Equivariant Tilting Modules, Pfaffian Varieties and Noncommutative Matrix
Factorizations

Equivariant Tilting Modules, Pfaffian Varieties
and Noncommutative Matrix Factorizations††This paper is a contribution to the
Special Issue on Primitive Forms and Related Topics in honor of Kyoji Saito
for his 77th birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Saito.html

Yuki HIRANO

Y. Hirano

Department of Mathematics, Kyoto University,
Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto, 606-8502, Japan
<EMAIL_ADDRESS>
Received September 29, 2020, in final form May 28, 2021; Published online June
02, 2021

We show that equivariant tilting modules over equivariant algebras induce
equivalences of derived factorization categories. As an application, we show
that the derived category of a noncommutative resolution of a linear section
of a Pfaffian variety is equivalent to the derived factorization category of a
noncommutative gauged Landau–Ginzburg model
$(\Lambda,\chi,w)^{\mathbb{G}_{m}}$, where $\Lambda$ is a noncommutative
resolution of the quotient singularity $W/\operatorname{GSp}(Q)$ arising from
a certain representation $W$ of the symplectic similitude group
$\operatorname{GSp}(Q)$ of a symplectic vector space $Q$.

equivariant tilting module; Pfaffian variety; matrix factorization

14F08; 18G80; 16E35

Dedicated to Professor Kyoji Saito
on the occasion of his 77th birthday

##  1 Introduction

###  1.1 Backgrounds

Derived categories of coherent sheaves on varieties are one of the important
invariants in algebraic geometry, and they have interesting links to other
fields of mathematics. For example, if a smooth variety $X$ admits a tilting
bundle $\mathcal{T}$, the derived category
$\operatorname{D^{b}}(\operatorname{coh}X)$ of coherent sheaves on $X$ is
equivalent to the derived category
$\operatorname{D^{b}}(\operatorname{mod}\Lambda)$ of finitely generated right
modules over the endomorphism ring $\Lambda\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{End}_{X}(\mathcal{T})$
of $\mathcal{T}$. Once we have such an equivalence, we can study the derived
category of coherent sheaves by the representation theory of noncommutative
algebras. However, if $X$ is a smooth projective Calabi–Yau variety, $X$ can
never admit a tilting bundle.

Recently, Okonek–Teleman proved that the derived category of a regular zero
section in a certain smooth variety is equivalent to the derived factorization
category of a noncommutative gauged Landau–Ginzbubrg (LG) model [13]. In
particular, they proved that the derived category
$\operatorname{D^{b}}(\operatorname{coh}Z)$ of a Calabi–Yau complete
intersection $Z\subset\mathbb{P}^{n}$ is equivalent to the derived
factorization category $\operatorname{Dmod}_{\mathbb{G}_{m}}(\Lambda,\chi,w)$
of a noncommutative gauged LG model $(\Lambda,\chi,w)^{\mathbb{G}_{m}}$, where
$\Lambda$ is a noncommutative crepant resolution of some quotient singularity.
This result gives a new approach to the study of the derived categories of
Calabi–Yau complete intersections. For example, it is interesting to interpret
autoequivalences of $\operatorname{D^{b}}(\operatorname{coh}Z)$ with
autoequivalences of $\operatorname{Dmod}_{\mathbb{G}_{m}}(\Lambda,\chi,w)$
induced by $\mathbb{G}_{m}$-equivariant tilting modules/complexes over
$\Lambda$, and it is expected that this interpretation enables us to study the
fundamental group action on $\operatorname{D^{b}}(\operatorname{coh}Z)$,
constructed in [5] using variations of GIT quotients, by equivariant tilting
theory over $\Lambda$.

