# A chain-level HKR-type map and a Chern character formula

Kuerak Chung  Department OF Mathematics Education
Korea National University of Education
250 Taeseongtabyeon-ro, Gangnae-myeon
Heungdeok-gu, Cheongju-si, Chungbuk 28173
Republic of Korea<EMAIL_ADDRESS>,  Bumsig Kim  Korea Institute for
Advanced Study
85 Hoegiro, Dongdaemun-gu
Seoul 02455
Republic of Korea<EMAIL_ADDRESS>and  Taejung Kim  Department OF
Mathematics Education
Korea National University of Education
250 Taeseongtabyeon-ro, Gangnae-myeon
Heungdeok-gu, Cheongju-si, Chungbuk 28173
Republic of Korea<EMAIL_ADDRESS>
###### Abstract.

We construct a Hochschild-Kostant-Rosenberg-type quasi-isomorphism for the
negative cyclic homology of the category of global matrix factorizations on a
smooth separated scheme of finite type over a field. The map is explicit
enough to yield a negative cyclic Chern character formula for global matrix
factorizations. We also extend these results to the equivariant case of a
finite group.

###### Key words and phrases:

Matrix factorizations, Negative cyclic homology, Hochschild homology,
Categorical Chern characters

###### 2010 Mathematics Subject Classification:

Primary 14A22; Secondary 16E40, 18E30

K. Chung and T. Kim are supported by NRF-2018R1D1A3B07043346. B. Kim is
supported by KIAS individual grant MG016404.

###### Contents

  1. 1 Introduction
  2. 2 Hochschild-type invariants
  3. 3 Categorical Chern characters
  4. 4 Global matrix factorizations
  5. 5 Connections and the map ${\mathrm{tr}}_{\nabla}$
  6. 6 Applications
  7. A Proof of Proposition 3.3
  8. B Proof of Theorem 5.2

##  1\. Introduction

###  1.1. Overview

Fix a field $k$ of characteristic zero and let $X$ be a smooth separated
scheme of finite type over $k$. Let $w$ be a global function on $X$ and let
${\mathbb{G}}$ be either the group ${\mathbb{Z}}$ or ${\mathbb{Z}}/2$. The
structure sheaf ${\mathcal{O}}_{X}$ of $k$-algebras is defined to be
concentrated in degree $0$ as a ${\mathbb{G}}$-graded sheaf. We require that
the degree of $w$ is 2. Hence if $w$ is nonzero, ${\mathbb{G}}$ is
${\mathbb{Z}}/2$. If $w=0$, we allow either ${\mathbb{G}}$. Assume that the
critical values of $w$ are possibly zero.

We consider a differential ${\mathbb{G}}$-graded (in short dg) category
$D_{dg}(X,w)$ of ${\mathbb{G}}$-graded matrix factorizations for $(X,w)$. By a
matrix factorization $(P,\delta_{P})$ for $(X,w)$ we mean a locally free
coherent ${\mathcal{O}}_{X}$-module $P$ with a ${\mathbb{G}}$-grading and
$\delta_{P}$ is a curved differential of $P$. By definition, $\delta_{P}$ is a
degree $1$ ${\mathcal{O}}_{X}$-endomorphism of $P$ satisfying
$\delta_{P}^{2}=w\cdot\mathrm{id}_{P}$. The Hom space from $P$ to $Q$ in
$D_{dg}(X,w)$ is taken to be a dg $k$-module
${\mathrm{Hom}}_{{\mathcal{O}}_{X}}^{\bullet}(I(P),I(Q))$ with the chosen
quasi-coherent curved injective replacements $I(P)$, $I(Q)$ of $P$, $Q$
respectively; see [1]. Note that when ${\mathbb{G}}={\mathbb{Z}}$, then $w=0$
and the homotopy category of $D_{dg}(X,w)$ is triangulated and equivalent to
the derived category of bounded complexes of coherent sheaves on $X$. When
$w\neq 0$, then ${\mathbb{G}}={\mathbb{Z}}/2$ and the homotopy category of
$D_{dg}(X,w)$ is the derived category of matrix factorizations on $X$. If
$w\neq 0$ in every component of $X$, the homotopy category is also equivalent
to the singularity category of the hypersurface $w^{-1}(0)$; see [17, Theorem
2] and [15, Proposition 2.13].

The periodic cyclic homology and the Hochschild homology of $D_{dg}(X,w)$ play
the roles of the de Rham cohomology and the Hodge cohomology of the “space”
$(X,w)$, respectively. The negative cyclic homology $HN_{*}(D_{dg}(X,w))$
encapsulates both homology theories, which is canonically isomorphic to

${\mathbb{H}}^{-*}(X,(\Omega^{\bullet}_{X/k}[[u]],-dw+ud));$

see [7, Theorem 1.4]. Here $u$ is a formal variable with degree $2$ and by
definition the exterior derivative $d$ has degree $-1$. Each $(P,\delta_{P})$
defines a tautological class in $HN_{0}(D_{dg}(X,w))$, the so-called
categorical Chern character $\mathrm{Ch}_{HN}(P)$. Under the isomorphism,
$\mathrm{Ch}_{HN}(P)$ can be written as an element $\mathrm{ch}_{HN}(P)$ of
${\mathbb{H}}^{0}(X,(\Omega^{\bullet}_{X/k}[[u]],-dw+ud))$. In this paper we
find a formula for $\mathrm{ch}_{HN}(P)$ by Chern-Weil theory and a Čech
resolution, which is a globalization of Brown – Walker’s formula [3].

###  1.2. A Chern character formula via Chern-Weil theory

We fix a finite open affine covering $\mathfrak{U}:=\\{U_{i}\\}_{i\in I}$ of
$X$ and a total ordering of the index set $I$. Let $(P,\delta_{P})$ be a
matrix factorization for $(X,w)$ as before. Since each homogeneous part of
$P|_{U_{i}}$ corresponds to a finitely generated projective module over a
noetherian ring $\Gamma(U_{i},{\mathcal{O}}_{X})$, $P|_{U_{i}}$ always admits
a connection

(1.1) $\nabla_{i}:P|_{U_{i}}\to
P|_{U_{i}}\otimes_{{\mathcal{O}}_{U_{i}}}\Omega^{1}_{U_{i}/k}.$

It is a $k$-linear degree $-1$ map satisfying
$\nabla_{i}(pa)=\nabla_{i}(p)a+(-1)^{|p|}p\otimes da$ for $p\in P|_{U_{i}}$,
$a\in{\mathcal{O}}_{U_{i}}$. We define the associated total curvature

(1.2) $R:=\prod_{i\in
I}(u\nabla_{i}^{2}+[\nabla_{i},\delta_{P}])+\prod_{i<j,i,j\in
I}(\nabla_{i}-\nabla_{j})$

as an element in the Čech complex
$\text{\rm\v{C}}(\mathfrak{U},{\mathcal{E}}nd(P)\otimes_{{\mathcal{O}}_{X}}\Omega_{X/k}^{\bullet}[[u]])$;
see (6.2) and Remark 5.6. Note that the total degree of each term in the
curvature $R$ is zero.

######  Theorem 1.1.

(Theorem 6.1) The following Chern character formula holds:

(1.3) $\mathrm{ch}_{HN}(P)={\mathrm{tr}}\exp(-R)$

in the ordered Čech cohomology of complexes of sheaves

$\check{H}^{0}(\mathfrak{U},(\Omega^{\bullet}_{X/k}[[u]],-dw+ud)).$

Here the expression

$\exp(-R)=\mathrm{id_{P}}-R+R^{2}/2-\cdots\cdots+(-1)^{\dim X}R^{\dim X}/(\dim
X)!$

uses the product formula (4.5) in the Čech cochain complex and ${\mathrm{tr}}$
denotes the supertrace (5.10).

When $|I|=1$ so that $X$ is affine, formula (1.3) is the main result of Brown
and Walker [3]. When $u=0$, $\mathrm{ch}_{HN}(P)$ becomes the image
$\mathrm{ch}_{HH}(P)$ of the Hochschild homology valued Chern character
$\mathrm{Ch}_{HH}(P)$. Formula (1.3) specialized to $u=0$ matches with those
of Căldăraru [4] for ${\mathbb{G}}={\mathbb{Z}}$; and Platt [18] and Kim and
Polishchuk [14] for ${\mathbb{G}}={\mathbb{Z}}/2$, respectively (up to a sign
convention which occurs in the choices of isomorphisms). We will also get a
localized Chern character formula; see § 6.2

###  1.3. A chain-level HKR-type map

We will prove Theorem 1.1 by an explicit quasi-isomorphism, which is a
globalization of the generalized Hochschild-Kostant-Rosenberg (in short HKR)
map in the affine case [3, Theorem 5.19]. The ordinary HKR map [16, § 1.3.15]
uses the canonical de Rham differential $d$ of ${\mathcal{O}}_{X}$. While $P$
does not admit a global connection in general, there is a connection on
$P|_{U_{i}}$. This leads us to consider the Čech model
$D_{\mathrm{\check{C}}}(X,w)$ of $D_{dg}(X,w)$; see §4.2.2. Let
$\mathrm{\check{C}}(P)$ denote the sheafified version of the ordered Čech
$k$-complex $\text{\rm\v{C}}(\mathfrak{U},P)$ for $P$; see (4.2). The objects
of $D_{\mathrm{\check{C}}}(X,w)$ are those of $D_{dg}(X,w)$. The Hom space
${\mathrm{Hom}}(P,Q)$ is defined to be the dg $k$-space

(1.4)
$\displaystyle{\mathrm{Hom}}_{\mathrm{\check{C}}({\mathcal{O}}_{X})}^{\bullet}(\mathrm{\check{C}}(P),\mathrm{\check{C}}(Q))$
$\displaystyle\cong\Gamma(X,{\mathcal{H}}om_{{\mathcal{O}}_{X}}^{\bullet}(P,Q)\otimes_{{\mathcal{O}}_{X}}\mathrm{\check{C}}({\mathcal{O}}_{X}))$
$\displaystyle=\bigoplus_{p}\text{\rm\v{C}}^{p}(\mathfrak{U},{\mathcal{H}}om_{{\mathcal{O}}_{X}}^{\bullet}(P,Q)).$

Its underlying graded $k$-space is the linear space of graded right
$\mathrm{\check{C}}({\mathcal{O}}_{X})$-linear sheaf homomorphisms from
$\mathrm{\check{C}}(P)$ to $\mathrm{\check{C}}(Q)$. Its differential is given
as follows. Let
$\delta_{\mathrm{\check{C}}(P)}=\delta_{P}+d_{\scriptscriptstyle\text{\v{C}ech}}$,
the sum of the differential $\delta_{P}$ of $P$ and the Čech differential
$d_{\scriptscriptstyle\text{\v{C}ech}}$. Then the differential of
${\mathrm{Hom}}(P,Q)$ is defined by sending $f\in{\mathrm{Hom}}(P,Q)$ to

$\delta_{\mathrm{\check{C}}(Q)}\circ
f-(-1)^{|f|}f\circ\delta_{\mathrm{\check{C}}(P)}\in{\mathrm{Hom}}(P,Q),$

where $|f|$ denotes the degree of $f$.

For each matrix factorization $P$ for $(X,w)$ and $i\in I$, choose, once and
for all, a connection $\nabla_{P,i}$ on $P|_{U_{i}}$. Hence we have a zero-th
global Čech element

$\nabla_{P}:=\prod_{i\in I}\nabla_{P|_{U_{i}}}$

of the $k$-sheaf
${\mathcal{H}}om_{k}(\text{\rm\v{C}}(P),\text{\rm\v{C}}(P)\otimes\Omega^{1}_{X})$.

When $P={\mathcal{O}}_{X}$, we choose the de Rham differential as a connection
on ${\mathcal{O}}_{U_{i}}$. Let $C(D_{\mathrm{\check{C}}}(X,w))$ denote the
Hochschild complex of the dg category $D_{\mathrm{\check{C}}}(X,w)$. Using the
supertrace ${\mathrm{tr}}$ (5.10) and the product (4.5) we define a
$k[[u]]$-linear map

(1.5)
${\mathrm{tr}}_{\nabla}:C(D_{\mathrm{\check{C}}}(X,w))[[u]]\to\text{\rm\v{C}}(\mathfrak{U},\Omega^{\bullet}_{X/k})[[u]]$

by sending

(1.6)
${\alpha}_{0}[{\alpha}_{1}|...|{\alpha}_{n}]\mapsto\sum_{(j_{0},...,j_{n})}(-1)^{J}{\mathrm{tr}}\big{(}\frac{{\alpha}_{0}R_{1}^{j_{0}}[\nabla,{\alpha}_{1}]R_{2}^{j_{1}}[\nabla,{\alpha}_{2}]...R_{n}^{j_{n-1}}[\nabla,{\alpha}_{n}]R_{0}^{j_{n}}}{(n+J)!}\big{)}$

for
${\alpha}_{j}\in\text{\rm\v{C}}(\mathfrak{U},{\mathcal{H}}om_{{\mathcal{O}}_{X}}(P_{j+1},P_{j}))$
with $P_{n+1}=P_{0}$, where $j_{i}\in{\mathbb{Z}}_{\geq 0}$ and
$J=\sum_{i=0}^{n}j_{i}$; and

$\displaystyle[\nabla,{\alpha}_{j}]$
$\displaystyle:=\nabla_{P_{j}}\cdot{\alpha}_{j}-(-1)^{|{\alpha}_{j}|}{\alpha}_{j}\cdot\nabla_{P_{j+1}};$
$\displaystyle R_{j}$
$\displaystyle:=u\nabla_{P_{j}}^{2}+[\nabla_{P_{j}},\delta_{P_{j}}]+\prod_{i_{0}<i_{1}}(\nabla_{P_{j},i_{0}}-\nabla_{P_{j},i_{1}});$

see (5.23) for unwinded expressions. Note that when $|I|=1$ and $P_{i}$ is
${\mathcal{O}}_{X}$ for $i=0,...,n$, formula (1.6) is the usual HKR map.

######  Theorem 1.2.

(Theorem 5.4) Let $b$ denote the sum of Hochschild differentials from
compositions and differentials; and let $B$ be Connes’ boundary map. Then the
$k[[u]]$-linear map

(1.7) ${\mathrm{tr}}_{\nabla}:\ \
(C(D_{\mathrm{\check{C}}}(X,w))[[u]],b+uB)\\\
\to(\text{\rm\v{C}}(\mathfrak{U},\Omega^{\bullet}_{X/k})[[u]],d_{\scriptscriptstyle\text{\v{C}ech}}-dw+ud)$

is a quasi-isomorphism compatible with the HKR-type isomorphism.

Again when $|I|=1$ (i.e., the affine case), this is a theorem of Brown and
Walker [3], which generalizes a work of Segal [21].

###  1.4. The equivariant case by a finite group

As an application of Theorem 1.2 we can get an equivariant Chern character
formula: Let $X$ come with a finite group $G$ action and assume that $w$ is
$G$-fixed. Furthermore, assume that $X$ is quasi-projective. We consider the
dg category $D_{\text{\rm\v{C}}}^{G}(X,w)$ of $G$-equivariant matrix
factorizations for $(X,w)$. We choose a finite open affine covering
$\mathfrak{U}=\\{U_{i}\\}_{i\in I}$ such that each $U_{i}$ is $G$-invariant.
For a finite open affine covering of the $g$-fixed locus $X^{g}$, let us use
$\\{X^{g}\cap U_{i}\\}_{i\in I}$. Let $\nabla_{P|_{U_{i}}}$ be a
$G$-equivariant connection on $P|_{U_{i}}$. Define a connection
$\nabla_{P|_{X^{g}},i}$ on $P|_{X^{g}\cap U_{i}}$ to be the restriction of
$\nabla_{P|_{U_{i}}}$ to $X^{g}\cap U_{i}$ followed by the natural map

$P|_{X^{g}\cap U_{i}}\otimes_{{\mathcal{O}}_{X^{g}\cap
U_{i}}}\Omega^{1}_{U_{i}/k}|_{X^{g}\cap U_{i}}\to P|_{X^{g}\cap
U_{i}}\otimes_{{\mathcal{O}}_{X^{g}\cap U_{i}}}\Omega^{1}_{X^{g}\cap U_{i}/k}\
.$

######  Theorem 1.3.

Let $w_{g}:=w|_{X^{g}}$.

  1. (1)

(Theorem 6.9) The mixed Hochschild complex
$(C(D_{\text{\rm\v{C}}}^{G}(X,w)),b,B)$ is naturally quasi-isomorphic to the
coinvariant mixed complex

$\left(\oplus_{g\in
G}(\Gamma(X^{g},\text{\rm\v{C}}(\Omega^{\bullet}_{X^{g}/k})),d_{\scriptscriptstyle\text{\v{C}ech}}-dw_{g},d)\right)_{G}.$

  2. (2)

(Formula (6.13)) For $P\in D_{\text{\rm\v{C}}}^{G}(X,w)$, its $G$-equivariant
Chern character $\mathrm{ch}^{G}_{HH}(P)$ becomes

$\mathrm{ch}^{G}_{HH}(P)=\frac{1}{|G|}\bigoplus_{g\in
G}{\mathrm{tr}}\left(\varphi_{g|_{X^{g}}}\exp(-[\prod_{i}\nabla_{P|_{X^{g}},i},\delta_{P|_{X^{g}}}+d_{\scriptscriptstyle\text{\v{C}ech}}])\right),$

where $\varphi_{g}\in\mathrm{End}_{{\mathcal{O}}_{X^{g}}}(P|_{X^{g}})$ is
defined by the multiplication by $g$.

