are minimum energy states of the system. NEB
will seek for the lowest energy cost path between two states. The path through
the potential energy surface show us the energy barrier which the atom needs
to overcome to go to a minimum energy state to another minimum energy state.

##  III Monolayers and nanopores of MoX2

The most stable structure among the MoS2 polytypes is 2H at ambient
conditions. Meanwhile, other polytypes of MX2, including 1T-MoS2 and 1T’-MoS2
have been found Zhao  _et al._ (2018); Huang  _et al._ (2018); Liu (2018).
These structures are metallic materials, while the 2H-MoS2 structure exhibits
semiconductor properties.

[figure]style=plain,subcapbesideposition=top

[] []

Figure 1: 48 H2 molecules adsorbed on a) 1T’-MoS2 structure and b) the 2H-MoS2
structure.

Gravimetric analysis is a quantitative analytical technique that enables the
determination of the amount of a substance in a given mixture, thus the
gravimetric hydrogen storage capacity for 2H-MoS2 reached 6.97wt%, it was seen
that the hydrogen molecules are inert on the 2H-MX2 layer, being more
energetically favorable the molecule perpendicular to the structures with type
2H-MX2, molybdenum disulfide (MoS2) demonstrates a structure with higher
storage capacity in relation to other transition metal dichalcogenides.
Adsorption energy of around 200 meV as shown in Fig. 2, for the 2H lies in the
physisorption range. For the 1T’ we see formation of bonds between hydrogen
and the surface atoms.

Figure 2: Adsorption energy of H2 molecules on 1T’ and 2H phases of MoS2 and
MoSe2 as a function of number of hydrogen molecules.

In order to find a new route for hydrogen storage, defective 2H-MoS2 was
investigated. In our previous work we have investigated several point defects
Onita  _et al._ (2023); Komsa and Krasheninnikov (2015). The most common
defects in MoS2 are S and Mo vacancies. Beyond spontaneous defects, ion beam
irradiation can also be used to promote larger defects Pantelides so we
calculated the 2H-MoS2 in vacancy defects in order to understand the behavior
of hydrogen molecules in the presence of these defects, as shown in the Fig.
3.

Figure 3: Adsorbed hydrogen molecules on MoS2 nanopores: a) molybdenum
monovacancy, b) monovacancy of molybdenum and four sulfur vacancies, c)
monovacancy of molybdenum and six sulfur vacancies and d) three vacancy of
molybdenum and six sulfur vacancies.

In order to estimate the possibility of additional H2 molecules within the
nanopores, the reaction of H2 molecules was investigated. We have performed
NEB Henkelman  _et al._ (2000); Henkelman and Jónsson (2000) calculations to
estimate the potential barrier for the molecule to diffuse through the pore.
The NEB was performed along a path perpendicumar to the MoS2 plane. The
intermediate states are shown in Fig. 4. Five images were chosen along the
diffusion path. Along the chosen path a barrier of 0.35 eV is found, as shown
in Fig. 4. This implies that additional molecules could reside within the
pore, thus enhancing the storage capacity of 2H-MoS2.

Figure 4: Energy as a function of reaction coordinate, where a hydrogen
molecule diffuses through the nanopore with a molybdenum vacancy.

The gravimetric storage of H2 adsorbed on the defective layers containing one
vacancy results in 7.09 wt$\%$. Defect engineering could be a path for
increasing hydrogen storage capacity.

##  IV Intercalation of H2 in bilayers of MoS2

Effective approaches to fabricate bandgap engineered bilayer MoS2 structures,
an important step to fabricate homo/hetero-structures, and thus advanced
electronic devices have been recently developed Zhou  _et al._ (2018); Cheng
(2023). This is specially interesting for energy storage applications, since
molecules can be trapped between the layers. We now discuss hydrogen
intercalation in MX2 bilayers. The relaxed configurations are shown in Fig. 5.
We calculated the gravimetric hydrogen storage capacity for the 2H-MoS2 and
1T’-MoS2 bilayers respectively. The gravimetric hydrogen storage varied from
3.39 wt$\%$ and 2.17wt$\%$, as shown in Tab. 1. We analyzed the adsorption
energy of the 2H-bilayer and 1T’-bilayer as a function of the H2 numbers. The
results are shown in Fig. 6