In this paper, for any reductive affine algebraic group $G$, we generalize
Okonek–Teleman’s result to a $G$-equivariant setting. More precisely, we prove
that $G$-equivariant tilting modules over $G$-equivariant algebras induce
equivalences of derived factorization categories of noncommutative gauged LG
models. Moreover, combining Rennemo–Segal’s results in [16] with our result,
we prove that the derived category of a noncommutative resolution of a generic
linear section of a Pfaffian variety is equivalent to the derived
factorization category of a noncommutative gauged LG model
$(\Lambda,\chi,w)^{\mathbb{G}_{m}}$, where $\Lambda$ is a non-commutative
resolution of the quotient singularity $W/\operatorname{GSp}(Q)$ arising from
a certain representation $W$ of the symplectic similitude group
$\operatorname{GSp}(Q)$ of a symplectic vector space $Q$.

###  1.2 Equivariant algebras and noncommutative LG models

Let $X$ be a scheme, and $G$ an algebraic group acting on $X$. Denote by
$\sigma\colon G\times X\to X$ the morphism defining the $G$-action on $X$, and
write $\pi\colon G\times X\to X$ for the natural projection. Let
$\uprho\colon\mathcal{O}_{X}\to\mathcal{A}$ be a (not necessarily commutative)
$\mathcal{O}_{X}$-algebra that is a coherent $\mathcal{O}_{X}$-module. A
$G$-equivariant structure on $\mathcal{A}$ is an isomorphism
$\uptheta^{\mathcal{A}}\colon\pi^{*}\mathcal{A}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\sigma^{*}\mathcal{A}$
of sheaves of $\mathcal{O}_{G\times X}$-algebras such that the pair
$\big{(}\mathcal{A},\uptheta^{\mathcal{A}}\big{)}$ defines the $G$-equivariant
coherent $\mathcal{O}_{X}$-module, and we call the pair
$\big{(}\mathcal{A},\uptheta^{\mathcal{A}}\big{)}$ a $G$-equivariant coherent
$X$-algebra. Similarly, we define a $G$-equivariant coherent
$\mathcal{A}$-module to be the pair
$\big{(}\mathcal{M},\uptheta^{\mathcal{M}}\big{)}$ of a right
$\mathcal{A}$-module $\mathcal{M}$ and an isomorphism
$\uptheta^{\mathcal{M}}\colon\pi^{*}\mathcal{M}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\sigma^{*}\mathcal{M}$
of $\pi^{*}\mathcal{A}$-modules such that the pair
$\big{(}\mathcal{M},\uptheta^{\mathcal{M}}\big{)}$ defines a $G$-equivariant
coherent $\mathcal{O}_{X}$-module. We write
$\operatorname{coh}_{G}\mathcal{A}$ for the category of $G$-equivariant
coherent $\mathcal{A}$-modules. Then we have an equivalence

$\displaystyle\operatorname{coh}_{G}\mathcal{A}\cong\operatorname{coh}[\mathcal{A}/G]$

of categories, where $[\mathcal{A}/G]$ is the associated coherent
$[X/G]$-algebra (see Appendix A).

Let $\chi\colon G\to\mathbb{G}_{m}$ be a character of $G$, and $w\colon
X\to\mathbb{A}^{1}$ a $\chi$-semi-invariant regular function on $X$, i.e.,
$W(g\cdot x)=\chi(g)W(x)$ for any $(g,x)\in G\times X$. We call the data
$(\mathcal{A},\chi,w)^{G}$ a gauged Landau–Ginzburg $(LG)$ model, and it is
said to be noncommutative if the algebra $\mathcal{A}$ is not commutative. We
define a factorization of $(\mathcal{A},\chi,w)^{G}$ to be a sequence

$\displaystyle\mathcal{M}_{1}\xrightarrow{\varphi_{1}}\mathcal{M}_{0}\xrightarrow{\varphi_{0}}\mathcal{O}_{X}(\chi)\otimes_{\mathcal{O}_{X}}\mathcal{M}_{1}$

consisting of $G$-equivariant coherent $\mathcal{A}$-modules $\mathcal{M}_{i}$
and $G$-equivariant $\mathcal{A}$-linear maps $\varphi_{i}$ such that the
compositions $\varphi_{0}\circ\varphi_{1}$ and
$\varphi_{1}(\chi)\circ\varphi_{0}$ are the multiplications by $w$. To a
gauged LG model $(\mathcal{A},\chi,w)^{G}$ we associate the derived
factorization category