Specializing to the case where X is affine space, this recovers the results
[19, Theorem 2.5.4 & Theorem 3.3.3] of Polishchuk and Vaintrob. When $w=0$ and
${\mathbb{G}}={\mathbb{Z}}$, Theorem 1.3 (1) is the main theorem of Baranovsky
[2]. In fact we will establish Theorem 1.3 combining Theorem 1.2 with his
works in [2]. Somewhere else we will treat the case of smooth separated DM
stacks of finite type over $k$.

###  1.5. Outline of the paper

In § 2 we collect definitions and fundamental facts of curved dg
$k$-categories; and their modules, mixed Hochschild complexes, and negative
cyclic complexes. In § 3 we recall the definitions of categorical Chern
characters and find their alternative expressions; see Proposition 3.3. In § 4
we discuss the Čech model of the dg category of global matrix factorizations
and introduce the sheafifications of mixed Hochschild complexes. We also
collect various invariance results which we will use later. In § 5 we
construct connections on a Čech resolution of global matrix factorizations.
Using those connections we define a cochain map ${\mathrm{tr}}_{\nabla}$ and
prove Theorem 1.2. In § 6 we prove Theorems 1.1 and 1.3.

###  1.6. Acknowledgements

K.C. and T.K. thank Chanyoung Sung and Hoil Kim for encouragement and support.
The authors thank Michael Brown, Jaeyoo Choy, David Favero, and Ed Segal for
careful reading and suggestions.

###  1.7. Conventions

Unless otherwise stated, “graded” means ${\mathbb{G}}$-graded. For a (graded)
sheaf ${\mathcal{F}}$ on a topological space and an open covering
$\mathfrak{U}=\\{U_{i}\\}_{i\in I}$, let
$U_{i_{0}...i_{p}}=U_{i_{0}}\cap...\cap U_{i_{p}}$ and let
$\text{\rm\v{C}}^{p}(\mathfrak{U},{\mathcal{F}})=\prod_{i_{0}<...<i_{p}}{\mathcal{F}}(U_{i_{0}...i_{p}})$.
For a $k$-category ${\mathcal{A}}$, by $x\in{\mathcal{A}}$ we mean an object
of ${\mathcal{A}}$. For $x,y\in{\mathcal{A}}$, by ${\mathcal{A}}(x,y)$,
${\mathrm{Hom}}(x,y)$ or ${\mathrm{Hom}}_{{\mathcal{A}}}(x,y)$ we mean the Hom
space from $x$ to $y$ in ${\mathcal{A}}$. For an element $a$ of a (bi-)graded
$k$-space, the (total) degree of $a$ is denoted by $|a|$. The commutator $[\
,\ ]$ is the graded commutator. Often we write $1_{P}$ for $\mathrm{id}_{P}$.
The symbol $\simeq$ indicates a quasi-isomorphism.

##  2\. Hochschild-type invariants

In this section we collect definitions of curved dg $k$-categories, their
mixed Hochschild complexes, and their negative cyclic complexes. The main
references of this section are [3, 20].

###  2.1. Curved dg $k$-categories

####  2.1.1. Curved dg algebras

Let $k$ be a field of characteristic zero. A $k$-category is a $k$-algebra
with several objects. A $k$-algebra is a $k$-module with a unital associative
$k$-linear multiplication. Let $A$ be a graded algebra over $k$ and let $h$ be
a degree $2$ element of $A$. Let $d_{A}$ be a degree $1$, $k$-linear map such
that (1) the Leibniz rule holds (i.e.,
$d_{A}(a_{1}a_{2})=d_{A}(a_{1})a_{2}+(-1)^{|a_{1}|}a_{1}d_{A}(a_{2})$ for
$a_{1},a_{2}\in A$), (2) $d_{A}^{2}=[h,?]$ (i.e., $d_{A}^{2}(a)=ha-ah$ for
$a\in A$), and (3) $d_{A}(h)=0$. Then we call $(A,d_{A},h)$ a curved dg (in
short cdg) $k$-algebra. Furthermore, if $h=0$, then we call $(A,d_{A})$ a dg
$k$-algebra.

####  2.1.2. Curved dg categories

There is the notion of a cdg $k$-category ${\mathcal{A}}$ as a cdg $k$-algebra
with several objects. It is a graded $k$-category with a differential
$d_{x,y}$ of ${\mathcal{A}}(x,y)$ and a degree $2$ element
$h_{x}\in{\mathcal{A}}(x,x)$ for every $x,y\in{\mathcal{A}}$ satisfying the
Leibniz rule, $d_{x,y}^{2}(f)=h_{y}\circ f-f\circ h_{x}$ for
$f\in{\mathcal{A}}(x,y)$, and $d_{x,x}(h_{x})=0$. The element $h_{x}$ is
called the curvature of $x$. If $h_{x}=0$ for all $x\in{\mathcal{A}}$, we call
${\mathcal{A}}$ a dg $k$-category.

####  2.1.3. Precomplexes

There is a cdg category $\mathrm{Com}_{cdg}(k)$, the category of precomplexes
of $k$-modules: an object is a graded $k$-module $C$ with a $k$-linear degree
$1$ map $\delta_{C}$ and the Hom space from $C$ to $D$ has a differential
$[\delta_{?},]$. The curvature of $C$ is by definition $\delta_{C}^{2}$. The
full subcategory $\mathrm{Com}_{dg}(k)$ of all objects with vanishing
curvatures is called the dg category of $k$-complexes.

####  2.1.4. Cdg functors

A pair $(F,{\alpha})$ is called a quasi-cdg functor from a cdg category
${\mathcal{A}}$ to another cdg category $\mathcal{B}$ if $F$ is a $k$-linear
degree $0$ homogeneous functor and ${\alpha}$ is an assignment of a degree $1$
element ${\alpha}_{x}\in\mathcal{B}(Fx,Fx)$ for each $x\in{\mathcal{A}}$ such
that

$F(d_{x,y}(f))=d_{Fx,Fy}(F(f))+{\alpha}_{y}\circ
F(f)-(-1)^{|f|}F(f)\circ{\alpha}_{x}$

for all $x,y\in{\mathcal{A}}$ and $f\in{\mathcal{A}}(x,y)$. Furthermore if
$F(h_{x})=h_{Fx}+d_{Fx,Fx}({\alpha}_{x})+{\alpha}_{x}^{2}$, $(F,{\alpha})$ is
called a cdg functor. We often say simply $F$ is a (quasi-)cdg functor for a
(quasi-)cdg functor $(F,{\alpha})$. If ${\alpha}_{x}=0$ for every $x$, we call
$F$ strict. In particular if ${\mathcal{A}}$, $\mathcal{B}$ are dg categories
and $F$ is a strict cdg functor, $F$ is just called a dg functor. Informally a
dg functor is a set of several “cochain” maps. If
$(G,\beta):\mathcal{B}\to{\mathcal{C}}$ is a quasi-cdg functor, then the
composition of $(G,{\beta})$ and $(F,{\alpha})$ is defined as $(G\circ
F,G\circ{\alpha}+{\beta}\circ F)$. Note that the composition is a quasi-cdg
functor and the composition of cdg functors becomes a cdg functor.

####  2.1.5. The opposite category

The opposite category ${\mathcal{A}}^{op}$ of ${\mathcal{A}}$ is a cdg
category whose objects are the objects of ${\mathcal{A}}$ and whose morphisms
are ${\mathcal{A}}^{op}(x,y):={\mathcal{A}}(y,x)$ with differential $d_{y,x}$.
The composition $g\circ f$ in ${\mathcal{A}}^{op}$ is $(-1)^{|f||g|}f\circ g$
in ${\mathcal{A}}$ and the curvature of $x$ in ${\mathcal{A}}^{op}$ is
$-h_{x}$.

####  2.1.6. Left and right (quasi-)cdg modules

A left (resp., right) quasi-cdg module over a cdg category ${\mathcal{A}}$ is
a strict quasi-cdg functor $F:{\mathcal{A}}\ (\text{resp.,
}{\mathcal{A}}^{op})\to\mathrm{Com}_{cdg}(k)$. A left (resp., right) cdg
module over ${\mathcal{A}}$ is a strict cdg functor $F:{\mathcal{A}}\
(\text{resp., }{\mathcal{A}}^{op})\to\mathrm{Com}_{cdg}(k)$. If
${\mathcal{A}}$ is a dg category and $F:{\mathcal{A}}\ (\text{resp.,
}{\mathcal{A}}^{op})\to Com_{dg}(k)$ is a dg functor, we call $F$ is a left
(resp., right) dg module over ${\mathcal{A}}$.

The category $q\mathrm{Mod}_{cdg}({\mathcal{A}})$ of right quasi-cdg modules
over ${\mathcal{A}}$ has a natural cdg structure: The Hom space
${\mathrm{Hom}}(F,G)$ of two quasi-cdg modules
$F,G:{\mathcal{A}}^{op}\to\mathrm{Com}_{cdg}(k)$ is the ${\mathbb{G}}$-graded
$k$-space of $k$-linear homogenous natural transformations
$\mathfrak{n}:F\Rightarrow G$. The differential $\delta$ of
${\mathrm{Hom}}(F,G)$ is given by
$(\delta\mathfrak{n})(x):=[d_{F(x),G(x)},\mathfrak{n}(x)]$ and the curvature
$h_{F}\in{\mathrm{Hom}}(F,F)$ of $F$ is given by $h_{F}(x)=h_{F(x)}-F(h_{x})$.
The full subcategory $\mathrm{Mod}_{cdg}({\mathcal{A}})$ of
$q\mathrm{Mod}_{cdg}({\mathcal{A}})$ consisting of right cdg modules over
${\mathcal{A}}$ has a natural dg structure. If ${\mathcal{A}}$ is a dg
category, we write $\mathrm{Mod}_{dg}({\mathcal{A}})$ for
$\mathrm{Mod}_{cdg}({\mathcal{A}})$.

####  2.1.7. Matrix factorizations

By unwinding the definition, a right quasi-cdg module $M$ over $(A,d_{A},-h)$
is a right graded $A$-module $M$ with a degree $1$, $k$-linear map
$\delta:M\to M$ such that $\delta(ma)=\delta(m)a+(-1)^{|m|}md_{A}(a)$ for
every $m\in M$, $a\in A$. Its curvature is defined to be
$\delta^{2}+\rho_{-h}\in\mathrm{End}_{A}(M)$ where $\rho_{-h}$ is the right
multiplication by $-h$. (1) For example, $(A,d_{A})$ can be regarded as a
right quasi-cdg module over $(A,d_{A},-h)$ with curvature $\lambda_{-h}$. (2)
Exactly when the curvature $\delta^{2}+\rho_{-h}$ of $M$ is zero, $M$ is
called a right cdg-module over $(A,d_{A},-h)$. Furthermore we call the right
cdg module $(M,\delta)$ over $(A,0,-h)$ a matrix factorization for $(A,h)$ or
for $({\mathrm{Spec}}A,h)$ when $A$ is a commutative $k$-algebra concentrated
in degree $0$, and $M$ is a finitely generated projective $A$-module. The full
subcategory of $\mathrm{Mod}_{cdg}(A,h)$ consisting of matrix factorizations
will be denoted by $D_{dg}(A,h)$ (and also by $D_{dg}({\mathrm{Spec}}A,h)$).

####  2.1.8. Quasi-Yoneda

There is a quasi-Yoneda embedding

${\mathcal{A}}\to q\mathrm{Mod}_{cdg}({\mathcal{A}}),\ x\mapsto{\mathcal{A}}(\
,x)$

generalizing the usual Yoneda embedding in the case of dg categories. We
define the full cdg-subcategory $q\mathrm{Perf}({\mathcal{A}})$ of
$q\mathrm{Mod}_{cdg}({\mathcal{A}})$ as the smallest cdg-subcategory
containing ${\mathcal{A}}$ and closed under finite operations of finite direct
sum, shift, twist, and passage to a direct summand; see [3, 20] for details.
We call an object of $q\mathrm{Perf}({\mathcal{A}})$ a perfect right
quasi-${\mathcal{A}}$-module. If its curvature vanishes, then call it a
perfect right ${\mathcal{A}}$-module. We define $\mathrm{Perf}({\mathcal{A}})$
as the full dg-subcategory of $\mathrm{Mod}_{cdg}({\mathcal{A}})$ consisting
of all perfect right ${\mathcal{A}}$-modules.

###  2.2. Mixed Hochschild complexes

####  2.2.1. The category of mixed complexes

Consider a ${\mathbb{G}}$-graded dg algebra $k[B]$ defined as a 2-dimensional
graded $k$-algebra generated by $B$ with relation $B^{2}=0$, degree $|B|=-1$,
and trivial differential. A dg $k[B]$-module is called a mixed complex. For
example, for a cdg category ${\mathcal{A}}$ we will have a mixed complex
$\mathrm{MC}({\mathcal{A}})$ as defined in (2.2) below. A morphism $\phi:M\to
M^{\prime}$ between mixed complexes is a dg $k[B]$-module homomorphism, i.e.,
a $k$-linear map preserving degree and both differentials. This defines the
category $\mathrm{Com}(k[B])$ of mixed complexes. We call $\phi$ a quasi-
isomorphism if it is so as a cochain map $(M,b)\to(M^{\prime},b^{\prime})$.

Let $u$ be a formal variable with degree $2$. Then we have the induced map
$\phi(u):(M\otimes_{k}k[[u]],b+uB)\to(M^{\prime}\otimes_{k}k[[u]],b^{\prime}+uB)$
in the category of complexes.

######  Lemma 2.1.

([12, Proposition 2.4]) If $\phi$ is a quasi-isomorphism, then so is
$\phi(u)$.

####  2.2.2. Hochschild complexes

For any cdg $k$-category ${\mathcal{A}}$, we may consider the Hochschild
complex $C({\mathcal{A}})$ of ${\mathcal{A}}$. Let ${\mathcal{A}}(x,y)[1]$
denote the degree shifted ${\mathcal{A}}(x,y)$ by $1$. The canonical degree
$-1$ map ${\mathcal{A}}(x,y)\to{\mathcal{A}}(x,y)[1]$ is denoted by
$\mathsf{s}$ so that $|\mathsf{s}a|=|a|-1$. Let

$C({\mathcal{A}})=\bigoplus_{x\in{\mathcal{A}}}{\mathcal{A}}(x,x)\oplus\\\
\bigoplus_{n\geq
1}\big{(}\bigoplus_{(x_{0},...,x_{n})\in{\mathcal{A}}^{\otimes
n+1}}{\mathcal{A}}(x_{1},x_{0})\otimes_{k}\underbrace{{\mathcal{A}}(x_{2},x_{1})[1]\otimes_{k}...\otimes_{k}{\mathcal{A}}(x_{0},x_{n})[1]}_{n}\big{)},$

with differential $b:=b_{2}+b_{1}+b_{0}$ defined as follows. Denote

(2.1)
$a_{0}[a_{1}|...|a_{n}]:=a_{0}\otimes\mathsf{s}a_{1}\otimes...\otimes\mathsf{s}a_{n}\\\
\in{\mathcal{A}}(x_{1},x_{0})\otimes{\mathcal{A}}(x_{2},x_{1})[1]\otimes...\otimes{\mathcal{A}}(x_{0},x_{n})[1].$

Then define

$\displaystyle b_{2}(a_{0}[a_{1}|...|a_{n}])$
$\displaystyle:=(-1)^{|a_{0}|}a_{0}a_{1}[a_{2}|...|a_{n}]+(-1)^{|a_{0}|+|a_{1}|-1}a_{0}[a_{1}a_{2}|a_{3}|...|a_{n}]$
$\displaystyle+\cdots+(-1)^{\sum_{i=0}^{n-1}|a_{i}|-(n-1)}a_{0}[a_{1}|...|a_{n-1}a_{n}]$
$\displaystyle+(-1)^{(|a_{n}|-1)(\sum_{i=0}^{n-1}|a_{i}|-(n-1))}a_{n}a_{0}[a_{1}|...|a_{n-1}];$
$\displaystyle b_{1}(a_{0}[a_{1}|...|a_{n}])$
$\displaystyle:=d(a_{0})[a_{1}|...|a_{n}]+(-1)^{|a_{0}|-1}a_{0}[d(a_{1})|a_{2}|...|a_{n}]$
$\displaystyle+\cdots+(-1)^{\sum_{i=0}^{n-1}|a_{i}|-n}a_{0}[a_{1}|...|a_{n-1}|d(a_{n})];$
$\displaystyle b_{0}(a_{0}[a_{1}|...|a_{n}])$
$\displaystyle:=(-1)^{|a_{0}|}a_{0}[h|a_{1}...|a_{n}]+\cdots+(-1)^{\sum_{i=0}^{n}|a_{i}|-n}a_{0}[a_{1}|...|a_{n}|h].$

The homology

$HH_{*}({\mathcal{A}})=H^{-*}(C({\mathcal{A}}),b)$

is called the Hochschild homology of ${\mathcal{A}}$.