Figure 5: Intercalated $H_{2}$ molecules in a) 2H-MoS2 and b) 1T’-bilayers. Figure 6: Adsorption energy of hydrogen molecules for 1T’ bilayer (blue) and 2H bilayer (red). structure | ML wt% | BL 2H(1T’) wt%
---|---|---
1T’(2H)-MoS2 | 6.43(6.97) | 3.39(2.17)
MoS2-V2S | 7.09 | -
MoS2-V1Mo | 7.11 | -
MoS2-V1Mo4S | 7.13 | -
MoS2-V1Mo6S | 7.14 | -
MoS2-V3Mo2S | 7.52 | -
1T’(2H)-MoSe2 | 4.67(5.22) | 1.55 (1.55)
WS2 | 4.66 (4.66) | 1.60 (1.60)
WSe2 | 3.42 (3.42) | 0.92 (0.92)
MoTe2 | 3.34 (3.34) | 1.13 (1.13)
WTe2 | 2.66 (2.66) | 0.91 (0.91)
Table 1: Hydrogen storage capacity, where the MX2(M=Mo,W and X=S,Se,Te). Note
that for vacancies, additional H2 storage capacity is considered within the
nanopore.

##  V Molecular dynamics of H2 adsorbed on MoS2

In order to investigate temperature effects of adsorption of H2 on both
polytypes, 1T’-MoS2 and 2H-MoS2 MD simulations were performed at a temperature
of 300 K with the Andersen thermostat bath in contact with the system. The RDF
revealed that for the 2H phase, the equidistances equivalent to the first and
second neighbors of a referential Mo, are illustrated in Ref. 8. Additionally,
the structure of 2H-MoS2 did not shown structural distortion.

[] []

Figure 7: Snapshots of AIMD simulations for 2H-MoS2 and 1T’ structure
performed at 300K.

[] []

Figure 8: Snapshots of AIMD simulations for 2H-MoS2 at 300K.

On the other hand, for 1T’-MoS2, owned to the asymetry between the S-Mo-S
bonds, additional peaks were seen, as depicted in Fig. 9(b).

[] []

Figure 9: AIMD simulations at 300K for 1T’-MoS2. a) Energy as a function of
time and b) RDF. Snapshots at 100, 2000 and 5000 fs simulation times are shown
in the inset.

##  VI MoX2 (X= Pd, Pd and Ni)

[] []
[] []
[] []

Figure 10: Total energy as a function of lattice parameter for a) NiS2, b)
PtS2, c) PdS2, d) NiSe2, e) PtSe2 and f) PdSe2.

[] []
[] []
[] []
[] []
[] []

Figure 11: Phonon dispersion for pristine a) 1T-PtS2, b) 1T-PtdS2, c)
1T-NidS2, d) 1T-PtSe2, e) 1T-PtdSe2, f) 1T-NidSe2, g) 2H-PtS2, h) 2H-PdS2, i)
2H-NiS2, j) 2H-PtSe2, k) 2H-PdSe2 and l) 2H-NiSe2.

Next we investigate XY2 nanostructures, namely (X = Ni,Pt,Pd) and (Y = S, Se)
in 1T and 2H polytypes. First we discuss the thermodynamic stability of
pristine structures. We find that the dichalcogenides, 1T-NiS2, 1T-PtS2, and
1T-PdS2 are higher in energy compared to their respective 2H polytypes
(2H-NiS2, 2H-PtS2, and 2H-PdS2). The lattice parameters for 1T-NiS2, 1T-PtS2,
and 1T-PdS2 are 3.35 $\AA$, 3.54 $\AA$, and 3.56 $\AA$, respectively. In the
2H polytype, 2H-NiS2 has a lattice parameter of 3.54 $\AA$, 2H-PtS2 has 3.48
$\AA$, and 2H-PdS2 has 3.92 $\AA$, as shown in Figs. 10(a)-(c). For
diselenides, a similar trend is found for NiSe2, PdSe2, and PtSe2. The 1T
polytypes show higher thermodynamic stability with lattice parameters of 3.53
$\AA$, 3.74 $\AA$, and 3.75 $\AA$, respectively, compared to the 2H polytypes
with lattice parameters of 3.56 $\AA$, 4.01 $\AA$, and 3.63 $\AA$, as depicted
in Figs. 10(d)-(e).

[] []

[] []

[] []