$\displaystyle\operatorname{Dcoh}_{G}(\mathcal{A},\chi,w)$

whose objects are factorizations of $(\mathcal{A},\chi,w)^{G}$. If
$X=\operatorname{Spec}R$, the $G$-equivariant $X$-algebra $\mathcal{A}$
corresponds to the $G$-equivariant $R$-algebra $A\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\Gamma(X,\mathcal{A})$,
and we write $\operatorname{Dmod}_{G}(A,\chi,w)$ for the corresponding derived
factorization category.

###  1.3 Results

Let $X$ be a variety, $G$ an affine algebraic group acting on $X$, and $H$ a
closed normal subgroup of $G$.

######  Definition 1.1.

A $G$-equivariant coherent $\mathcal{A}$-module
$\mathcal{T}\in\operatorname{coh}_{G}\mathcal{A}$ is called a $(G,H)$-tilting
(resp. partial $(G,H)$-tilting) if the restriction
$\mathcal{T}_{H}\in\operatorname{coh}_{H}\mathcal{A}$ satisfies the following
conditions (1), (2) and (3) (resp. (1) and (2)):

  * $1.$

We have
$\operatorname{Ext}_{\operatorname{coh}_{H}\mathcal{A}}^{i}(\mathcal{T}_{H},\mathcal{T}_{H})=0$
for all $i>0$.

  * $2.$

The object $\mathcal{T}_{H}$ is compact in
$\operatorname{D}\big{(}\operatorname{Qcoh}_{H}\mathcal{A}\big{)}$.

  * $3.$

The object $\mathcal{T}_{H}$ is a generator of
$\operatorname{D}\big{(}\operatorname{Qcoh}_{H}\mathcal{A}\big{)}$, i.e., for
any complex $A^{\bullet}$ in $\operatorname{Qcoh}_{H}\mathcal{A}$,
$\operatorname{{\mathbb{R}}Hom}(T,A^{\bullet})\cong 0$ in
$\operatorname{D}(\operatorname{Mod}\operatorname{End}(\mathcal{T}_{H}))$
implies $A^{\bullet}\cong 0$ in
$\operatorname{D}\big{(}\operatorname{Qcoh}_{H}\mathcal{A}\big{)}$.

Let $\operatorname{Spec}R$ be an affine variety with a $G$-action such that
the induced $H$-action is trivial so that $\operatorname{Spec}R$ has the
induced $G/H$-action. Let $f\colon X\to\operatorname{Spec}R$ be a
$G$-equivariant morphism such that the associated morphism
$\overline{f}\colon[X/H]\to\operatorname{Spec}R$ is proper. Let
$\mathcal{T}\in\operatorname{coh}_{G}\mathcal{A}$ be a $G$-equivariant
coherent $\mathcal{A}$-module, and write $\Lambda\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{End}_{\operatorname{coh}\mathcal{A}}(\mathcal{T})$
for the endomorphism ring of the underlying $\mathcal{A}$-module
$\mathcal{T}$. Then $\Lambda$ has a natural $H$-action, and the $H$-invariant
ring $\Lambda^{H}$ is a $G/H$-equivariant coherent $R$-algebra. Let
$\chi\colon G/H\to\mathbb{G}_{m}$ be a character of $G/H$, and define the
character $\widehat{\chi}\colon G\to\mathbb{G}_{m}$ to be the composition of
$\chi$ and the natural projection $G\to G/H$. Let
$w_{R}\colon\operatorname{Spec}R\to\mathbb{A}^{1}$ be a $\chi$-semi-invariant
regular function on $\operatorname{Spec}R$, and write $w\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=f^{*}w_{R}$.
Then we have two gauged LG models

$\displaystyle\big{(}\mathcal{A},\widehat{\chi},w\big{)}^{G}\qquad\text{and}\qquad\big{(}\Lambda^{H},\chi,w_{R}\big{)}^{G/H}.$

The following is a generalization of Okonek–Teleman’s result [13, Theorem
1.11].