When ${\mathcal{A}}$ is a curved dg algebra $(A,d,h)$, we write $C(A,d,h)$ for
$C({\mathcal{A}})$.

####  2.2.3. Connes’ boundary map

On the graded $k$-module $C({\mathcal{A}})$, there is another boundary map,
the Connes boundary map, $B=(1-t^{-1})sN$, where

$\displaystyle t(a_{0}[a_{1}|...|a_{n}])$
$\displaystyle:=(-1)^{(|a_{0}|-1)\sum_{i=1}^{n}(|a_{i}|-1)}a_{1}[a_{2}|...|a_{n}|a_{0}];$
$\displaystyle s(a_{0}[a_{1}|...|a_{n}])$
$\displaystyle:=1[a_{0}|a_{1}|...|a_{n}];$ $\displaystyle
N(a_{0}[a_{1}|...|a_{n}])$
$\displaystyle:=\sum_{i=0}^{n}t^{i}(a_{0}[a_{1}|...|a_{n}]).$

Consider

(2.2) $\mathrm{MC}({\mathcal{A}}):=(C({\mathcal{A}}),b,B).$

Since $bB+Bb=0$, $\mathrm{MC}({\mathcal{A}})$ is a mixed complex, called the
mixed Hochschild complex of ${\mathcal{A}}$.

It is known that $HH_{*}({\mathcal{A}})$ vanishes if the cdg category
${\mathcal{A}}$ contains an object with a nonzero curvature; see [5, 20]. This
motivates the following.

####  2.2.4. Hochschild complexes of the second kind

For a cdg category ${\mathcal{A}}$ one can take the underlying graded
$k$-module to be

$C^{II}({\mathcal{A}})=\bigoplus_{x\in{\mathcal{A}}}{\mathcal{A}}(x,x)\oplus\\\
\prod_{n\geq 1}\Big{(}\bigoplus_{(x_{0},...,x_{n})\in{\mathcal{A}}^{\otimes
n+1}}{\mathcal{A}}(x_{1},x_{0})\otimes_{k}\underbrace{{\mathcal{A}}(x_{2},x_{1})[1]\otimes_{k}...\otimes_{k}{\mathcal{A}}(x_{0},x_{n})[1]}_{n}\Big{)},$

which has the corresponding differentials $b_{i}$, $i=0,1,2$. This complex is
called the Hochschild complex of the second kind.

####  2.2.5. Normalized Hochschild complexes

Consider a subcomplex $D$ of $C({\mathcal{A}})$ generated by elements
$a_{0}[a_{1}|...|a_{n}]$ for which $a_{i}=t\cdot\mathrm{id}_{x}$ for
$x\in{\mathcal{A}}$, $t\in k$ and some $i\geq 1$. Define the normalized
Hochschild complex

$\overline{C}({\mathcal{A}})=(C({\mathcal{A}}),b)/D=:\bigoplus_{n\geq
0}\overline{C}_{n}({\mathcal{A}}),$

where $n$ denotes the tensor degree as in (2.1). The Connes operator descends
to an operator on $\overline{C}({\mathcal{A}})$, which will be also denoted by
$B$ by the abuse of notation. We let
$\overline{\mathrm{MC}}({\mathcal{A}}):=(\overline{C}({\mathcal{A}}),b,B)$.

######  Proposition 2.2.

([16, §5.3] and [9, §3].)  For a dg category ${\mathcal{A}}$ the natural map

(2.3)
$\mathrm{MC}({\mathcal{A}})\xrightarrow{\sim}\overline{\mathrm{MC}}({\mathcal{A}})$

is a quasi-isomorphism.

Likewise, there is the normalized Hochschild complex
$\overline{C}^{II}({\mathcal{A}})$ of the second kind. There are corresponding
Connes’ operators $B$ and mixed complexes
$\mathrm{MC}^{II}({\mathcal{A}}):=(C^{II}({\mathcal{A}}),b,B)$,
$\overline{\mathrm{MC}}^{II}({\mathcal{A}}):=(\overline{C}^{II}({\mathcal{A}}),b,B)$.

######  Proposition 2.3.

([3, Proposition 3.15])  For a cdg category ${\mathcal{A}}$ the quotient map

(2.4)
$quot^{II}:\mathrm{MC}^{II}({\mathcal{A}})\xrightarrow{\sim}\overline{\mathrm{MC}}^{II}({\mathcal{A}})$

is a quasi-isomorphism.

####  2.2.6. Negative cyclic complexes

Let $u$ be a formal variable with degree $2$. Then we can make a total complex

$(C({\mathcal{A}})[[u]]:=C({\mathcal{A}})\otimes_{k}k[[u]],b+uB)$

It is called the negative cyclic complex of ${\mathcal{A}}$ and its homology

$HN_{*}({\mathcal{A}})=H^{-*}(C({\mathcal{A}})[[u]],b+uB)$

is called the negative cyclic homology of ${\mathcal{A}}$. Similarly, one may
define its variants

$\overline{HN}_{*}({\mathcal{A}}),HN_{*}^{II}({\mathcal{A}}),\overline{HN}^{II}_{*}({\mathcal{A}}).$

####  2.2.7. Functoriality

Let $\text{cdg-cat}_{k}$ denote the category of small cdg $k$-categories. The
morphisms between them are cdg functors. We define a functor

$\text{cdg-
cat}_{k}\xrightarrow{\overline{\mathrm{MC}}^{II}}\mathrm{Com}(k[B]);{\mathcal{A}},(F,{\alpha})\mapsto(\overline{C}^{II}({\mathcal{A}}),b,B),(F,{\alpha})_{*}$

by letting

$(F,{\alpha})_{*}(a_{0}[a_{1}|...|a_{n}]):=\sum_{n=0}^{\infty}\sum_{(j_{0},...,j_{n})\in{\mathbb{Z}}_{\geq
0}^{n+1}}(-1)^{j_{0}+...+j_{n}}\\\
F(a_{0})[\underbrace{{\alpha}_{0}|...|{\alpha}_{0}}_{j_{0}}|F(a_{1})|\underbrace{{\alpha}_{1}|...|{\alpha}_{1}}_{j_{1}}|F(a_{2})|...|F(a_{n})|\underbrace{{\alpha}_{n}|...|{\alpha}_{n}}_{j_{n}}].$

Here ${\alpha}_{i}:={\alpha}_{\text{domain of }F(a_{i})}$. This is indeed a
functor; see [3, 20].

######  Remark 2.4.

For a cdg functor $(F,{\alpha})$, it induces a cochain map
$(C^{II}({\mathcal{A}}),b)\to(\overline{C}^{II}({\mathcal{A}}^{\prime}),b^{\prime})$
but not necessarily a morphism of mixed complexes
$\mathrm{MC}^{II}({\mathcal{A}})\to\mathrm{MC}^{II}({\mathcal{A}}^{\prime})$;
see [3, Remark 3.21].

Let $\text{cdg-cat}^{st}_{k}$ denote the category of small cdg $k$-categories
whose morphisms are strict cdg functors. Let $\text{dg-cat}_{k}$ be the full
subcategory of $\text{cdg-cat}^{st}_{k}$ whose objects are dg categories. In
the same way we have natural functors

$\displaystyle\text{cdg-cat}^{st}_{k}$
$\displaystyle\xrightarrow{\overline{\mathrm{MC}}}\mathrm{Com}(k[B]);{\mathcal{A}},F\mapsto(\overline{C}({\mathcal{A}}),b,B),F_{*}$
$\displaystyle\text{dg-cat}_{k}$
$\displaystyle\xrightarrow{\mathrm{MC}}\mathrm{Com}(k[B]);{\mathcal{A}},F\mapsto(C({\mathcal{A}}),b,B),F_{*}$

where $F_{*}:=(F,0)_{*}$.

###  2.3. Some invariances

####  2.3.1. Morita invariances

Let $F:{\mathcal{A}}\to\mathcal{B}$ be a dg functor between dg categories
${\mathcal{A}}$, $\mathcal{B}$. We say that it is cohomologically full and
faithful if for every $x,y\in{\mathcal{A}}$, the cochain map
$F:{\mathcal{A}}(x,y)\to\mathcal{B}(Fx,Fy)$ is a quasi-isomorphism.
Furthermore, if $\mathcal{B}$ is split-generated by the image of $F$, then $F$
induces a Morita equivalence between ${\mathcal{A}}$ and $\mathcal{B}$. Thus
the morphism

(2.5)
$F_{*}:\mathrm{MC}({\mathcal{A}})\xrightarrow{\sim}\mathrm{MC}(\mathcal{B})$

is a quasi-isomorphism; see [23, Lemma 4.12] and references therein. The
Yoneda embedding ${\mathcal{A}}\to\mathrm{Perf}{\mathcal{A}}$ is such an
example. Furthermore if $F$ is essentially surjective, $F$ is called a quasi-
equivalence.

####  2.3.2. Invariance under Pseudo-equivalences

Let $(F,{\alpha}):{\mathcal{A}}\to\mathcal{B}$ be a cdg functor between cdg
categories ${\mathcal{A}}$, $\mathcal{B}$. We say that $(F,{\alpha})$ is a
pseudo-equivalence if ${\mathcal{A}}(x,y)\to\mathcal{B}(Fx,Fy)$ is an
isomorphism for all $x,y\in{\mathcal{A}}$ and every object of $\mathcal{B}$
can be constructed from the image of $F$ by a finite sequence of operations of
finite direct sum, passage to a direct summand, twist, and shift.

######  Theorem 2.5.

(Polishchuk-Positselski [20]) For a pseudo-equivalence $(F,{\alpha})$

(2.6)
$(F,{\alpha})_{*}:\overline{\mathrm{MC}}^{II}({\mathcal{A}})\xrightarrow{\sim}\overline{\mathrm{MC}}^{II}(\mathcal{B})$

is a quasi-isomorphism.

The embeddings ${\mathcal{A}}\to q\mathrm{Perf}{\mathcal{A}}$ and
$\mathrm{Perf}{\mathcal{A}}\to q\mathrm{Perf}{\mathcal{A}}$ are pseudo-
equivalences; see [20].

####  2.3.3. Comparison

Let $X$ be an affine $k$-scheme and let $w:X\to\mathbb{A}^{1}_{k}$ be a
regular function. Then we consider the dg category $D_{dg}(X,w)$ of matrix
factorizations for $(X,w)$; see § 2.1.7.

######  Proposition 2.6.

([20, §4.8 Corollary A])  If $X$ is a smooth affine scheme finite type over
$k$ and $w$ has no other critical values but zero, then the natural map

(2.7) $C(D_{dg}(X,w))\xrightarrow{\sim}C^{II}(D_{dg}(X,w))$

is a quasi-isomorphism.

###### Proof.

Without loss of generality we may assume that $X$ is connected. When $w\neq
0$, this follows from [20, Corollary A], since $k$ is a perfect field. When
$w=0$, this follows from considering the tensor grading of
$C^{II}(\Gamma(X,{\mathcal{O}}_{X}))$ and the HKR isomorphism of [10]. ∎

##  3\. Categorical Chern characters

In this section we recall the definition of categorical Chern characters and
discuss alternative expressions of them which we will use in § 6.3 for Chern
character formulae.

###  3.1. Chern characters $\mathrm{Ch}_{HH}(P)$ and $\mathrm{Ch}_{HN}(P)$

For every object $P$ of a dg category ${\mathcal{A}}$, note that the identity
map $1_{P}$ is a degree $0$ cocycle of the Hochschild complex of
${\mathcal{A}}$. Hence each $P$ determines the class $[1_{P}]$ in
$HH_{0}({\mathcal{A}})$. We call $[1_{P}]$ the Hochschild homology valued
Chern character of $P$, denoted by $\mathrm{Ch}_{HH}(P)$. Likewise, since
$1_{P}$ is a cocycle of the normalized negative cyclic complex of
${\mathcal{A}}$, we simply define the negative cyclic homology valued Chern
character $\mathrm{Ch}_{HN}(P)$ of $P$ to be the class $[1_{P}]$ in
$\overline{HN}_{0}({\mathcal{A}})\cong HN_{0}({\mathcal{A}})$.

For a cdg category ${\mathcal{A}}$ and $P\in q\mathrm{Perf}{\mathcal{A}}$ we
define Chern characters $\mathrm{Ch}^{II}_{HH}(P)$, $\mathrm{Ch}^{II}_{HN}(P)$
of the second kind in the same manner using $1_{P}$ in the normalized
complexes of the second kind.

###  3.2. Alternative expression via direct summands

Let $P$ and $N$ be objects of a dg category ${\mathcal{A}}$. Suppose that $P$
is a direct summand of $N$, i.e., there are $g:P\to N$ and $f:N\to P$ degree
$0$ closed homomorphisms such that $f\circ g=1_{P}$. Note that $1_{P}$ and the
idempotent $\pi:=g\circ f$ are homologous

(3.1) $1_{P}\sim\pi$

in the Hochschild complex of ${\mathcal{A}}$; see [24]. If $N$ is simpler than
$P$, then $\pi$ is often easier to handle than $1_{P}$.

#### 3.2.1.

More generally, every element of $C(\mathrm{End}_{{\mathcal{A}}}P)$ is
homologous to an element of $C(\mathrm{End}_{{\mathcal{A}}}N)$ in the
Hochschild complex of ${\mathcal{A}}$: Consider the inclusion functor $inc$
from $\\{N\\}$ to $\\{P,N\\}$ between the full dg subcategories of
${\mathcal{A}}$ consisting only one indicated object, two indicated objects
respectively. As a semifunctor (i.e., a dg functor not necessarily preserving
identities between dg categories), $inc$ has a left inverse $F$.

$\\{N\\}\xleftarrow{F}\\{P,N\\}\xleftarrow{inc}\\{N\\}.$

The functor $F:\\{P,N\\}\to\\{N\\}$ is determined by sending ${\alpha}\mapsto
g\circ{\alpha}\circ f$ for ${\alpha}\in\mathrm{End}P$,
${\alpha}^{\prime}\mapsto g\circ{\alpha}^{\prime}$ for
${\alpha}^{\prime}\in{\mathrm{Hom}}(N,P)$,
${\alpha}^{\prime\prime}\mapsto{\alpha}^{\prime\prime}\circ f$ for
${\alpha}^{\prime\prime}\in{\mathrm{Hom}}(P,N)$, and $\beta\mapsto\beta$ for
$\beta\in\mathrm{End}(N)$.

######  Lemma 3.1.

In the Hochschild complex $(C\\{P,N\\},b)$, every element
${\alpha}_{0}[{\alpha}_{1}|....|{\alpha}_{n}]$ is homologous to
$F({\alpha}_{0})[F({\alpha}_{1})|...|F({\alpha}_{n})]$.

###### Proof.

The semifunctor $F$ induces a cochain map
$F_{*}:(C\\{P,N\\},b)\to(C\\{N\\},b)$, which is a left inverse of the quasi-
isomorphism $inc_{*}:(C\\{N\\},b)\to(C\\{P,N\\},b)$ induced from $inc$.
Therefore for every $m\in C\\{P,N\\}$, $m$ is homologous to
$inc_{*}(F_{*}(m))$, since $F_{*}(m)=F_{*}(inc_{*}(F_{*}(m)))$. ∎

######  Remark 3.2.

We can generalize Lemma 3.1: Consider two full dg subcategories
${\mathcal{C}}$ and $\mathcal{D}$ of $\mathrm{Mod}_{cdg}{\mathcal{A}}$ such
that ${\mathcal{C}}$ is a subcategory of $\mathcal{D}$ and any object of
$\mathcal{D}$ is a direct summand of an object of ${\mathcal{C}}$. Then we can
construct a semifunctor from $\mathcal{D}\to{\mathcal{C}}$ which is a left
inverse of the inclusion ${\mathcal{C}}\to\mathcal{D}$ and hence a left
inverse of the quasi-isomorphism $(C({\mathcal{C}}),b)\to(C(\mathcal{D}),b)$.

#### 3.2.2.

There is a negative cyclic homology version of (3.1): Consider a cycle

$\gamma_{P}:=1_{P}+\sum_{i=1}^{\infty}(-1)^{i}\frac{(2i)!}{2(i!)}2\cdot
1_{P}[\underbrace{1_{P}|\cdots|1_{P}}_{2i}]u^{i}$

in $(C\\{P\\}[[u]],b+uB)$. It projects to $1_{P}$ in
$(\overline{C}\\{P\\}[[u]],b+uB)$. Let

(3.2) $\displaystyle\eta_{\pi}$
$\displaystyle:=\pi+\sum_{i=1}^{\infty}(-1)^{i}\frac{(2i)!}{2(i!)}(2\pi-1_{N})[\underbrace{\pi|\cdots|\pi}_{2i}]u^{i}$
$\displaystyle=\pi-(2\pi-1_{N})[\pi|\pi]u+6(2\pi-1_{N})[\pi|\pi|\pi|\pi]u^{2}+\cdots.$

It is straightforward to check that $\eta_{\pi}$ is also a cycle in
$(\overline{C}\\{N\\}[[u]],b+uB)$.