Figure 12: Hydrogen adsorption on NiS2 (clockwise) : a) bridge (side A), b)
bridge (side B), c) top-X (side A), d) bottom-X (side B), e) hollow (side
A)and f) bottom (side B). a) sideview with H2 perpendicular to the layer, b)
sideview with H2 parallel to the layer, c) topview with H2 perpendicular to
the layer, d) topview with H2 parallel to the layer. NiS2
---
| 1T | 2H
site | Eads | dH-S | d${}_{\rm H-S_{2}}$ | dH-X | Eads | dH-S | dH-Se | dH-X
| (meV) | (Å) | (Å) | (Å) | (meV) | (Å) | (Å) | (Å)
Top bridge | 30(30) | 2.86(3.24) |  | 3.90(4.14) | -0.64(-0.63) | 2.62(2.94) | 4.62(4.51) |
Top hollow | 70(70) | 2.98(3.22) |  | 3.92(4.27) | -0.65(-0.62) | 3.11(2.97) | 3.96(3.98) |
Top X | 60(60) | 2.98(3.37) |  | 3.38(3.83) | -0.66(-0.64) | 3.11(3.10) | 3.41(3.62) |
Top S | 20(20) | 2.92(3.30) |  | 4.51(4.87) | -0.65(-0.64) | 2.94(3.01) | 4.52(4.33) |
Bottom bridge | 30(30) |  | 3.15(3.14) | 3.95(4.14) | -0.64(-0.62) |  | 2.62(2.88) | 3.57(3.78)
Bottom X | 30(30) |  | 5.75(3.05) | 4.26(3.7) | -0.65(-0.62) |  | 3.14(2.96) | 3.96(3.97)
Bottom S | 60(60) |  | 2.95(3.16) | 3.40(4.34) | -0.65(-0.63) |  | 3.13(3.05) | 3.40(3.61)
Bottom hollow | 30(30) |  | 4.35(3.12) | 4.38(4.61) | -0.65(-0.66) |  | 2.71(3.55) | 4.31(4.92)
Table 2: Adsorption energy and atomic distances of the material NiS2. in
brackets for $H_{2}$ molecule perpendicular to the monolayer and without
brackets parallel to the monolayer, note disordered refer to a structural
disorder. NiSe2
---
site | 1T | 2H
| Eads | dH-S | dH-Se | dH-X | Eads | dH-S | dH-Se | dH-X
| (meV) | (Å) | (Å) | (Å) | (meV) | (Å) | (Å) | (Å)
Top bridge | -0.57 | 3.05 |  | 4.09 | dis. |  |  |
Top hollow | -0.56 | 3.33 |  | 4.38 | -0.53 | 3.01 |  | 3.99
Top X | -0.53 | 3.03 |  | 3.47 | -0.53 | 2.96 |  | 3.41
Top S | -0.57 | 2.90 |  | 4.62 | dis. |  |  |
Bottom bridge | -0.57 |  | 3.38 | 4.24 | dis. |  |  |
Bottom X | -0.57 |  | 3.52 | 4358 | -0.57 |  | 3.47 | 5.79
Bottom S | -0.57 |  | 3.38 | 3.94 | -0.56 |  | 3.20 | 3.73
Bottom Se | -0.56 |  | 2.87 | 4.59 | dis. |  |  |
PdSe2
| 1T | 2H
site | Eads | dH-S | dH-Se | dH-X | Eads | dH-S | dH-Se | dH-X
| (meV) | (Å) | (Å) | (Å) | (meV) | (Å) | (Å) | (Å)
Top bridge | -0.57 | 3.10 |  | 4.16 | dis. |  |  |
Top hollow | -0.52 | 2.99 |  | 4.05 | -0.58 | 3.36 |  | 4.25
Top X | -0.53 | 3.02 |  | 5.05 | -0.56 | 3.03 |  | 3.39
Top S | -0.57 | 2.88 |  | 4.71 | dis. |  |  |
Bottom bridge | -0.57 |  | 3.39 | 4.27 | dis. |  |  |
Bottom X | -0.52 |  | 3.02 | 4.04 | -0.55 |  | 2.95 | 5.52
Bottom S | -0.57 |  | 3.39 | 3.90 | -0.58 |  | 3.28 | 3.52
Bottom Se | -0.57 |  | 2.88 | 4.71 | dis. |  |  |
PtSe2
| 1T | 2H
site | Eads | dH-S | dH-Se | dH-dis. | Eads | dH-S | dH-Se | dH-X
| (meV) | (Å) | (Å) | (Å) | (meV) | (Å) | (Å) | (Å)
Top bridge 40 | 2.87 |  | 3.95 | dis. |  |  |  |
Top hollow | 20 | 3.50 |  | 4.62 | 20 | 3.06 |  | 4.99
Top dis. | 50 | 3.13 |  | 3.53 | 10 | 3.90 |  | 4.78
Top S | 20 | 3.02 |  | 4.84 | dis. |  |  |
Bottom bridge | 20 |  | 3.44 | 4.32 | dis. |  |  |
Bottom dis. | 20 |  | 3.43 | 4.52 | 20 |  | 3.64 | 6.12
Bottom S | 20 |  | 3.57 | 4.15 | diss. |  |  |
Bottom Se | 20 |  | 2.99 | 4.81 | dis. |  |  |
Table 3: Adsorption energy and interatomic distances for NiSe2, PdSe2 and
PtSe2. H2 molecule perpendicular (parallel) to the monolayer. Disordered
refers to a structural disorder.