######  Theorem 1.2 (Theorem 4.7).

Assume that $G/H$ is reductive and $\Lambda^{H}$ is of finite global
dimension.

  * $1.$

If $\mathcal{T}$ is partial $(G,H)$-tilting, then we have a fully faithful
functor

$\displaystyle\operatorname{Dmod}_{G/H}\big{(}\Lambda^{H},\chi,w_{R}\big{)}\hookrightarrow\operatorname{Dcoh}_{G}\big{(}\mathcal{A},\widehat{\chi},w\big{)}.$

  * $2.$

If $\mathcal{T}$ is $(G,H)$-tilting and every quasi-coherent
$\mathcal{A}$-module has a finite injective resolution, we have an equivalecne

$\displaystyle\operatorname{Dcoh}_{G}\big{(}\mathcal{A},\widehat{\chi},w\big{)}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\operatorname{Dmod}_{G/H}\big{(}\Lambda^{H},\chi,w_{R}\big{)}.$

Let $V$ be a vector space of dimension $v$. For an integer $q$ with $0\leq
2q\leq v$, we have a Pfaffian variety

$\displaystyle\operatorname{Pf}_{q}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\big{\\{}x\in\raisebox{1.0pt}[0.0pt][0.0pt]{$\textstyle\bigwedge^{\raisebox{-1.0pt}[0.0pt][0.0pt]{\tiny\\!$2$\,}}$}V^{*}\mid\operatorname{\sf
rk}(x)\leq
2q\big{\\}}\subseteq\mathbb{P}\big{(}\raisebox{1.0pt}[0.0pt][0.0pt]{$\textstyle\bigwedge^{\raisebox{-1.0pt}[0.0pt][0.0pt]{\tiny\\!$2$\,}}$}V^{*}\big{)},$

and we consider its linear section

$\displaystyle\operatorname{Pf}_{q}\\!|_{L}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{Pf}_{q}\cap\,\,\mathbb{P}(L),$

where
$L\subset\raisebox{1.0pt}[0.0pt][0.0pt]{$\textstyle\bigwedge^{\raisebox{-1.0pt}[0.0pt][0.0pt]{\tiny\\!$2$\,}}$}V^{*}$
is a subspace of $V$ such that $\operatorname{Pf}_{q}\\!|_{L}\neq\varnothing$.
If we choose a generic $L$, there is a noncommutative resolution
$\mathcal{B}_{L}$ of $\operatorname{Pf}_{q}\\!|_{L}$ [17, 18], and if
$\operatorname{Pf}_{q}\\!|_{L}$ is smooth
$\operatorname{D^{b}}(\operatorname{coh}\mathcal{B}_{L})$ is equivalent to
$\operatorname{D^{b}}(\operatorname{coh}\operatorname{Pf}_{q}\\!|_{L})$.
Combining Theorem 1.2 and the results in [16], we have the following.

######  Corollary 1.3 (Corollary 5.1).

There is a noncommutative gauged LG model $(\Lambda,\chi,w)^{\mathbb{G}_{m}}$
such that we have an equivalence

$\displaystyle\operatorname{D^{b}}(\operatorname{coh}\mathcal{B}_{L})\cong\operatorname{Dmod}_{\mathbb{G}_{m}}(\Lambda,\chi,w)$

and that $\Lambda$ is a noncommutative resolution of the affine quotient
$W/\operatorname{GSp}(Q)$ of a certain representation $W$ of the symplectic
similitude group $\operatorname{GSp}(Q)$ of a symplectic vector space $Q$.

###  1.4 Notation and convention

  * •

Unless stated otherwise, categories and stacks we consider are over an
algebraically closed field $k$ of characteristic zero.