######  Proposition 3.3.

(Proposition A.1)The two cycles $1_{P}$ and $\eta_{\pi}$ are homologous in
$(\overline{C}\\{P,N\\}[[u]],b+uB)$.

Proposition 3.3 might be known to the experts. Due to lack of a suitable
reference, we will give a proof in Appendix A. Another proof will be given in
[6].

######  Remark 3.4.

A combination of Proposition 3.3 and [3, Theorem 5.7] immediately provides an
answer to a question in [3, Remark 5.23].

##  4\. Global matrix factorizations

Let $X$ be a smooth separated scheme of finite type over $k$. In this section
we introduce the notion of global matrix factorizations as certain cdg modules
over the sheaf $({\mathcal{O}}_{X},0,-w)$ of cdg algebras. And we define the
Čech model for a dg enhancement of the derived category of global matrix
factorizations. We introduce the sheafifications of mixed Hochschild complexes
that are defined in § 2. All of these will be used in § 5.

###  4.1. Matrix factorizations and injective models

We may sheafify the notion of cdg algebras and (quasi-)cdg modules. For
example, we have

(4.1) $\mathscr{A}_{w}:=({\mathcal{O}}_{X},0,-w)$

a sheaf of cdg $k$-algebras: $U\mapsto({\mathcal{O}}_{X}(U),0,-w|_{U})$ for
open subsets $U$ of $X$. A global matrix factorization, or simply matrix
factorization for $(X,w)$ amounts to a cdg $\mathscr{A}_{w}$-module
$(P,\delta_{P})$ such that $P$ is a locally free and coherent as an
${\mathcal{O}}_{X}$-module. There is a notion of the derived category of
matrix factorizations for $(X,w)$; see [17]. It is equivalent to the homotopy
category of $D_{dg}(X,w)$ if we take $D_{dg}(X,w)$ to be the dg category of
matrix factorizations whose objects are matrix factorizations and whose Hom
from $P$ to $Q$ is defined to be
${\mathrm{Hom}}_{{\mathcal{O}}_{X}}(I_{P},I_{Q})$ using quasi-coherent curved
injective replacements $I_{P}$, $I_{Q}$ of $P$, $Q$, respectively.

###  4.2. Čech models

We will consider another model using Čech resolutions of matrix factorizations
instead of injective resolutions. We fix a finite open affine covering
$\mathfrak{U}=\\{U_{i}\\}_{i\in I}$ of $X$. Fix a total ordering of $I$. For
an ${\mathcal{O}}_{X}$-sheaf ${\mathcal{F}}$, let
$\mathrm{\check{C}}({\mathcal{F}})$ the sheafified version of the (ordered)
Čech complex of ${\mathcal{F}}$ with respect to the covering $\mathfrak{U}$:

(4.2)
$\mathrm{\check{C}}^{p}({\mathcal{F}}):=\prod_{i_{0}<...<i_{p}}f_{*}({\mathcal{F}}|_{U_{i_{0},...,i_{p}}})={\mathcal{F}}\otimes_{{\mathcal{O}}_{X}}\prod_{i_{0}<...<i_{p}}f_{*}{\mathcal{O}}_{U_{i_{0},...,i_{p}}}$

where $f$ denotes the immersions $U_{i_{0},...,i_{p}}:=U_{i_{0}}\cap...\cap
U_{i_{p}}\to X$. Here the second equality follows from the projection formula.
The Čech differential $1_{\mathcal{F}}\otimes
d_{\scriptscriptstyle\text{\v{C}ech}}$ for

(4.3)
$\text{\rm\v{C}}({\mathcal{F}})={\mathcal{F}}\otimes\text{\rm\v{C}}({\mathcal{O}}_{X})$

will be written simply $d_{\scriptscriptstyle\text{\v{C}ech}}$ by abuse of
notation. We follow the Koszul sign rule so that
$d_{\scriptscriptstyle\text{\v{C}ech}}(x\otimes a)=(-1)^{|x|}x\otimes
d_{\scriptscriptstyle\text{\v{C}ech}}(a)$ for $x\otimes
a\in{\mathcal{F}}\otimes\text{\rm\v{C}}({\mathcal{O}}_{X})$.

####  4.2.1.  Alexander-Čech-Whitney products

The Alexander-Čech-Whitney product (see [26]) is an
${\mathcal{O}}_{X}$-homomorphism

$\cdot:\text{\rm\v{C}}({\mathcal{O}}_{X})\otimes_{{\mathcal{O}}_{X}}\text{\rm\v{C}}({\mathcal{O}}_{X})\to\text{\rm\v{C}}({\mathcal{O}}_{X})$

defined by setting

(4.4) $\displaystyle(a\cdot b)_{i_{0},...,i_{p+q}}$
$\displaystyle:=a_{i_{0},...,i_{p}}|_{U_{i_{0},...,i_{p+q}}}b_{i_{p},...,i_{p+q}}|_{U_{i_{0},...,i_{p+q}}}$

for
$a\in\text{\rm\v{C}}^{p}({\mathcal{O}}_{X}),b\in\text{\rm\v{C}}^{q}({\mathcal{O}}_{X})$.

If ${\mathcal{F}}$ is a sheaf of graded ${\mathcal{O}}_{X}$-modules, we have
the right $\mathrm{\check{C}}({\mathcal{O}}_{X})$-module structure on the Čech
resolution
$\text{\rm\v{C}}({\mathcal{F}})={\mathcal{F}}\otimes\text{\rm\v{C}}({\mathcal{O}}_{X})$.
If ${\mathcal{F}}$ is a sheaf of graded ${\mathcal{O}}_{X}$-algebras, we
define the product structure on
$\text{\rm\v{C}}({\mathcal{F}})={\mathcal{F}}\otimes\text{\rm\v{C}}({\mathcal{O}}_{X})$
by the formula

(4.5) $\displaystyle(x\otimes a)\cdot(y\otimes b):=(-1)^{p|y|}xy\otimes a\cdot
b$

for $x,y\in{\mathcal{F}}$
$a\in\text{\rm\v{C}}^{p}({\mathcal{O}}_{X}),b\in\text{\rm\v{C}}^{q}({\mathcal{O}}_{X})$,
and the degree $|y|$ of $y$.

Note that
$d_{\scriptscriptstyle\text{\v{C}ech}}(\alpha\cdot\beta)=(d_{\scriptscriptstyle\text{\v{C}ech}}\alpha)\cdot\beta+(-1)^{|\alpha|}\alpha\cdot(d_{\scriptscriptstyle\text{\v{C}ech}}\beta)$
for $\alpha,\beta\in\text{\rm\v{C}}({\mathcal{F}})$. Therefore

(4.6)
$\mathscr{A}_{\mathrm{\check{C}},w}:=(\mathrm{\check{C}}({\mathcal{O}}_{X}),d_{\scriptscriptstyle\text{\v{C}ech}},-w)$

is a sheaf of cdg $k$-algebras on $X$ as
$[-w,]=d_{\scriptscriptstyle\text{\v{C}ech}}^{2}$.

####  4.2.2. The category $D_{\mathrm{\check{C}}}(X,w)$

For each $P\in D_{dg}(X,w)$, consider $\mathrm{\check{C}}(P)$. It comes with
the total curved differential $\delta_{P}\otimes 1+1\otimes
d_{\scriptscriptstyle\text{\v{C}ech}}$ by the identification
$\mathrm{\check{C}}(P)=P\otimes_{{\mathcal{O}}_{X}}\mathrm{\check{C}}({\mathcal{O}}_{X})$.
Abusing notation we simply write

$\delta_{P}+d_{\scriptscriptstyle\text{\v{C}ech}}\ \text{ for }\
\delta_{P}\otimes 1+1\otimes d_{\scriptscriptstyle\text{\v{C}ech}},$

but we need to remember the Koszul sign rule when we apply
$d_{\scriptscriptstyle\text{\v{C}ech}}$. Hence $\mathrm{\check{C}}(P)$ can be
regarded as a certain cdg-module over $\mathscr{A}_{\mathrm{\check{C}},w}$. In
the dg category $\mathrm{Mod}_{cdg}(\mathscr{A}_{\mathrm{\check{C}},w})$ of
right cdg-modules over $\mathscr{A}_{\mathrm{\check{C}},w}$, let us take all
the objects of form
$(\mathrm{\check{C}}(P),\delta_{P}+d_{\scriptscriptstyle\text{\v{C}ech}})$,
for some $P\in D_{dg}(X,w)$. This yields the full subcategory

$D_{\mathrm{\check{C}}}(X,w),$

which is easily seen to be quasi-equivalent to $D_{dg}(X,w)$. We call
$D_{\mathrm{\check{C}}}(X,w)$ the Čech model for $D_{dg}(X,w)$ with respect to
$\mathfrak{U}$. We may regard the objects of $D_{dg}(X,w)$ as the objects of
$D_{\mathrm{\check{C}}}(X,w)$.

####  4.2.3. The category $qD_{\mathrm{\check{C}}}(X,w)$

We will also consider a full subcategory $qD_{\mathrm{\check{C}}}(X,w)$ of
$q\mathrm{Mod}_{cdg}(\mathscr{A}_{\mathrm{\check{C}},w})$ of right quasi-cdg-
modules over $\mathscr{A}_{\mathrm{\check{C}},w}$, consisting of
$(\mathrm{\check{C}}(P),\delta_{P}+d_{\scriptscriptstyle\text{\v{C}ech}})$
such that $(P,\delta_{P})$ is a locally free coherent right quasi-cdg module
over $\mathscr{A}_{w}$. Here a locally free coherent right quasi-cdg module
$(P,\delta_{P})$ means that $P$ is a ${\mathbb{G}}$-graded locally free
coherent ${\mathcal{O}}_{X}$-module and $\delta_{P}$ is an
${\mathcal{O}}_{X}$-linear, degree $1$ map. It is not required that
$\delta_{P}^{2}=-\rho_{-w}$. The curvature element of $(P,\delta_{P})$ is
defined to be $\delta_{P}^{2}+\rho_{-w}$.

###  4.3. Sheafification

#### 4.3.1.

Let $\mathscr{P}$ be a presheaf of cdg categories on $X$:

$U\mapsto\mathscr{P}(U)\in\mathrm{cdg}_{k}$

for each open subset $U$ of $X$. For example, $\mathscr{A}_{w}$ and
$\mathscr{A}_{\mathrm{\check{C}},w}$ are sheaves of cdg $k$-algebras on $X$,
that is $U\mapsto({\mathcal{O}}(U),0,-w|_{U})$ and
$U\mapsto(\mathrm{\check{C}}({\mathcal{O}}_{X})(U),d_{\scriptscriptstyle\text{\v{C}ech}},-w|_{U})$,
respectively. Also, we have a presheaf

$U\mapsto D_{\mathrm{\check{C}}}(U,w|_{U})$

(resp., $qD_{\mathrm{\check{C}}}(U,w|_{U})$) of dg (resp., cdg) categories,
which will be denoted by $\underline{D}_{\mathrm{\check{C}}}(X,w)$ (resp.,
$q\underline{D}_{\mathrm{\check{C}}}(X,w)$). Here for each $U$ we use the
induced covering $\\{U_{i}\cap U\\}_{i\in I}$ of $U$ to construct
$\mathrm{\check{C}}(Q)$ for $Q\in(q)D_{dg}(U,w|_{U})$. Note that for
$P\in(q)D_{dg}(X,w)$,
$\mathrm{\check{C}}(P)|_{U}\cong\mathrm{\check{C}}(P|_{U})$.

#### 4.3.2.

We may sheafify the mixed Hochschild complex of $\mathscr{P}(U)$: the sheaf
$\underline{C}(\mathscr{P})$ associated to the presheaf

$U\mapsto C(\mathscr{P}(U)).$

Let $\underline{\mathrm{MC}}(\mathscr{P})$ denote this complex of sheaves,
i.e., $\underline{C}(\mathscr{P})$ with the induced differential and Connes’
operator, denoted by $b$, $B$ by abuse of notation. Similarly we have the
complex $\underline{\overline{\mathrm{MC}}}^{II}(\mathscr{P})$ of sheaves
associated to $U\mapsto\overline{C}^{II}(\mathscr{P}(U))$. For example, a
sheafification of complexes $U\mapsto C(D_{\mathrm{\check{C}}}(U,w|_{U}))$ is
denoted by $\underline{C}(\underline{D}_{\mathrm{\check{C}}}(X,w))$.

#### 4.3.3.

Let $G$ denote the Godement resolution functor so that for a sheaf
${\mathcal{F}}$ of abelian groups, the canonical flasque resolution of
${\mathcal{F}}$ is denoted by $G({\mathcal{F}})$. Let $G_{\leq d}$ denote the
Godement resolution functor canonically truncated at the amplitude $d$ with
$d:=\dim X$. Since $H^{i}(X,{\mathcal{F}})=0$ for all $i>d$ by a vanishing
theorem of Grothendieck ([8, III. Theorem 2.7]), $G_{\leq d}({\mathcal{F}})$
is a $\Gamma$-acyclic resolution of ${\mathcal{F}}$. For a sheaf
$(\underline{C},b,B)$ of mixed (unbounded) complexes on $X$ we define a mixed
complex

(4.7) ${\mathbb{R}}\Gamma(\underline{C},b,B):=(\Gamma(X,G_{\leq
d}\underline{C}),G_{\leq d}b,G_{\leq d}B)$

following [7, Definition 3.22]. The functor ${\mathbb{R}}\Gamma$ preserves
quasi-isomorphisms.

Condition $(\star)$: $X$ is a smooth separated scheme finite type over $k$ and
$w$ has no other critical values but zero.

######  Proposition 4.1.

([7, Proposition 5.1 & Proposition 3.24])

  1. (1)

Assume $(\star)$ and let $C^{\prime}$ denote either $C,\overline{C}$ $C^{II}$
or $\overline{C}^{II}$. The natural morphism

$(C^{\prime}(D_{\mathrm{\check{C}}}(X,w)),b,B)\to{\mathbb{R}}\Gamma(\underline{C}^{\prime},b,B)$

is a quasi-isomorphism between the mixed complexes.

  2. (2)

For a sheaf ${\mathcal{F}}$ of $k$-vector spaces the natural morphism

$G_{\leq d}({\mathcal{F}}\otimes k[[u]])\to(G_{\leq d}{\mathcal{F}})\otimes
k[[u]]$

is a quasi-isomorphism between the complexes of $\Gamma$-acyclic sheaves.

###### Proof.

(1) The proof of Proposition 5.1 of Efimov [7] works for the various
$C^{\prime}$. (2) This is the contents of [7, Proposition 3.24 & Lemma 3.25].
∎

Combining Proposition 2.6 and Proposition 4.1 (1) we may remove the affine
condition on $X$ in Proposition 2.6.

######  Corollary 4.2.

Under the condition $(\star)$ the natural map

(4.8)
$C(D_{\mathrm{\check{C}}}(X,w))\xrightarrow{\sim}C^{II}(D_{\mathrm{\check{C}}}(X,w))$

is a quasi-isomorphism.

From now on we will assume the condition $(\star)$ unless otherwise stated.

###  4.4. Applications of invariances

Note that for each open affine subset $U$ of $X$, we have the following.

  1. (1)

The HKR map

$\displaystyle
I_{\scriptscriptstyle\text{HKR}}:\overline{\mathrm{MC}}^{II}(\mathscr{A}_{w}(U))$
$\displaystyle\to(\Omega^{\bullet}_{U/k},d,-dw);$ $\displaystyle
a_{0}[a_{1}|...|a_{n}]$ $\displaystyle\mapsto\frac{1}{n!}a_{0}da_{1}...da_{n}$

is a quasi-isomorphism of mixed complexes; see [5, 21].

  2. (2)

The natural morphism

$\overline{\mathrm{MC}}^{II}(\mathscr{A}_{w}(U))\to\overline{\mathrm{MC}}^{II}(\mathscr{A}_{\mathrm{\check{C}},w}(U))$

is a quasi-isomorphism by the spectral sequence argument with the filtration
of the complex
$(\overline{C}^{II}(\mathrm{\check{C}}({\mathcal{O}}_{X})(U)),b)$ by the Čech
grading.

  3. (3)

The quasi-Yoneda embedding

$\mathscr{A}_{\mathrm{\check{C}},w}(U)\to qD_{\mathrm{\check{C}}}(U,w|_{U})$

is a pseudo-equivalence and hence induces a quasi-isomorphism in the mixed
normalized Hochschild complexes of the second kind by (2.6).

  4. (4)

The natural embedding

$D_{\mathrm{\check{C}}}(U,w|_{U})\to qD_{\mathrm{\check{C}}}(U,w|_{U})$

is a pseudo-equivalence and hence induces a quasi-isomorphism in the mixed
normalized Hochschild complexes of the second kind again by (2.6).