Adsoprtion energy for a single molecule on 3. It was found that due to the low
concentration of H2 in the network, it caused the dissociation of H2 on PdS2
and PtS2, forming S-H bonds. When compared to adsorption at high
concentrations of H2, the adsorption energy shows a dependency on the
concentration. As the concentration of H2 increases, the available adsorption
sites on the surface become unavailable. As all surface sites become occupied,
the addition of further H2 molecules does not lead to a significant decrease
in adsorption energy.

[]
[]

Figure 13: Adsorption energy Eads of H2 for MS2 as a function of the number of
H2 molecules.

The adsorption energy of the investigated structures reached the order of 100
meV. A close-up view of the region with high H2 concentration reveals that as
the H2 concentration increases, the available adsorption sites on the surface
become saturated, forming layers of H2 as shown in Fig. 6.

For 1T-NiS2, it was seen that for a quantity of H2 adsorption molecules higher
than 28 molecules, the nickel disulfide (NiS2) structure exhibits structural
disorder caused by the absorption of H2, forming S-H bonds as demonstrated in
Fig. 14. A storage capacity of 10.8 wt% was obtained. Alternatively,
considering the quantity of 28 H2 molecules, the gravimetry is 8.83 wt%.
Calculations show that for 2H-NiS2, the gravimetric value is 11.44 wt%, and
there is no significant structural distortion.

[]
[]

Figure 14: a) Adsorption of 32 molecules on a) 1T-NiS2 structure and b)
2H-NiS2 structure.

For 1T-PdS2, a similar behavior was obtained from 28 H2 molecules. This result
is independent on the molecule orientation. A storage capacity of 8.04 wt% is
found with a small amount of H2 dissociating on the surface sulphur atoms
(0.3%). Due to larger weight of Pd compared to Ni, 2H-PdS2 was found to have a
storage capacity of 8.53 wt%.

[]
[]

Figure 15: Adsorption of 32 H2 molecules on a) 1T-PdS2 and b) 2H-PdS2.

It was found that for the 1T-PtS2 polytype, the H2 adsorption behavior was
similar to 1T-PdS2. However, due to the fact that Pt is heavier than Pd, the
storage capacity of 1T-PtS2 was 5.45 wt%, whereas for 2H-PtS2, it was found to
be 5.79 wt%, as shown in Tab. 4.

| Gravimetry (wt%) | Eads (meV)
---|---|---
1T-NiS2 | 10.8 | 30
2H-NiS2 | 11.44 | 10
1T-PtS2 | 5.45 | 30
2H-PtS2 | 5.79 | -160
1T-PdS2 | 8.04 | 20
2H-PdS2 | 8.53 | 210
1T-NiSe2 | 6.87 | -110
2H-NiSe2 | 6.87 | -130
1T-PtSe2 | 4.33 | 15
2H-PtSe2 | 3.28 | 27
1T-PdSe2 | 5.71 | 24
2H-PdSe2 | 3.99 | 32
Table 4: Gravimetric analysis for disulphides and diselenides for H2
adsorption at maximum hydrogen concentration for each structure.

Adsorption on dichalcogenides resulted in the dissociation of H2 and a
modification of the structure, leading to structural changes. Nonetheless,
when compared with the 1T-NiS2 structures, an increase in storage capacity
from 8.61 wt% to up to 11.44 wt% was found. Therefore, both NiS2 polytypes and
PdS2 polytypes can be considered promising candidates for hydrogen storage.

##  VII Conclusions

In this work we have investigated porous MoS2 and XS2 (X = Pd, Ni and Pt) as
alternative materials for hydrogen storage. Nickel selenide appears as a
viable candidate for hydrogen storage, boasting a gravimetry of 6.87% wt. We
also show that MoS2 nanopores could enhance the gravimetry for hydrogen
storage. This enhancement occurs through the adsorption of H2 molecules within
the pore. AIMD suggested that the structures identified in this work could be
stable even at room temperature.