  * •

For a character $\chi\colon G\to\mathbb{G}_{m}$ of an algebraic group $G$, we
denote by $\mathcal{O}_{X}(\chi)$, or simply $\mathcal{O}(\chi)$, the
$G$-equivariant invertible sheaf on a $G$-scheme $X$ associated to $\chi$.

  * •

For an exact category $\mathcal{E}$, we denote by
$\operatorname{Ch}(\mathcal{E})$ (resp. $\operatorname{Ch^{b}}(\mathcal{E})$)
the category of cochain complexes (resp. bounded cochain complexes) in
$\mathcal{E}$, and we write $\operatorname{D}(\mathcal{E})$ (resp.
$\operatorname{D^{b}}(\mathcal{E})$) for the derived category (resp. the
bounded derived category) of $\mathcal{E}$.

##  2 Preliminaries

In this section, we recall definitions and basic properties about derived
factorization categories, equivariant quasi-coherent sheaves and tilting
objects.

###  2.1 Derived factorization categories

We recall the basics of derived factorization categories mainly to fix
notation. See, for example, [2, 8, 15] for more details. Throughout this
section, $\mathcal{A}$ is an abelian category with small coproducts such that
the small coproducts of families of short exact sequences are exact. In what
follows, we fix a triple

$(\mathcal{E},\Phi,w)$ (2.1)

consisting of an exact subcategory $\mathcal{E}\subseteq\mathcal{A}$ of
$\mathcal{A}$, an exact autoequivalence $\Phi\colon\mathcal{A}\to\mathcal{A}$
preserving $\mathcal{E}$ and a functor morphism
$w\colon\operatorname{id}\to\Phi$ that is compatible with $\Phi$, i.e., the
equality $w({\Phi(E)})=\Phi(w(E))$ holds for every object $E\in\mathcal{E}$.
The functor morphism $w\colon\operatorname{id}\to\Phi$ is called a potential
of $\mathcal{E}$.

######  Definition 2.1.

A factorization of $(\mathcal{E},\Phi,w)$ is a sequence

$\displaystyle
E=\Bigl{(}E_{1}\xrightarrow{\varphi^{E}_{1}}E_{0}\xrightarrow{\varphi^{E}_{0}}\Phi(E_{1})\Bigr{)}$

in $\mathcal{E}$ such that $\varphi^{E}_{0}\circ\varphi^{E}_{1}=w(E_{1})$ and
$\Phi\big{(}\varphi^{E}_{1}\big{)}\circ\varphi^{E}_{0}=w(E_{0})$. Objects
$E_{1}$ and $E_{0}$ in the above sequence are called the components of $E$.

######  Definition 2.2.

Let $(\mathcal{E},\Phi,w)$ be the above triple.

  * $1.$

For two factorizations $E,F$ of $(\mathcal{E},\Phi,w)$, we define the complex
$\operatorname{Hom}(E,F)^{\bullet}$ with differential
$d^{\bullet}_{(E,F)}\colon\operatorname{Hom}(E,F)^{\bullet}\to\operatorname{Hom}(E,F)^{\bullet+1}$
by

$\displaystyle{\rm Hom}(E,F)^{2n}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}={\rm
Hom}_{\mathcal{E}}(E_{1},\Phi^{n}(F_{1}))\oplus{\rm
Hom}_{\mathcal{E}}(E_{0},\Phi^{n}(F_{0})),$ $\displaystyle{\rm
Hom}(E,F)^{2n+1}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}={\rm
Hom}_{\mathcal{E}}(E_{1},\Phi^{n}(F_{0}))\oplus{\rm
Hom}_{\mathcal{E}}(E_{0},\Phi^{n+1}(F_{1})),$

where $E_{i}$ and $F_{i}$ are the components of $E$ and $F$ respectively, and

$\displaystyle d^{\bullet}_{(E,F)}(f):=\varphi^{F}\circ f-(-1)^{{\rm
deg}(f)}f\circ\varphi^{E}\qquad\text{if}\quad f\in{\rm Hom}(E,F)^{{\rm
deg}(f)}.$

This defines the dg category

$\displaystyle\operatorname{\sf{Fact}}(\mathcal{E},\Phi,w)$

of factorizations of $(\mathcal{E},\Phi,w)$.