Thus we have quasi-isomorphisms between sheaves of mixed complexes

(4.9)
$\displaystyle\underline{\overline{\mathrm{MC}}}^{II}(\underline{D}_{\mathrm{\check{C}}}(X,w))$
$\displaystyle\simeq\underline{\overline{\mathrm{MC}}}^{II}(q\underline{D}_{\mathrm{\check{C}}}(X,w))$
by (4)
$\displaystyle\simeq\underline{\overline{\mathrm{MC}}}^{II}(\mathscr{A}_{\mathrm{\check{C}},w})$
by (3) $\displaystyle\simeq(\Omega^{\bullet}_{X/k},-dw,d)$ by (2) & (1)
$\displaystyle\simeq(\text{\rm\v{C}}(\Omega^{\bullet}_{X/k}),-dw,d).$

##  5\. Connections and the map ${\mathrm{tr}}_{\nabla}$

###  5.1. $V$-Connections

Let $(A,d_{A},-h)$ be a curved dg $k$-algebra. Let $V$ be a subalgebra of the
$k$-algebra $A$ such that

$d_{A}|_{V}=0,\quad h\in V,$

and $V$ is contained in the even degree part of $A$. Furthermore assume that

$va=av\quad\text{ for every }v\in V,\ a\in A.$

Note that $d_{A}^{2}=[-h,?]=0$. Consider a $k$-linear map

$d:A\to\Omega^{1}_{V/k}\otimes_{V}A$

satisfying the Leibniz rule

(5.1) $d(a_{0}a_{1})=(da_{0})a_{1}+(-1)^{|a_{0}|}a_{0}da_{1}$

for $a_{0},a_{1}\in A$. Here if $da_{i}=dv_{i}\otimes a_{i}^{\prime}$ with
$v_{i}\in V,a_{i}^{\prime}\in A$, we define

$(da_{0})a_{1}:=dv_{0}\otimes a_{0}^{\prime}a_{1}\text{ and
}a_{0}da_{1}:=(-1)^{|a_{0}|}dv_{1}\otimes a_{0}a_{1}^{\prime}.$

On $\Omega^{1}_{V/k}\otimes_{V}A$ we define a differential (again denoted by)
$d_{A}$ by extension of scalars:

$d_{A}(dv\otimes a)=-dv\otimes d_{A}(a).$

Assume that for every $a\in A$

(5.2) $d_{A}d(a)+dd_{A}(a)=0.$

######  Example 1.

When $V=A$, then $d$ is the usual $k$-derivation of $V$.

######  Example 2.

From the de Rham differential ${\mathcal{O}}_{X}\to\Omega^{1}_{X/k}$ we have

(5.3)
$d:\mathrm{\check{C}}({\mathcal{O}}_{X})\to\Omega^{1}_{X/k}\otimes\mathrm{\check{C}}({\mathcal{O}}_{X})$

by the projection formula. Let $U$ be an affine open subset of $X$ and let

$V=\Gamma(U,{\mathcal{O}}_{X}),A=\Gamma(U,\mathrm{\check{C}}({\mathcal{O}}_{X})),d_{A}=d_{\scriptscriptstyle\text{\v{C}ech}},h=w.$

Then taking $\Gamma(U,-)$ at (5.3) (and slightly abusing notation) we obtain a
canonical $k$-derivation

(5.4)
$d:\Gamma(U,\mathrm{\check{C}}({\mathcal{O}}_{X}))\to\Omega^{1}_{U/k}\otimes_{\Gamma(U,{\mathcal{O}}_{X})}\Gamma(U,\mathrm{\check{C}}({\mathcal{O}}_{X}))$

satisfying (5.1) and (5.2).

######  Definition 5.1.

Let $M$ be a graded $A$-module. We call a $k$-linear degree $-1$ map

$\nabla:M\to\Omega^{1}_{V/k}\otimes_{V}M$

a $V$-connection with respect to $d$ above if

$\nabla(ma)=\nabla(m)a+(-1)^{|m|}m\otimes da$

for every $a\in A$. Here $m\otimes da\in
M\otimes_{A}(A\otimes_{V}\Omega^{1}_{V/k})\cong M\otimes_{V}\Omega^{1}_{V/k}$.

######  Example 3.

Note that if $\nabla^{\prime}$ is another $V$-connection on $M$, then
$\nabla-\nabla^{\prime}$ is an $A$-linear map
$M\to\Omega^{1}_{V/k}\otimes_{V}M$ as usual. If $M=A^{\oplus n}$, then $M$ has
the obvious connection $d_{F}$ induced from $d$. So a connection $\nabla$ on
$A^{\oplus n}$ can be written as $d_{F}+C$ for some $n\times n$ matrix $C$
whose entries are elements of $\Omega^{1}_{V}\otimes_{V}A$.

###  5.2. The map ${\mathrm{tr}}_{\nabla}$

Consider a perfect right quasi-cdg module $(P,\delta_{P})$ over $(V,0,-h)$.
Note that $\delta_{P}$ is $V$-linear. Let ${\mathcal{P}}:=P\otimes_{V}A$ and
note that $\mathrm{End}_{A}({\mathcal{P}})=\mathrm{End}_{V}(P)\otimes_{V}A$.
Then we have a cdg algebra

$(\mathrm{End}_{A}({\mathcal{P}}),[\delta_{{\mathcal{P}}},],\delta_{{\mathcal{P}}}^{2}+\rho_{-h}),$

where $\delta_{{\mathcal{P}}}:=\delta_{P}\otimes 1+1\otimes d_{A}$. Suppose
that $\delta_{{\mathcal{P}}}^{2}=\rho_{\nu}$ for some $\nu\in V$. Here
$\rho_{\nu}$ denotes the right multiplication by $\nu$.

Assume that a $V$-connection $\nabla$ on ${\mathcal{P}}$ is given and let

(5.5) $R:=u\nabla^{2}+[\nabla,\delta_{{\mathcal{P}}}]\
\in\Omega_{V}^{\bullet}\otimes_{V}\mathrm{End}_{A}({\mathcal{P}})[[u]].$

Here $\nabla^{2}$ denotes $(\mathrm{id}\otimes\nabla)\circ\nabla$ followed by
the wedge operation:

${\mathcal{P}}\xrightarrow{\nabla}\Omega^{1}_{V/k}\otimes_{V}{\mathcal{P}}\xrightarrow{\mathrm{id}\otimes\nabla}\Omega^{1}_{V/k}\otimes_{V}(\Omega^{1}_{V/k}\otimes_{V}{\mathcal{P}})\xrightarrow{\wedge\otimes
1}\Omega^{2}_{V/k}\otimes_{V}{\mathcal{P}}.$

The composition $\nabla^{2}$ is $A$-linear. The $A$-linearity of
$[\nabla,\delta_{{\mathcal{P}}}]$ requires (5.2). We regard
$\Omega^{\bullet}_{V/k}\otimes_{V}\mathrm{End}_{A}({\mathcal{P}})[[u]]$ as a
$k[[u]]$-algebra defined by

$(\gamma_{1}\otimes{\alpha}_{1})\cdot(\gamma_{2}\otimes{\alpha}_{2})=(-1)^{|{\alpha}_{1}||\gamma_{2}|}(\gamma_{1}\wedge\gamma_{2})\otimes({\alpha}_{1}\circ{\alpha}_{2})$

for $\gamma_{i}\in\Omega^{\bullet}_{V/k},\
{\alpha}_{i}\in\mathrm{End}_{A}({\mathcal{P}})$.

Define a $k[[u]]$-linear map ${\mathrm{tr}}_{\nabla}$ as the composition

(5.6) $\displaystyle
C(\mathrm{End}_{A}({\mathcal{P}}))[[u]]\to\Omega^{\bullet}_{V/k}\otimes_{V}\mathrm{End}_{A}({\mathcal{P}})[[u]]\xrightarrow{{\mathrm{tr}}}\Omega^{\bullet}_{V/k}[[u]]\otimes_{V}A,$

where the first map is defined by sending
${\alpha}_{0}[{\alpha}_{1}|...|{\alpha}_{n}]$ to

(5.7)
$\sum_{(j_{0},...,j_{n})}(-1)^{J}{\mathrm{tr}}\big{(}\frac{{\alpha}_{0}R^{j_{0}}[\nabla,{\alpha}_{1}]R^{j_{1}}[\nabla,{\alpha}_{2}]...R^{j_{n-1}}[\nabla,{\alpha}_{n}]R^{j_{n}}}{(n+J)!}\big{)},$

with $j_{i}\in{\mathbb{Z}}_{\geq 0}$ and $J=\sum_{i=0}^{n}j_{i}$; and the
second map ${\mathrm{tr}}$ is the $\Omega^{\bullet}_{V/k}[[u]]$-linear
extension the supertrace

(5.8)
$\mathrm{End}_{A}({\mathcal{P}})\underset{\text{canonically}}{\cong}({\mathrm{Hom}}_{A}({\mathcal{P}},A)\otimes_{A}{\mathcal{P}})\xrightarrow{\mathrm{ev}}A.$

Here the canonical isomorphism $\cong$ follows from the fact that
${\mathcal{P}}$ is a finitely generated projective $A$-module. Note that in
the sum (5.7), the terms for $j_{0}+...+j_{n}>\dim X$ vanish.

Similarly we may define the version of ${\mathrm{tr}}_{\nabla}$ for the
negative cyclic complex of the second kind as

(5.9)
${\mathrm{tr}}_{\nabla}^{II}:C^{II}(\mathrm{End}_{A}({\mathcal{P}}))[[u]]\to
A\otimes_{V}\Omega^{\bullet}_{V/k}[[u]]$

by the same formula (5.7).

######  Theorem 5.2.

(Theorem B.1) The map ${\mathrm{tr}}_{\nabla}$ satisfies

$(ud+d_{A}-dh)\circ{\mathrm{tr}}_{\nabla}={\mathrm{tr}}_{\nabla}\circ(uB+b_{2}+b_{1}+b_{0}).$

The same cochain map equality also holds for ${\mathrm{tr}}_{\nabla}^{II}$.

The proof of Theorem 5.2 is rather technical and will be given in Appendix B
closely following the proof of [3, Theorem 5.19].

###  5.3. Proof of Theorem 1.2

As in § 4.2 we fix a finite open affine covering
$\mathfrak{U}=\\{U_{i}\\}_{i\in I}$ of $X$ and fix a total ordering of $I$.

From Example 2 for each affine open subset $U$ of $X$ we consider the case:

$V=\Gamma(U,{\mathcal{O}}_{X}),A=\Gamma(U,\mathrm{\check{C}}({\mathcal{O}}_{X})),d_{A}=d_{\scriptscriptstyle\text{\v{C}ech}},h=w$

with the $V$-connection $d$ on $A$ from the de Rham differential of $V$.

For each $P\in qD_{\mathrm{\check{C}}}(X,w)$, we choose, once and for all, a
$k$-linear sheaf homomorphism

$\nabla_{P}:\mathrm{\check{C}}(P)\to\mathrm{\check{C}}(P)\otimes_{{\mathcal{O}}_{X}}\Omega^{1}_{X/k}$

such that $\nabla_{P}(U)$ is an ${\mathcal{O}}_{X}(U)$-connection with respect
to $d$. Such a $\nabla_{P}$ we call an ${\mathcal{O}}_{X}$-connection on
$\mathrm{\check{C}}(P)$ and will construct in § 5.4 as $\nabla_{E}$ in (5.24)
for $E=P$. The supertrace map (5.8) becomes

(5.10)
${\mathrm{tr}}:\Gamma(U,\Omega^{\bullet}_{X/k}\otimes{\mathcal{E}}nd_{{\mathcal{O}}_{X}}(P)\otimes\text{\rm\v{C}}({\mathcal{O}}_{X}))\xrightarrow{}\Gamma(U,\Omega^{\bullet}_{X/k}\otimes\text{\rm\v{C}}({\mathcal{O}}_{X})).$

Since any finite sum of objects of $qD_{\mathrm{\check{C}}}(U,w|_{U}))$ is
allowed, the definition of ${\mathrm{tr}}_{\nabla}^{II}$ in (5.9) can be
modified to give the map:

$\overline{{\mathrm{tr}}}^{II}_{\nabla,U}:\overline{C}^{II}(qD_{\mathrm{\check{C}}}(U,w|_{U}))[[u]]\to\Omega^{\bullet}_{X/k}(U)\otimes_{{\mathcal{O}}_{X}(U)}\mathrm{\check{C}}({\mathcal{O}}_{X})(U)[[u]].$

By Theorem 5.2, it is a cochain map. Hence its sheafified version

$\overline{\underline{{\mathrm{tr}}}}^{II}_{\nabla}:\overline{\underline{C}}^{II}(q\underline{D}_{\mathrm{\check{C}}}(X,w))[[u]]\to\Omega^{\bullet}_{X/k}\otimes_{{\mathcal{O}}_{X}}\mathrm{\check{C}}({\mathcal{O}}_{X})[[u]]$

is also a cochain map between the sheaves of complexes over $k[[u]]$.

Taking Lemma 2.1 into account, we see that
$\overline{\underline{{\mathrm{tr}}}}^{II}_{\nabla}$ fits into the following
commuting diagram of cochain maps between the sheaves of complexes over
$k[[u]]$:

(5.18)

######  Proposition 5.3.

The map $\overline{\underline{{\mathrm{tr}}}}^{II}_{\nabla}$ and the map

${\mathbb{R}}\Gamma(\overline{\underline{{\mathrm{tr}}}}^{II}_{\nabla}\circ
inc):{\mathbb{R}}\Gamma(\underline{\overline{C}}^{II}(D_{\mathrm{\check{C}}}(X,w)))[[u]]\to{\mathbb{R}}\Gamma(\text{\rm\v{C}}(\Omega^{\bullet}_{X/k}))[[u]]$

are quasi-isomorphisms.

###### Proof.

It follows from Proposition 4.1 (2) and the fact that in diagram
(LABEL:diag:_tr_Hoch) all the arrows but
$\overline{\underline{{\mathrm{tr}}}}^{II}_{\nabla}$ are shown to be quasi-
isomorphisms in § 4.4. ∎

We define a $k[[u]]$-linear cochain map

(5.19)
${\mathrm{tr}}_{\nabla}:C(D_{\mathrm{\check{C}}}(X,w))[[u]]\to\text{\rm\v{C}}(\mathfrak{U},\Omega^{\bullet}_{X/k})[[u]]$

by composing

(5.20) $\displaystyle C(D_{\mathrm{\check{C}}}(X,w))[[u]]$
$\displaystyle\xrightarrow{}\overline{C}^{II}(D_{\mathrm{\check{C}}}(X,w))[[u]]$
$\displaystyle\xrightarrow{natural}\Gamma(X,\overline{\underline{C}}^{II}(\underline{D}_{\text{\rm\v{C}}}(X,w)))[[u]]$
$\displaystyle\xrightarrow{\Gamma(\underline{\overline{{\mathrm{tr}}}}^{II}_{\nabla}\circ
inc)}\Gamma(X,\text{\rm\v{C}}(\Omega^{\bullet}_{X/k}))[[u]].$

######  Theorem 5.4.

(Theorem 1.2) The $k[[u]]$-linear map

(5.21) ${\mathrm{tr}}_{\nabla}:\ \
(C(D_{\mathrm{\check{C}}}(X,w))[[u]],b+uB)\\\
\to(\text{\rm\v{C}}(\mathfrak{U},\Omega^{\bullet}_{X/k})[[u]],d_{\scriptscriptstyle\text{\v{C}ech}}-dw+ud)$

is a quasi-isomorphism compatible with the HKR-type isomorphism.