## References

  * Sen (2023) _Statistical Review of World Energy_ (2023).
  * Schlapbach and Zuttel (2001) L. Schlapbach and A. Zuttel, Nature 414, 353 (2001).
  * Züttel (2004) A. Züttel, Naturwissenschaften 91, 157–172 (2004).
  * Zhou (2005) L. Zhou, Renewable and Sustainable Energy Reviews 9, 395 (2005).
  * Tarhan and Çil (2021) C. Tarhan and M. A. Çil, Journal of Energy Storage 40, 102676 (2021).
  * Graetz (2009) J. Graetz, Chem. Soc. Rev. 38, 73 (2009).
  * Lynch (2011) S. Lynch, in  _Stress Corrosion Cracking_ , Woodhead Publishing Series in Metals and Surface Engineering, edited by V. Raja and T. Shoji (Woodhead Publishing, 2011) pp. 90–130.
  * Cao (2021) Y. Cao, ACS Nano 15, 11014 (2021).
  * Geim and Novoselov (2007) A. K. Geim and K. S. Novoselov, Nature Materials 6, 183 (2007).
  * Chen and Kuriyama (2001) J. Chen and Kuriyama, J. Am. Chem. Soc. (2001).
  * Chen J (2001) Y. H. T. H. S. T. Chen J, Kuriyama N, J Am Chem Soc. 123, 11813 (2001).
  * Chen  _et al._ (2020) J. Chen, J. Cao, J. Zhou, Y. Zhang, M. Li, W. Wang, J. Liu, and X. Liu, Phys. Chem. Chem. Phys. 22, 430 (2020).
  * Yang  _et al._ (2021a) S. Yang, X. Wang, G. Lei, H. Xu, Z. Wang, J. Xiong, and H. Gu, Surfaces and Interfaces 26, 101329 (2021a).
  * Yang  _et al._ (2021b) S. Yang, Y. Liu, G. Lei, Y. Xie, L. Peng, H. Xu, Z. Wang, and H. Gu, International Journal of Hydrogen Energy 46, 24233 (2021b).
  * Pham and Yeom (2016) V. P. Pham and G. Y. Yeom, Advanced Materials 28, 9024 (2016).
  * Altuntepe (2024) E. S. O. M. e. a. Altuntepe, A., J Mater Sci: Mater Electron 35, 69 (2024).
  * Zhu  _et al._ (2016) Z. Zhu, H. Peelaers, and C. G. Van de Walle, Phys. Rev. B 94, 085426 (2016).
  * Altuntepe  _et al._ (2024) A. Altuntepe, S. Erkan, M. A. Olğar, S. Çelik, and R. Zan, International Journal of Hydrogen Energy 56, 690 (2024).
  * Moujaes and Diery (2021) E. A. Moujaes and W. Diery, Physica E: Low-dimensional systems and nanostructures 128, 114611 (2021).
  * Deng  _et al._ (2018) S. Deng, L. Li, and Y. Zhang, ACS Applied Nano Materials 1 (2018), 10.1021/acsanm.8b00363.
  * Zhao (2020) Y. Z. G. J. e. a. Zhao, X.W., Sci. Rep. 10, 4028 (2020).
  * Kresse and Joubert (1999) G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).
  * Kresse and Furthmüller (1996) G. Kresse and J. Furthmüller, Computational Materials Science 6, 15 (1996).
  * Blöchl (1994) P. E. Blöchl, Phys. Rev. B 50, 17953 (1994).
  * Perdew (1996) K. E. M. Perdew, J. P.; Burke, Phys. Rev. Lett. 77, 3865–3868 (1996).
  * Zhao  _et al._ (2018) W. Zhao, J. Pan, Y. Fang, X. Che, D. Wang, K. Bu, and F. Huang, Chemistry – A European Journal 24, 15942 (2018).
  * Huang  _et al._ (2018) F.-Q. Huang, W. Zhao, J. Pan, F. Yuqiang, X. Che, D. Wang, and K. Bu, Chemistry 24 (2018), 10.1002/chem.201801018.
  * Liu (2018) W. J. W. L. e. a. Liu, L., Nature Mater. 17, 1108–1114 (2018).
  * Onita  _et al._ (2023) V. Onita, M. Kendjy, F. B. de Oliveira, and A. L. da Rosa, Brazilian Journal of Physics 54, 7 (2023).
  * Komsa and Krasheninnikov (2015) H.-P. Komsa and A. V. Krasheninnikov, Phys. Rev. B 91, 125304 (2015).
  * Henkelman  _et al._ (2000) G. Henkelman, B. P. Uberuaga, and H. Jónsson, The Journal of Chemical Physics 113, 9901 (2000).
  * Henkelman and Jónsson (2000) G. Henkelman and H. Jónsson, The Journal of Chemical Physics 113, 9978 (2000).
  * Zhou  _et al._ (2018) W. Zhou, C. Yuan, A. Hong, X. Luo, and W. Lei, Nanoscale 10, 1145 (2018).
  * Cheng (2023) X. W. W. H. e. a. Cheng, X., Front. Phys. 18, 53303 (2023).