  * $2.$

The dg category $\operatorname{\sf{Fact}}(\mathcal{E},\Phi,w)$ defines the
additive categories

$\displaystyle\operatorname{Fact}(\mathcal{E},\Phi,w)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=Z^{0}(\operatorname{\sf{Fact}}(\mathcal{E},\Phi,w)),$
$\displaystyle\operatorname{K}(\mathcal{E},\Phi,w)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=H^{0}(\operatorname{\sf{Fact}}(\mathcal{E},\Phi,w)).$

We call $\operatorname{K}(\mathcal{E},\Phi,w)$ the homotopy category of
$(\mathcal{E},\Phi,w)$.

The category $\operatorname{Fact}(\mathcal{E},\Phi,w)$ is an exact category,
and the homotopy category $\operatorname{K}(\mathcal{E},\Phi,w)$ is a
triangulated category (see [8, Propositions 3.5 and 3.9]). To define the
derived factorization categories, we define the totalizations of complexes of
factorizations: For an object $E\in\operatorname{Fact}(\mathcal{E},\Phi,w)$,
let us set

$\displaystyle{\rm Com}(E)^{2i}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\Phi^{i}(E_{0}),\qquad{\rm
Com}(E)^{2i-1}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\Phi^{i}(E_{1}),$
$\displaystyle d_{E}^{2i}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\Phi^{i}\big{(}\varphi_{0}^{E}\big{)},\qquad
d_{E}^{2i-1}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\Phi^{i}\big{(}\varphi_{1}^{E}\big{)}.$

Then a periodic (up to twists by $\Phi$) infinite sequence

$\displaystyle{\rm Com}(E)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\big{(}{\rm
Com}(E)^{\bullet},d_{E}^{\bullet}\big{)}$

in $\mathcal{E}$ satisfies $d_{E}^{i+1}\circ d_{E}^{i}=w({\rm Com}(E)^{i})$
for all $i\in\mathbb{Z}$.

######  Definition 2.3.

Let $E^{\bullet}=\big{(}\cdots\rightarrow
E^{i}\xrightarrow{\delta^{i}}E^{i+1}\rightarrow\cdots\big{)}$ be a bounded
complex of $\operatorname{Fact}(\mathcal{E},\Phi,w)$. We define the
totalization
$\operatorname{Tot}(E^{\bullet})\in\operatorname{Fact}(\mathcal{E},\Phi,w)$ of
$E^{\bullet}$ by

$\displaystyle{\rm Tot}(E^{\bullet})\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\big{(}T_{1}\xrightarrow{t_{1}}T_{0}\xrightarrow{t_{0}}\Phi(T_{1})\big{)},$

where

$\displaystyle T_{l}:=\bigoplus_{i+j=-l}{\rm Com}(E^{i})^{j},$ $\displaystyle
t_{l}|_{{\rm Com}(E^{i})^{j}}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}={\rm
Com}(\delta^{i})^{j}+(-1)^{i}d_{E^{i}}^{j}.$

Taking totalizations defines an exact functor

$\displaystyle\operatorname{Tot}\colon\
\operatorname{Ch^{b}}(\operatorname{Fact}(\mathcal{E},\Phi,w))\rightarrow\operatorname{Fact}(\mathcal{E},\Phi,w).$

######  Definition 2.4.

Let $\operatorname{A}(\mathcal{E},\Phi,w)$ be the smallest thick subcategory
of $\operatorname{K}(\mathcal{E},\Phi,w)$ that contains totalizations of all
short exact sequences in $\operatorname{Fact}(\mathcal{E},\Phi,w)$. Then we
define the derived factorization category
$\operatorname{D}(\mathcal{E},\Phi,w)$ of the triple $(\mathcal{E},\Phi,w)$ by
the Verdier quotient