###### Proof.

The proof follows from the following commuting diagram of cochain maps

(5.22)
$\textstyle{C(D_{\mathrm{\check{C}}}(X,w))[[u]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\scriptstyle{\eqref{eqn:
quot
I}}$$\scriptstyle{{\mathrm{tr}}_{\nabla}}$$\textstyle{\overline{C}(D_{\mathrm{\check{C}}}(X,w))[[u]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim\quad}$Prop.
(4.1)
$\textstyle{{\mathbb{R}}\Gamma(\underline{\overline{C}}(D_{\mathrm{\check{C}}}(X,w)))[[u]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\scriptstyle{\eqref{eqn:
BW 3.9}\&\eqref{eqn: I to
II}}$$\textstyle{\Gamma(\underline{\overline{C}}^{II}(D_{\mathrm{\check{C}}}(X,w)))[[u]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Gamma(\overline{\underline{{\mathrm{tr}}}}_{\nabla}^{II}\circ
inc)}$$\textstyle{{\mathbb{R}}\Gamma(\underline{\overline{C}}^{II}(D_{\mathrm{\check{C}}}(X,w)))[[u]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$Prop.5.3$\textstyle{\Gamma(\text{\rm\v{C}}(\Omega^{\bullet}_{X/k}))[[u]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{{\mathbb{R}}\Gamma(\text{\rm\v{C}}(\Omega^{\bullet}_{X/k}))[[u]]}$

where the arrows with $\sim$ are meant to be quasi-isomorphisms. ∎

For
${\alpha}_{j}\in\text{\rm\v{C}}(\mathfrak{U},{\mathcal{H}}om_{{\mathcal{O}}_{X}}(P_{j+1},P_{j}))$
with $P_{n+1}=P_{0}$, we note that
${\mathrm{tr}}_{\nabla}({\alpha}_{0}[{\alpha}_{1}|...|{\alpha}_{n}])$ is as in
formula (1.6)

$\sum_{(j_{0},...,j_{n})}(-1)^{J}{\mathrm{tr}}\big{(}\frac{{\alpha}_{0}R_{1}^{j_{0}}[\nabla,{\alpha}_{1}]R_{2}^{j_{1}}[\nabla,{\alpha}_{2}]...R_{n}^{j_{n-1}}[\nabla,{\alpha}_{n}]R_{0}^{j_{n}}}{(n+J)!}\big{)},$

where $j_{i}\in{\mathbb{Z}}_{\geq 0}$ and $J=\sum_{i=0}^{n}j_{i}$. From Remark
5.5 we see that

(5.23) $\displaystyle[\nabla,{\alpha}_{j}]$
$\displaystyle=\prod_{i_{0}<...<i_{p_{j}}}\big{(}\nabla_{P_{j},i_{0}}|_{U_{i_{0},...,i_{p_{j}}}}\circ({\alpha}_{j})_{i_{0},...,i_{p_{j}}}-$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
(-1)^{|{\alpha}_{j}|}({\alpha}_{j})_{i_{0},...,i_{p_{j}}}\circ\nabla_{P_{j+1},i_{0}}|_{U_{i_{0},...,i_{p_{j}}}}\big{)};$
$\displaystyle R_{j}$
$\displaystyle=\prod_{i}\left(u\nabla_{P_{j},i}^{2}+[\nabla_{P_{j},i},\delta_{P_{j}}|_{U_{i}}]\right)+\prod_{i_{0}<i_{1}}(\nabla_{P_{j},i_{0}}-\nabla_{P_{j},i_{1}}).$

###  5.4. Construction of ${\mathcal{O}}_{X}$-connections on
$\mathrm{\check{C}}(E)$

Let $E$ be a vector bundle on $X$. Choose a connection $\nabla_{i}$ of
$E|_{U_{i}}$ for $i\in I$:

$\nabla_{i}:E|_{U_{i}}\to\Omega^{1}_{U_{i}/k}\otimes_{{\mathcal{O}}_{U_{i}}}E|_{U_{i}},$

which exists, since $U_{i}$ is affine. We consider

(5.24) $\nabla_{E}:=\prod_{i\in
I}\nabla_{i}\in\text{\rm\v{C}}^{0}(\mathfrak{U},{\mathcal{H}}om_{k}(\text{\rm\v{C}}(E),\Omega^{1}_{X/k}\otimes\text{\rm\v{C}}(E)))$

a Čech element of a $k$-sheaf
${\mathcal{H}}om_{k}(\text{\rm\v{C}}(E),\Omega^{1}_{X/k}\otimes\text{\rm\v{C}}(E))$
on $X$. Note that $\nabla_{E}$ satisfies

$\nabla_{E}(sa)=\nabla_{E}(s)a+(-1)^{|s|}s\otimes da$

for $s\in\mathrm{\check{C}}(E)$, $a\in\mathrm{\check{C}}({\mathcal{O}}_{X})$.
Thus for each affine $U$ open subset of $X$, $\nabla_{E}(U)$ is an
${\mathcal{O}}_{X}(U)$-connection of the ${\mathcal{O}}_{X}(U)$-module
$\text{\rm\v{C}}(E)(U)$ with respect to $d$. We have constructed an
${\mathcal{O}}_{X}$-connection on $\mathrm{\check{C}}(E)$.

######  Remark 5.5.

Note that

$\displaystyle\nabla_{E}^{2}$
$\displaystyle\in{\mathrm{Hom}}_{\mathrm{\check{C}}({\mathcal{O}}_{X})}(\mathrm{\check{C}}(E),\Omega^{2}_{X/k}\otimes\text{\rm\v{C}}(E))={\mathrm{Hom}}_{{\mathcal{O}}_{X}}(E,\Omega^{2}_{X/k}\otimes\text{\rm\v{C}}(E));$
$\displaystyle[\nabla_{E},d_{\scriptscriptstyle\text{\v{C}ech}}]$
$\displaystyle\in{\mathrm{Hom}}_{\mathrm{\check{C}}({\mathcal{O}}_{X})}(\mathrm{\check{C}}(E),\Omega^{1}_{X/k}\otimes\text{\rm\v{C}}(E))={\mathrm{Hom}}_{{\mathcal{O}}_{X}}(E,\Omega^{1}_{X/k}\otimes\text{\rm\v{C}}(E)).$

In fact they are elements in
${\mathrm{Hom}}_{{\mathcal{O}}_{X}}(E,\Omega^{2}_{X/k}\otimes_{{\mathcal{O}}_{X}}\mathrm{\check{C}}^{0}(E))$
and
${\mathrm{Hom}}_{{\mathcal{O}}_{X}}(E,\Omega^{1}_{X/k}\otimes_{{\mathcal{O}}_{X}}\mathrm{\check{C}}^{1}(E))$,
respectively. Now it is easy to see that they are expressed as

(5.25) $\displaystyle\prod_{i}\nabla_{i}^{2}$
$\displaystyle\in\text{\rm\v{C}}^{0}(\mathfrak{U},\Omega_{X/k}^{2}\otimes_{{\mathcal{O}}_{X}}{\mathcal{E}}nd_{{\mathcal{O}}_{X}}(E));$
(5.26) $\displaystyle\prod_{i<j}(\nabla_{i}-\nabla_{j})$
$\displaystyle\in\text{\rm\v{C}}^{1}(\mathfrak{U},\Omega_{X/k}^{1}\otimes_{{\mathcal{O}}_{X}}{\mathcal{E}}nd_{{\mathcal{O}}_{X}}(E)).$

Let
${\alpha}\in\text{\rm\v{C}}^{p}(\mathfrak{U},{\mathcal{E}}nd_{{\mathcal{O}}_{X}}(E))\subseteq\mathrm{End}_{\mathrm{\check{C}}({\mathcal{O}}_{X})}(\text{\rm\v{C}}(E))$.
Then $[\nabla_{E},{\alpha}]$ is an element in
${\mathrm{Hom}}_{\mathrm{\check{C}}({\mathcal{O}}_{X})}(\mathrm{\check{C}}(E),\Omega^{1}_{X/k}\otimes_{{\mathcal{O}}_{X}}\mathrm{\check{C}}(E))$,
which is expressed as

$\prod_{i_{0}<...<i_{p}}[\nabla_{i_{0}}|_{U_{i_{0},...,i_{p}}},{\alpha}_{i_{0},...,i_{p}}]\in\text{\rm\v{C}}^{p}(\mathfrak{U},\Omega_{X/k}^{1}\otimes_{{\mathcal{O}}_{X}}{\mathcal{E}}nd_{{\mathcal{O}}_{X}}(E)).$

######  Remark 5.6.

Note that (5.25) is the curvature $\nabla_{E}^{2}$ of the connection
$\nabla_{E}$ on $\text{\rm\v{C}}(E)$ and (5.26) is the Čech representative of
the Atiyah class of the vector bundle $E$ (see [11] for example).

##  6\. Applications

In this section we will apply Theorem 1.2 to prove a Chern character formula
(1.3). We also generalize the formula to the localized case as well as the
equivariant case of a finite group action.

###  6.1. A Chern character formula

Denote by $\mathrm{ch}_{HN}(P)$ and $\mathrm{ch}_{HH}(P)$ the classes in

${\mathbb{H}}^{-*}(X,(\Omega^{\bullet}_{X/k}[[u]],ud-dw))\text{ and }\
{\mathbb{H}}^{-*}(X,(\Omega^{\bullet}_{X/k},-dw))$

corresponding to $\mathrm{Ch}_{HN}(P)$ and $\mathrm{Ch}_{HH}(P)$, respectively
via the isomorphism induced from (5.21).

####  6.1.1. Proof of Theorem 1.1

Recall that under the natural cochain map there is an isomorphism

$\overline{HN}_{*}(D_{\mathrm{\check{C}}}(X,w))\xrightarrow{\cong}\overline{HN}_{*}^{II}(D_{\mathrm{\check{C}}}(X,w)).$

Now considering the class $[1_{P}]$ of
$\overline{HN}_{*}^{II}(D_{\mathrm{\check{C}}}(X,w))$ and denoting by
$\overline{{\mathrm{tr}}}_{\nabla,X}^{II}$ the composition of the last two
maps in (5.20), we get

(6.1) $\mathrm{ch}_{HN}(P)=\overline{{\mathrm{tr}}}_{\nabla,X}^{II}(1_{P})$

in the cohomology $\check{H}^{0}(\mathfrak{U},(\Omega^{\bullet}_{X}[[u]],ud-
dw)).$ For an ${\mathcal{O}}_{X}$-connection $\nabla_{P}$ on
$\mathrm{\check{C}}(P)$ let

(6.2) $\displaystyle R$
$\displaystyle:=u\nabla_{P}^{2}+[\nabla_{P},\delta_{P}+d_{\scriptscriptstyle\text{\v{C}ech}}],$

which we call the total curvature of $\nabla_{P}$. Then by formula (5.7) we
have

$\overline{{\mathrm{tr}}}_{\nabla,X}^{II}(1_{P})={\mathrm{tr}}\exp(-R).$

Therefore we have established the following.

######  Theorem 6.1.

(Theorem 1.1) The Chern character $\mathrm{ch}_{HN}(P)$ is representable by

(6.3) $\mathrm{ch}_{HN}(P)={\mathrm{tr}}\exp(-R).$

Remark 5.5 shows expression (1.2).

####  6.1.2. When $k={\mathbb{C}}$

Assume furthermore that the critical locus of $w$ is proper over
${\mathbb{C}}$. Let $X^{an}$ denote the complex manifold associated to $X$.
Due to GAGA, instead of $\mathrm{\check{C}}(P)$, we may use the Dolbeault
resolution

$(\mathscr{A}^{0,\bullet}_{X^{an}},\bar{\partial})\otimes_{{\mathcal{O}}_{X^{an}}}P^{an}$

to construct the Dolbeault dg enhancement $D_{\mathrm{Dol}}(X,w)$ of the
derived category $D(X,w)$. Here $P^{an}$ denotes the holomorphic version of
the matrix factorization $P$. The objects of $D_{\mathrm{Dol}}(X,w)$ are the
objects of $D_{dg}(X,w)$. However ${\mathrm{Hom}}(P,Q)$ in
$D_{\mathrm{Dol}}(X,w)$ is, by definition, a complex

$\Gamma(X,\mathscr{A}^{0,\bullet}_{X^{an}}\otimes_{{\mathcal{O}}_{X^{an}}}{\mathcal{H}}om_{{\mathcal{O}}_{X^{an}}}(P^{an},Q^{an})).$

It is straightforward to check that the Dolbeault version of Theorem 5.4 holds
once we choose a differentiable $(1,0)$-connection $\nabla$ for each $P\in
D_{\mathrm{Dol}}(X,w)$. Hence we also have a Chern character formula

(6.4)
$\mathrm{ch}_{HN}(P)={\mathrm{tr}}\exp(-u\nabla^{2}-[\nabla,\delta_{P}+\bar{\partial}])$

in the cohomology

$H^{0}(\Gamma(X^{an},\mathscr{A}^{\bullet,\bullet}_{X^{an}})[[u]],u\partial+\bar{\partial}-\partial
w\wedge).$

###  6.2. Cohomology with supports

A matrix factorization $P$ for $(X,w)$ is called acyclic if $1_{P}$ is null-
homotopic in $\mathrm{End}_{D_{dg}(X,w)}(P)$. Let $Z$ be a closed subset of
$X$. We say that a matrix factorization $P$ for $(X,w)$ is supported on $Z$ if
$P|_{X-Z}$ is an acyclic matrix factorization for $(X-Z,w|_{X-Z})$. We may
consider a dg category $D_{dg}(X,w)_{Z}$ of matrix factorizations for $(X,w)$
supported on $Z$. It is a full subcategory of $D_{dg}(X,w)$.

Let $\mathfrak{U}_{1}=\\{U_{i}\\}_{i\in I_{1}}$ be a finite collection of open
affine subsets $U_{i}$ of $X$ such that

$\bigcup_{i\in I_{1}}U_{i}\supseteq Z$

and let $\mathfrak{U}_{2}=\\{U_{i}\\}_{i\in I_{2}}$ be a finite open affine
covering of $X-Z$. Let $I:=I_{1}\coprod I_{2}$ and
$\mathfrak{U}:=\\{U_{i}\\}_{i\in I}$, a finite open affine covering of $X$. As
before $\mathrm{\check{C}}(P)$ and $\text{\rm\v{C}}(P|_{X-Z})$ denote the
ordered Čech complexes of $P$ and $P|_{X-Z}$ with respect to $\mathfrak{U}$
and $\mathfrak{U}_{2}$, respectively. Let $\mathrm{\check{C}}(P)_{Z}$ denote
the kernel of the natural projection cochain map $\mathrm{\check{C}}(P)\to
j_{*}\text{\rm\v{C}}(P|_{X-Z})$ where $j:X-Z\to X$ is the open inclusion.

With respect to $\mathfrak{U}$, we have the Čech model
$D_{\mathrm{\check{C}}}(X,w)$ for $D_{dg}(X,w)$; see § 4.2.2. We define
$D_{\mathrm{\check{C}}}(X,w)_{Z}$ as the full subcategory of
$D_{\mathrm{\check{C}}}(X,w)$ consisting all objects supported on $Z$. We will
define another version $D_{\mathrm{\check{C}}}(X,w)^{rel}_{Z}$ of
$D_{dg}(X,w)_{Z}$, which is a relative version. The objects remain the same.
However the Hom space ${\mathrm{Hom}}(P,Q)$ for $P,Q\in
D_{\mathrm{\check{C}}}(X,w)^{rel}_{Z}$ is defined to be

(6.5)
${\mathrm{Hom}}_{\mathrm{\check{C}}({\mathcal{O}}_{X})_{Z}}^{\bullet}(\mathrm{\check{C}}(P)_{Z},\mathrm{\check{C}}(Q)_{Z})\cong\Gamma(X,{\mathcal{H}}om_{{\mathcal{O}}_{X}}^{\bullet}(P,Q)\otimes_{{\mathcal{O}}_{X}}\mathrm{\check{C}}({\mathcal{O}}_{X})_{Z}).$

This is a relative version of (1.4). Note that the natural inclusion

${\mathrm{Hom}}_{\mathrm{\check{C}}({\mathcal{O}}_{X})_{Z}}^{\bullet}(\mathrm{\check{C}}(P)_{Z},\mathrm{\check{C}}(Q)_{Z})\to{\mathrm{Hom}}_{\mathrm{\check{C}}({\mathcal{O}}_{X})}^{\bullet}(\mathrm{\check{C}}(P),\mathrm{\check{C}}(Q))$

is a quasi-isomorphism. The relative version
$D_{\mathrm{\check{C}}}(X,w)^{rel}_{Z}$ is not necessarily unital, which is
required in the definition of dg categories at §2.1. It is however
cohomologically unital, i.e., it has units in its cohomological category; for
the definition see [22, §1,1].

For a not-necessarily unital dg category $\mathscr{C}$, with no changes the
definition of Hochschild complex $(C(\mathscr{C}),b)$ works. There is another
version, the so-called non-unital Hochschild complex

$(C^{e}(\mathscr{C}),b)$

which includes $(C(\mathscr{C}),b)$ as a subcomplex and the inclusion is a
quasi-isomorphism. The complex $(C^{e}(\mathscr{C}),b)$ however has a suitable
Connes differential $B^{e}$ making a mixed complex
$(C^{e}(\mathscr{C}),b^{e},B^{e})$; see [23, §3.5] where $C^{e}(\mathscr{C})$
is denoted by $CC^{nu}_{\bullet}(\mathscr{C})$. On the other hand for a unital
dg category ${\mathcal{A}}$, there is a quasi-isomorphism

$p(u):(C^{e}({\mathcal{A}})[[u]],b^{e}+uB^{e})\xrightarrow{\sim}(C({\mathcal{A}})[[u]],b+uB)$

defined in [25].

Therefore we may consider a natural commutative diagram in the category of
cochain complexes

(6.6)
$\textstyle{C^{e}(D_{\mathrm{\check{C}}}(X,w)^{rel}_{Z})[[u]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{inc}$$\scriptstyle{{\mathrm{tr}}_{\nabla}^{Z}}$$\textstyle{C^{e}(D_{\mathrm{\check{C}}}(X,w)_{Z})[[u]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p(u)}$$\textstyle{C^{e}(D_{\mathrm{\check{C}}}(X,w))[[u]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p(u)}$$\textstyle{C^{e}(D_{\mathrm{\check{C}}}(U,w|_{U}))[[u]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p(u)}$$\textstyle{C(D_{\mathrm{\check{C}}}(X,w)_{Z})[[u]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C(D_{\mathrm{\check{C}}}(X,w))[[u]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathrm{tr}}_{\nabla,X}}$$\textstyle{C(D_{\mathrm{\check{C}}}(U,w|_{U}))[[u]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathrm{tr}}_{\nabla,U}}$$\textstyle{\Gamma(X,\mathrm{\check{C}}(\Omega^{\bullet}_{X/k})_{Z}[[u]])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Gamma(X,\mathrm{\check{C}}(\Omega^{\bullet}_{X/k})[[u]])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Gamma(U,\mathrm{\check{C}}(\Omega^{\bullet}_{U/k})[[u]]).}$

Here ${\mathrm{tr}}_{\nabla,X}$ (resp. ${\mathrm{tr}}_{\nabla,U}$) denotes
${\mathrm{tr}}_{\nabla}$ for $X$ with respect to $\mathfrak{U}$ (resp. for $U$
with respect to $\mathfrak{U}_{2}$) and the dotted arrow
${\mathrm{tr}}_{\nabla}^{Z}$ denotes ${\mathrm{tr}}_{\nabla,X}$ restricted to
$C^{e}(D_{\mathrm{\check{C}}}(X,w)^{rel}_{Z})[[u]]$, which lands on the
subcomplex $\Gamma(X,\mathrm{\check{C}}(\Omega^{\bullet}_{X/k})_{Z}[[u]])$.

######  Theorem 6.2.

The cochain map

(6.7) ${\mathrm{tr}}_{\nabla}^{Z}:\ \
(C^{e}(D_{\mathrm{\check{C}}}(X,w)^{rel}_{Z})[[u]],b^{e}+uB^{e})\\\
\to(\Gamma(X,\Omega^{\bullet}_{X/k}\otimes_{{\mathcal{O}}_{X}}\mathrm{\check{C}}({\mathcal{O}}_{X})_{Z})[[u]],d_{\scriptscriptstyle\text{\v{C}ech}}-dw+ud)$

is a quasi-isomorphism compatible with the HKR-type isomorphism
${\mathrm{tr}}_{\nabla,X}$. Furthermore, in the relative Čech cohomology

$H^{0}(\Gamma(X,\Omega^{\bullet}_{X/k}\otimes_{{\mathcal{O}}_{X}}\mathrm{\check{C}}({\mathcal{O}}_{X})_{Z})[[u]],d_{\scriptscriptstyle\text{\v{C}ech}}-dw+ud),$

the localized Chern character $\mathrm{ch}^{Z}_{HN}(P)$ for $P\in
D_{\mathrm{\check{C}}}(X,w)_{Z}$ with local connections (1.1) is representable
by

(6.8) ${\mathrm{tr}}\left(\prod_{i\in
I_{1}}1_{P|_{U_{i}}}\cdot\exp\left(-u(\nabla_{i}^{2})_{i\in
I}-([\nabla_{i},\delta_{P}])_{i\in I}-(\nabla_{i}-\nabla_{j})_{i<j,\ i,j\in
I}\right)\right).$

###### Proof.

First we invoke two facts that $inc$ in (6.6) is a quasi-isomorphism; see [23,
Corollary 4.13] and the third line of (6.6) is a distinguished triangle; see
[13, §5.7]. Note that the last line of (6.6) is obviously a short exact
sequence. By these facts together with the quasi-isomorphisms $p(u)$ it
follows that ${\mathrm{tr}}_{\nabla}^{Z}$ is also a quasi-isomorphism. Since
the unit $\prod_{i\in I}1_{P|_{U_{i}}}$ for $P$ in
$D_{\mathrm{\check{C}}}(X,w)_{Z}$ is homologous to $1_{P}^{Z}:=\prod_{i\in
I_{1}}1_{P|_{U_{i}}}$ and, by the definition of $p(u)$,

$p(u)\circ inc(1_{P}^{Z})=1_{P}^{Z},$

we conclude (6.8). ∎

###  6.3. Global quotients by a finite group

Let a finite group $G$ act on a smooth quasi-projective variety $X$ from the
left. Suppose that $w$ is $G$-fixed. We consider the dg category
$D_{\text{\rm\v{C}}}^{G}(X,w)$ of $G$-equivariant matrix factorizations for
$(X,w)$. The Hom dg-space from $P$ to $Q$ is the $G$-fixed part of
${\mathrm{Hom}}_{D_{\mathrm{\check{C}}}(X,w)}(P,Q)$.

Denote by $\mathfrak{t}$ the action map and by $\mathfrak{s}$ the projection
map

$\mathfrak{t}:G\times X\to X,\ (g,x)\mapsto gx;\ \ \ \mathfrak{s}:G\times X\to
X,\ (g,x)\mapsto x.$

For $E\in D_{\mathrm{\check{C}}}(X,w)$ denote $(g^{-1})^{*}E$ by $gE$ and let

$\widetilde{E}:=\mathfrak{s}_{*}\mathfrak{t}^{*}E=\bigoplus_{g\in
G}gE=E\otimes\bigoplus_{g}g{\mathcal{O}}_{X}.$

We will consider $\widetilde{E}$ as an object in
$D_{\text{\rm\v{C}}}^{G}(X,w)$ endowed with the natural $G$-equivariant
structure.

Let $\widetilde{D_{\text{\rm\v{C}}}^{G}}(X,w)$ be the full dg-subcategory of
$D_{\text{\rm\v{C}}}^{G}(X,w)$ consisting of all the objects of form
$\widetilde{E}$ for some $E\in D_{\mathrm{\check{C}}}(X,w)$. In fact this
gives rise to a functor
$D_{\mathrm{\check{C}}}(X,w)\to\widetilde{D_{\text{\rm\v{C}}}^{G}}(X,w)$. We
will see that every object $P$ of $D_{\text{\rm\v{C}}}^{G}(X,w)$ is a direct
summand of $\widetilde{P}$ as follows.

For $P\in D_{\text{\rm\v{C}}}^{G}(X,w)$, its $G$-equivariant structure is a
compatible isomorphism $\varphi:\mathfrak{t}^{*}P\to\mathfrak{s}^{*}P$. Let
$\varphi_{g}$ be the $g$-component of $\varphi$,

$\varphi_{g}:gP\xrightarrow{\cong}P.$

Then we get a map $\iota:P\to\mathfrak{s}_{*}\mathfrak{t}^{*}P$ given by the
composition

$\iota:P\xrightarrow{\text{natural}}\mathfrak{s}_{*}\mathfrak{s}^{*}P\xrightarrow{\mathfrak{s}_{*}\varphi^{-1}}\mathfrak{s}_{*}\mathfrak{t}^{*}P.$

Let $\varpi$ be a map $\mathfrak{s}_{*}\mathfrak{t}^{*}P\to P$ given by the
composition

$\varpi:\mathfrak{s}_{*}\mathfrak{t}^{*}P\xrightarrow{\mathfrak{s}_{*}\varphi}\mathfrak{s}_{*}\mathfrak{s}^{*}P\xrightarrow{\frac{\text{sum}}{|G|}}P,$

making $\varpi\circ\iota=1_{P}$. We impose a $G$-equivariant structure on
$\mathfrak{s}_{*}\mathfrak{t}^{*}P$ by letting

$\mathfrak{s}_{*}\mathfrak{t}^{*}P=\widetilde{P^{\sharp}}=P\otimes\bigoplus_{g}g{\mathcal{O}}_{X}$

where $P^{\sharp}$ denotes the object in $D_{\mathrm{\check{C}}}(X,w)$
corresponding to $P$. Then it is straightforward to check that $\iota$,
$\varpi$ are closed maps in $D_{\text{\rm\v{C}}}^{G}(X,w)$. By Morita
invariance (2.5) we have the following lemma.

######  Lemma 6.3.

The inclusion

$\mathrm{MC}(D_{\text{\rm\v{C}}}^{G}(X,w))\xleftarrow{inc}\mathrm{MC}(\widetilde{D_{\text{\rm\v{C}}}^{G}}(X,w))$

is a quasi-isomorphism.

For $E,F\in D_{\mathrm{\check{C}}}(X,w)$ every map
$\phi:\widetilde{F}\to\widetilde{E}$ in
$\widetilde{D_{\text{\rm\v{C}}}^{G}}(X,w)$ is expressed as a square matrix
$(\phi_{g,h})_{g,h\in G}$ whose entries $\phi_{g,h}:gF\to hE$ is a morphism in
$D_{\mathrm{\check{C}}}(X,w)$. Since $\phi$ is $G$-equivariant,
$h(\phi_{h^{-1}g,\mathrm{id}})=\phi_{g,h}$ if we let
$h(\phi_{h^{-1}g,\mathrm{id}}):=(h^{-1})^{*}(\phi_{h^{-1}g,\mathrm{id}})$.
Hence $\phi$ is uniquely determined by the components

$\phi_{g}:=\phi_{g,\mathrm{id}}:gF\to E,\ g\in G.$

We formally write $\phi=\sum_{g\in G}\phi_{g}\otimes g$.

######  Example 4.

For ${\alpha}\in\mathrm{End}^{G}_{{\mathcal{O}}_{X}}(P)$, let
$\widetilde{{\alpha}}:=\iota\circ{\alpha}\circ\varpi$, which is an
endomorphism of $\widetilde{P^{\sharp}}=\mathfrak{s}_{*}\mathfrak{t}^{*}P$ in
$D_{\text{\rm\v{C}}}^{G}(X,w)$. Since $\widetilde{{\alpha}}$ is a
$G$-equivariant map, it is determined by each component $gP\to P$, which is

(6.9) $\frac{1}{|G|}{\alpha}\circ\varphi_{g}.$

For $E_{i}\in D_{\mathrm{\check{C}}}(X,w)$ the composition $(a_{0}\otimes
g_{0})\circ(a_{1}\otimes g_{1})$ of $a_{0}\otimes
g_{0}:\widetilde{E_{1}}\to\widetilde{E_{0}}$ and $a_{1}\otimes
g_{1}:\widetilde{E_{2}}\to\widetilde{E_{1}}$ is written as $a_{0}\circ
g_{0}(a_{1})\otimes g_{0}g_{1}$, which is illustrated in the diagram

$E_{0}\xleftarrow{a_{0}}g_{0}E_{1}\xleftarrow{g_{0}(a_{1})}g_{0}g_{1}E_{2}$

where $g_{0}(a_{1}):=(g_{0}^{-1})^{*}a_{1}$.

Let $X^{g}$ denote the $g$-fixed loci of $X$ and let $w_{g}:=w|_{X^{g}}$.
Consider a $G$-action on the disjoint union $\coprod_{g\in G}X^{g}$ by

$X^{g}\to X^{hgh^{-1}},\ x\mapsto hx,\text{ for }h\in G.$

It induces an action on the Hochschild complex of $\coprod_{g\in G}X^{g}$ and
its $G$-coinvariants are denoted by $(\oplus_{g\in
G}C(D_{\mathrm{\check{C}}}(X^{g},w_{g})))_{G}$. In the category of mixed
complexes, following Baranovsky [2], we define a morphism

$C(\widetilde{D_{\text{\rm\v{C}}}^{G}}(X,w))\xrightarrow{\Psi}(\oplus_{g\in
G}C(D_{\mathrm{\check{C}}}(X^{g},w_{g})))_{G}$

by sending

$a_{0}\otimes g_{0}[a_{1}\otimes g_{1}|\cdots|a_{n}\otimes g_{n}]\mapsto
a_{0}{{}_{|_{X^{g}}}}[g_{0}(a_{1})_{|_{X^{g}}}|g_{0}g_{1}(a_{2})_{|_{X^{g}}}|\cdots|g_{0}...g_{n}(a_{n})_{|_{X^{g}}}]$

with $g:=g_{0}...g_{n}$.

######  Proposition 6.4.

The morphism $\Psi$ between the mixed complexes is a quasi-isomorphism.

###### Proof.

When $w=0$, the map $\Psi$ was defined and shown to be a quasi-isomorphism in
[2]. For general $w$ we may assume, by the Mayer-Vietoris sequence argument,
that $X$ is affine. In this case, for the Hochschild homology of the second
kind, it is a quasi-isomorphism by a spectral sequence argument as in [5,
Theorem 6.3]. Now again by (2.7), we conclude that it is a quasi-isomorphism
in the Hochschild complex of the first kind. Taking Lemma 2.1 into account, we
complete the proof. ∎

Fix a finite open affine covering $\mathfrak{U}=\\{U_{i}\\}_{i\in I}$ of $X$
such that each $U_{i}$ is $G$-invariant. This induces a finite open affine
covering $\\{X^{g}\cap U_{i}\\}_{i\in I}$ of $X^{g}$. For each $g$ and each
$Q\in D_{\mathrm{\check{C}}}(X^{g},w)$, we choose, once and for all, a
connection $\nabla_{Q,i}$ of $Q|_{X^{g}\cap U_{i}}$ such that the chosen
connection $\nabla_{h^{*}Q,i}$ of $(h^{*}Q)|_{X^{h^{-1}gh}\cap U_{i}}$ is
isomorphic to $h^{*}(\nabla_{Q,i})$. This is possible by Zorn’s lemma. Now we
have a quasi-isomorphism

(6.10) $\oplus_{g\in
G}C(D_{\mathrm{\check{C}}}(X^{g},w_{g}))[[u]]\xrightarrow{{\mathrm{tr}}_{\nabla}}\oplus_{g\in
G}\Gamma(X^{g},\mathrm{\check{C}}({\mathcal{O}}_{X^{g}})\otimes\Omega^{\bullet}_{X^{g}/k})[[u]]$

defined by applying ${\mathrm{tr}}_{\nabla}$ component-wisely for each $g$.
The cochain map ${\mathrm{tr}}_{\nabla}$ in (6.10) is $G$-equivariant. Hence
it induces a cochain map in $G$-coinvariants, also denoted by
${\mathrm{tr}}_{\nabla}$ by abuse of notation.

######  Theorem 6.5.

There is a natural isomorphism

(6.11) $HN_{*}(D_{\text{\rm\v{C}}}^{G}(X,w))\cong(\oplus_{g\in
G}H^{-*}(\Gamma(X^{g},\text{\rm\v{C}}(\Omega^{\bullet}_{X^{g}/k}))[[u]],d_{\scriptscriptstyle\text{\v{C}ech}}-dw_{g}+ud))_{G}.$

Under the isomorphism, the $G$-equivariant Chern character becomes

(6.12) $\mathrm{ch}^{G}_{HN}(P)={\mathrm{tr}}_{\nabla}\Psi(\eta_{\pi}),$

where $\pi:=\iota\circ\varpi$ and $\eta_{\pi}$ is from (3.2). In particular,

(6.13) $\mathrm{ch}^{G}_{HH}(P)=\frac{1}{|G|}\bigoplus_{g\in
G}{\mathrm{tr}}\left(\varphi_{g|_{X^{g}}}\exp(-[\prod_{i\in
I}\nabla_{P|_{X^{g}},i},d_{\scriptscriptstyle\text{\v{C}ech}}+\delta_{P|_{X^{g}}}])\right)$

for $G$-equivariant connections $\nabla_{P|_{U_{i}}}$ of $P|_{U_{i}}$.

###### Proof.

The isomorphism (6.11) follows by isomorphisms

$HN_{*}(D_{\text{\rm\v{C}}}^{G}(X,w))\xleftarrow{}HN_{*}(\widetilde{D_{\text{\rm\v{C}}}^{G}}(X,w))\xrightarrow{({\mathrm{tr}}_{\nabla}\circ\Psi)_{*}}\text{RHS
of \eqref{eqn: equiv HN}}.$

The specialization of (6.12) at $u=0$ becomes (6.13) by (6.9). Thus it is
enough to show (6.12). However this is clear from Proposition 3.3 which says
that $\eta_{\pi}$ is homologous to $1_{P}$ in
$(\overline{C}(D_{\text{\rm\v{C}}}^{G}(X,w))[[u]],b+uB)$. ∎

######  Remark 6.6.

Since ${\alpha}$ is homologous to $\widetilde{{\alpha}}=\sum_{g\in
G}\frac{1}{|G|}({\alpha}\circ\varphi_{g})\otimes g$, we also get a boundary
bulk map formula for Hochschild homology:

(6.14) ${\alpha}\mapsto\frac{1}{|G|}\bigoplus_{g\in
G}{\mathrm{tr}}\left(({\alpha}\circ\varphi_{g})_{|_{X^{g}}}\exp(-[(\nabla_{P|_{X^{g}},i})_{i\in
I},d_{\scriptscriptstyle\text{\v{C}ech}}+\delta_{P|_{X^{g}}}])\right).$

######  Remark 6.7.

Let $X$ be an open subscheme of $\mathbb{A}^{n}_{k}$ containing the origin and
let $w$ have only one critical point at the origin. Under the natural
isomorphism $(\bigoplus_{g\in
G}\Omega^{\bullet}_{X^{g}/k})^{G}\to(\bigoplus_{g\in
G}\Omega^{\bullet}_{X^{g}/k})_{G}$ and the completion, (6.14) coincides with
the corresponding formula (3.16) in [19, Theorem 3.3.3] up to a normalization
factor $|G|$ and the sign convention in HKR-type isomorphisms.

######  Remark 6.8.

When $k={\mathbb{C}}$, using a $G$-equivariant differentiable
$(1,0)$-connection on $P$ we get a $G$-equivariant Chern character formula of
$\mathrm{ch}^{G}_{HH}(P)$ in the Dolbeault cohomology. It is the
$G$-equivariant version of (6.4).

####  6.3.1. The equivariant mixed complex

Let $\overline{\Psi}^{II}$ be the version of $\Psi$ for the mixed normalized
Hochschild complex of the second kind. We have a diagram of quasi-isomorphisms
of mixed complexes

(6.15)
$\textstyle{\overline{C}^{II}(D_{\text{\rm\v{C}}}^{G}(X,w))}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\overline{C}^{II}(\widetilde{D_{\text{\rm\v{C}}}^{G}}(X,w))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{inc}$$\scriptstyle{\overline{\Psi}^{II}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces(\oplus_{g\in
G}\overline{C}^{II}(D_{\mathrm{\check{C}}}(X^{g},w_{g})))_{G}}$$\scriptstyle{inc}$$\textstyle{(\oplus_{g\in
G}\overline{C}^{II}(qD_{\mathrm{\check{C}}}(X^{g},w_{g})))_{G}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces(\oplus_{g\in
G}\overline{C}^{II}(\Gamma(X^{g},\text{\rm\v{C}}({\mathcal{O}}_{X^{g}}))))_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{inc}$$\scriptstyle{I_{\scriptscriptstyle\text{HKR}}}$$\textstyle{(\oplus_{g\in
G}(\Gamma(X^{g},\text{\rm\v{C}}(\Omega_{X^{g}/k})))_{G}.}$

The third inclusion map $inc$ is a quasi-isomorphism, since it is the case for
the affine case.

######  Theorem 6.9.

The natural morphism
$\overline{\mathrm{MC}}(D_{\text{\rm\v{C}}}^{G}(X,w))\to\overline{\mathrm{MC}}^{II}(D_{\text{\rm\v{C}}}^{G}(X,w))$
is a quasi-isomorphism and
$\overline{\mathrm{MC}}^{II}(D_{\text{\rm\v{C}}}^{G}(X,w)))$ is quasi-
isomorphic to

$\left(\oplus_{g\in
G}(\Gamma(X^{g},\text{\rm\v{C}}(\Omega^{\bullet}_{X^{g}/k})),d_{\scriptscriptstyle\text{\v{C}ech}}-dw_{g},d)\right)_{G}.$

###### Proof.

Since the second statement follows from diagram (6.15), it remains to prove
the first statement. When $X$ is affine, its proof follows from §2.2.5, (2.7),
and the first two quasi-isomorphisms in diagram (6.15). For the general $X$,
note first that there exists a finite collection of $G$-invariant open affine
schemes that cover $X$, since $X$ is quasi-projective. We may now apply the
Mayer-Vietoris exact triangle argument as in [2, 7] to reduce the proof to the
affine case. ∎

##  Appendix A Proof of Proposition 3.3

######  Proposition A.1.

(Proposition 3.3) The two cycles $1_{P}$ and $\eta_{\pi}$ are homologous in
$(\overline{C}\\{P,N\\}[[u]],b+uB)$.

###### Proof.

Let ${\mathcal{M}}$ be a $k$-category consisting of two formal objects, say
$P$ and $N$, and arrows generated by morphisms $1_{P}:=\mathrm{id}_{P},g:P\to
N,f:N\to P,1_{N}:=\mathrm{id}_{N},\pi:N\to N$, together with the obvious
relations by $1_{P},1_{N}$ and retract relations

$g\circ f=\pi,\ f\circ g=1_{P}.$

Let $\overline{{\mathcal{M}}}(x,y)$ denote ${\mathcal{M}}(x,y)$ if $x\neq y$
or ${\mathcal{M}}(x,y)/k\cdot 1_{x}$ if $x=y$. The proof of Proposition A.1
amounts to finding $\\{\xi_{i}\\}_{i=1}^{\infty}$ in

$\xi_{i}\in\overline{C}_{2i-1}({\mathcal{M}}):=\bigoplus_{x_{0},...,x_{2i-1}\in{\mathcal{M}}}{\mathcal{M}}(x_{1},x_{0})\otimes\overline{{\mathcal{M}}}(x_{2},x_{1})[1]\otimes\cdots\otimes\overline{{\mathcal{M}}}(x_{0},x_{2i-1})[1]$

such that $\xi_{1}:=g[f]$ and the $\xi_{i}$ satisfy the following relation

(A.1) $b(\xi_{i+1})=\eta_{i}-B(\xi_{i})\text{ where
}\eta_{\pi}:=\sum_{i=0}^{\infty}\eta_{i}u^{i}.$

Let

$\overline{D^{\prime}}:=\bigoplus_{n=0}^{\infty}\\{a_{0}[a_{1}|...|a_{n}]\mid\begin{aligned}
&a_{0}=t_{k_{1}}\pi+t_{k_{2}}1_{N}\text{ and }\\\ &a_{i}=t_{k_{i}}\pi\text{
for }t_{k_{i}}\in k,\text{ all }i\geq
1\end{aligned}\\}\subseteq\overline{C}({\mathcal{M}}).$

One may check that it is an acyclic subcomplex in
$\overline{C}({\mathcal{M}})$. Letting
$\overline{C}^{2}({\mathcal{M}}):=\overline{C}({\mathcal{M}})/\overline{D^{\prime}}$,
define

$\overline{D^{\prime\prime}}^{2}:=\bigoplus_{n=0}^{\infty}\\{a_{0}[a_{1}|...|a_{n}]\mid\begin{aligned}
&a_{i}=t_{i}f\text{ and }a_{i+1}=t_{i+1}\pi\\\ &\text{ for some }i+1\text{ mod
}n+1\text{ and }t_{j}\in
k\end{aligned}\\}\subseteq\overline{C}^{2}({\mathcal{M}}).$

By some computation, it can be also seen that it is an acyclic subcomplex in
$\overline{C}^{2}({\mathcal{M}})$. In the quotient space

$\overline{C}^{3}({\mathcal{M}}):=\overline{C}^{2}({\mathcal{M}})/\overline{D^{\prime\prime}}^{2},$

relation (A.1) becomes

(A.2) $b(\xi_{i+1})=-B(\xi_{i})\text{ for }i\geq 1.$

In $\overline{C}^{3}({\mathcal{M}})$, letting $\xi_{1}=g[f]$ and for $i\geq 2$

$\displaystyle\xi_{i}:$
$\displaystyle=(i-1)!(-1)^{i-1}\bigg{(}g[f|\underbrace{g|...|f}_{(g|f)^{i-1}}]-\big{(}1_{N}[\pi|g|...|f]+1_{N}[\pi|\pi|\pi|\underbrace{g|...|f}_{(g|f)^{i-2}}]+\cdots$
$\displaystyle\cdots+1_{N}[\underbrace{\pi|...|\pi}_{2i-3}|g|f]\big{)}\bigg{)}\in\overline{C}_{2i-1}^{3}({\mathcal{M}}),$

one may check that they satisfy the recursive relation (A.2). ∎

##  Appendix B Proof of Theorem 5.2

######  Theorem B.1.

(Theorem 5.2)The map ${\mathrm{tr}}_{\nabla}$ satisfies

$(ud+d_{A}-dh)\circ{\mathrm{tr}}_{\nabla}={\mathrm{tr}}_{\nabla}\circ(uB+b_{2}+b_{1}+b_{0}).$

The same cochain map equality also holds for ${\mathrm{tr}}_{\nabla}^{II}$.

###### Proof.

It is enough to show the equality for ${\mathrm{tr}}_{\nabla}^{II}$. The proof
of [3, Theorem 5.19] for when $A=V$, $V$ is commutative, and $\delta_{P}=0$
also works for the general case. We provide some details following the proof.
In what follows, let

(B.1) ${\alpha}^{\prime}:=[\nabla,{\alpha}]\text{ for
}{\alpha}\in\mathrm{End}_{A}({\mathcal{P}}),$

and let $\nabla\\!\\!\\!\\!\nabla:=u\nabla+\delta_{{\mathcal{P}}}$ and
$J:=\sum_{i=0}^{n}j_{i}$. Note that
$[\delta_{{\mathcal{P}}},{\alpha}]\in\mathrm{End}_{A}({\mathcal{P}})$.
Recalling $\delta_{{\mathcal{P}}}=\delta_{P}\otimes 1+1\otimes d_{A}$ and the
$V$-linearity of $\delta_{P}$, we see that

$\displaystyle(ud+d_{A})\circ\mathrm{tr}_{\nabla}^{II}({\alpha}_{0}[{\alpha}_{1}|\cdots|{\alpha}_{n}])$
$\displaystyle=\mathrm{tr}([u\nabla+\delta_{{\mathcal{P}}},\sum_{(j_{0},...,j_{n})}(-1)^{J}\frac{{\alpha}_{0}R^{j_{0}}{\alpha}^{\prime}_{1}\cdots{\alpha}^{\prime}_{n}R^{j_{n}}}{(J+n)!}])$
$\displaystyle=\mathrm{tr}(\sum_{(j_{0},...,j_{n})}(-1)^{J}\frac{[\nabla\\!\\!\\!\\!\nabla,{\alpha}_{0}]R^{j_{0}}{\alpha}^{\prime}_{1}\cdots{\alpha}^{\prime}_{n}R^{j_{n}}}{(J+n)!})$
$\displaystyle+\mathrm{tr}(\sum_{i=1}^{n}(-1)^{i-1+\sum_{k=0}^{i-1}|{\alpha}_{k}|}\sum_{(j_{0},...,j_{n})}(-1)^{J}\frac{{\alpha}_{0}R^{j_{0}}{\alpha}^{\prime}_{1}\cdots
R^{j_{i-1}}[\nabla\\!\\!\\!\\!\nabla,{\alpha}^{\prime}_{i}]R^{j_{i}}\cdots{\alpha}^{\prime}_{n}R^{j_{n}}}{(J+n)!})$
$\displaystyle+\mathrm{tr}(\sum_{i=0}^{n}(-1)^{i+\sum_{k=0}^{i}|{\alpha}_{k}|}\sum_{(j_{0},...,j_{n})}(-1)^{J}\frac{{\alpha}_{0}R^{j_{0}}{\alpha}^{\prime}_{1}\cdots[\nabla\\!\\!\\!\\!\nabla,R^{j_{i}}]{\alpha}^{\prime}_{i+1}R^{j_{i+1}}\cdots{\alpha}^{\prime}_{n}R^{j_{n}}}{(J+n)!}).$

In the last equality, by (B.7) the first two terms become
$\mathrm{tr}_{\nabla}^{II}\circ(b_{2}+b_{1}+uB)$ and by (B.4) the last term
becomes

(B.2)
$-\mathrm{tr}(\sum_{i=0}^{n}j_{i}(d\nu)\sum_{(j_{0},...,j_{n})}(-1)^{J}\frac{{\alpha}_{0}R^{j_{0}}{\alpha}^{\prime}_{1}\cdots
R^{j_{i}-1}{\alpha}^{\prime}_{i+1}R^{j_{i+1}}\cdots{\alpha}^{\prime}_{n}R^{j_{n}}}{(J+n)!}).$

After letting $K:=J-1$ and reindexing by $K=k_{0}+...+k_{n}$, (B.2) becomes

$d\nu\wedge\mathrm{tr}(\sum_{(k_{0},...,k_{n})}(-1)^{K}\frac{{\alpha}_{0}R^{k_{0}}{\alpha}^{\prime}_{1}\cdots
R^{k_{i}}{\alpha}^{\prime}_{i+1}R^{k_{i+1}}\cdots{\alpha}^{\prime}_{n}R^{k_{n}}}{(K+n)!}).$

Writing $\nu$ as the sum of $h$ and the curvature element $(\nu-h)$ of
${\mathcal{P}}$, we conclude the proof. ∎

######  Lemma B.2.

${\mathrm{tr}}:\mathrm{End}_{A}({\mathcal{P}})\otimes_{V}\Omega^{\bullet}_{V/k}[[u]]\to
A\otimes_{V}\Omega^{\bullet}_{V/k}[[u]]$

$(ud+d_{A}){\mathrm{tr}}(\gamma)={\mathrm{tr}}[u\nabla+\delta_{{\mathcal{P}}},\gamma]$

###### Proof.

As a $V$-module, $P$ is a direct summand of a finite sum $N$ of shifted $V$’s.
Hence we have $A$-linear maps $g:P\otimes_{V}A\to N\otimes_{V}A$ and
$f:N\otimes_{V}A\to P\otimes_{V}A$ such that $f\circ g=1_{P\otimes_{V}A}$.
Then we have

$ud{\mathrm{tr}}_{P}(\gamma)=ud{\mathrm{tr}}_{N}(g\circ\gamma\circ
f)={\mathrm{tr}}_{N}[ud,g\circ\gamma\circ f]\\\
={\mathrm{tr}}_{N}(g\circ[uf\circ d\circ g,\gamma]\circ
f)={\mathrm{tr}}_{N}(g\circ[u\nabla,\gamma]\circ
f)={\mathrm{tr}}_{P}[u\nabla,\gamma],$

where ${\mathrm{tr}}_{P}$, ${\mathrm{tr}}_{N}$ indicate the supertrace for
$\mathrm{End}_{V}(P)$, $\mathrm{End}_{V}(N)$, respectively. Similarly,
$d_{A}{\mathrm{tr}}_{P}(\gamma)={\mathrm{tr}}_{P}[d_{A},\gamma]={\mathrm{tr}}_{P}[\delta_{{\mathcal{P}}},\gamma]$.
∎

######  Lemma B.3.

With the notation from (B.1), in the algebra
$\mathrm{End}_{A}({\mathcal{P}})\otimes_{V}\Omega^{\bullet}_{V}[[u]]$ we have

(B.3)
$[u\nabla+\delta_{{\mathcal{P}}},{\alpha}^{\prime}]=[R,{\alpha}]-([\delta_{{\mathcal{P}}},{\alpha}])^{\prime};$
(B.4) $[u\nabla+\delta_{{\mathcal{P}}},R^{j}]=-j(d\nu)R^{j-1}.$

###### Proof.

The proofs are straightforward. ∎

######  Lemma B.4.

In $A\otimes_{V}\Omega^{\bullet}_{V/k}[[u]]$ we have

(B.5)
$\mathrm{tr}\Big{(}\sum_{i=1}^{n}(-1)^{i-1+\sum_{k=0}^{i-1}|{\alpha}_{k}|}\sum_{(j_{0},...,j_{n})}(-1)^{J}\frac{{\alpha}_{0}R^{j_{0}}{\alpha}^{\prime}_{1}\cdots
R^{j_{i-1}}[R,{\alpha}_{i}]R^{j_{i}}\cdots{\alpha}^{\prime}_{n}R^{j_{n}}}{(J+n)!}\Big{)}\\\
=\mathrm{tr}_{\nabla}^{II}\circ
b_{2}({\alpha}_{0}[{\alpha}_{1}|\dots|{\alpha}_{n}]);$ (B.6)
$\mathrm{tr}\big{(}\sum_{(j_{0},...,j_{n})}(-1)^{J}\frac{u{\alpha}^{\prime}_{0}R^{j_{0}}{\alpha}^{\prime}_{1}\cdots{\alpha}^{\prime}_{n}R^{j_{n}}}{(J+n)!}\big{)}=\mathrm{tr}_{\nabla}^{II}\circ(uB)({\alpha}_{0}[{\alpha}_{1}|\dots|{\alpha}_{n}]);\text{
and }$ (B.7)
$\displaystyle\mathrm{tr}(\sum_{(j_{0},...,j_{n})}(-1)^{J}\frac{[\nabla\\!\\!\\!\\!\nabla,{\alpha}_{0}]R^{j_{0}}{\alpha}^{\prime}_{1}\cdots{\alpha}^{\prime}_{n}R^{j_{n}}}{(J+n)!})$
$\displaystyle+\mathrm{tr}(\sum_{i=1}^{n}(-1)^{i-1+\sum_{k=0}^{i-1}|{\alpha}_{k}|}\sum_{(j_{0},...,j_{n})}(-1)^{J}\frac{{\alpha}_{0}R^{j_{0}}{\alpha}^{\prime}_{1}\cdots
R^{j_{i-1}}[\nabla\\!\\!\\!\\!\nabla,{\alpha}^{\prime}_{i}]R^{j_{i}}\cdots{\alpha}^{\prime}_{n}R^{j_{n}}}{(J+n)!})$
$\displaystyle=\mathrm{tr}_{\nabla}^{II}\circ(b_{2}+b_{1}+uB),$

where $\nabla\\!\\!\\!\\!\nabla:=u\nabla+\delta_{{\mathcal{P}}}$.

###### Proof.

Equations (B.5) and (B.6) are straightforward up to some combinatorics which
are checked in the Appendix in [3]. Equation (B.7) follows from (B.3), (B.5),
and (B.6). ∎

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