$\displaystyle\operatorname{D}(\mathcal{E},\Phi,w)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{K}(\mathcal{E},\Phi,w)/\operatorname{A}(\mathcal{E},\Phi,w).$

If $\mathcal{E}$ is closed under coproducts in $\mathcal{A}$, we denote by
$\operatorname{A^{co}}(\mathcal{E},\Phi,w)$ the smallest thick subcategory of
$\operatorname{K}(\mathcal{E},\Phi,w)$ that contains totalizations of all
short exact sequences in $\operatorname{Fact}(\mathcal{E},\Phi,w)$ and closed
under coproducts. Then we define the coderived factorization category
$\operatorname{D^{co}}(\mathcal{E},\Phi,w)$ by

$\displaystyle\operatorname{D^{co}}(\mathcal{E},\Phi,w)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{K}(\mathcal{E},\Phi,w)/\operatorname{A^{co}}(\mathcal{E},\Phi,w).$

Let $\mathcal{F}$ be an exact subcategory of another abelian category
$\mathcal{B}$ satisfying the same properties of $\mathcal{A}$, and let

$\displaystyle(\mathcal{F},\Psi,v)$

be a triple as in (2.1).

######  Definition 2.5.

An additive functor

$\displaystyle F\colon\ \mathcal{E}\to\mathcal{F}$

is factored with respect to the potentials $(\Phi,w)$ and $(\Psi,v)$ if there
is a functor isomorphism

$\displaystyle\alpha\colon\
F\circ\Phi\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\Psi\circ
F$

such that for every object $E\in\mathcal{E}$, the following diagram commutes:

${F(E)}$${F(\Phi(E))}$${\Psi(F(E)).}$$\scriptstyle{F(w(E))}$$\scriptstyle{v(F(E))}$$\scriptstyle{\alpha(E)}$

If $F\colon\mathcal{E}\to\mathcal{F}$ is a factored functor with respect to
$(\Phi,w)$ and $(\Psi,v)$, then it induces a dg functor

$\displaystyle F\colon\
\operatorname{\sf{Fact}}(\mathcal{E},\Phi,w)\to\operatorname{\sf{Fact}}(\mathcal{F},\Psi,v).$

This dg functor defines the additive functors

$\displaystyle F\colon\
\operatorname{Fact}(\mathcal{E},\Phi,w)\to\operatorname{Fact}(\mathcal{F},\Psi,v),$
$\displaystyle F\colon\
\operatorname{K}(\mathcal{E},\Phi,w)\to\operatorname{K}(\mathcal{F},\Psi,v),$

where the latter functor is an exact functor. To define the derived functors
of exact functors between homotopy categories, we need the following results
due to [2].

######  Proposition 2.6 ([2, Corollary 2.25]).

Assume that $\mathcal{A}$ has enough injectives and that the coproducts of
injectives are injective. Let $\mathcal{I}\subset\mathcal{A}$ be the
subcategory of injective objects. Then the natural functor

$\displaystyle\operatorname{K}(\mathcal{I},\Phi,w)\to\operatorname{D^{co}}(\mathcal{A},\Phi,w)$

is an equivalence.

######  Proposition 2.7.

Let $\mathcal{P}\subset\mathcal{A}$ be the subcategory of projective objects
in $\mathcal{A}$, and assume that $\mathcal{A}$ has enough projectives. Let
$\mathcal{C}\subset\mathcal{A}$ be an abelian subcategory that is preserved by
$\Phi$, and let $\mathcal{Q}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\mathcal{C}\cap\mathcal{P}$.

  * $1.$

Assume that all objects in $\mathcal{C}$ are compact in $\mathcal{A}$. Then
for any $P\in\operatorname{K}(\mathcal{Q},\Phi,w)$ and
$A\in\operatorname{A^{co}}(\mathcal{A},\Phi,w)$ we have

$\displaystyle\operatorname{Hom}_{\operatorname{K}(\mathcal{A},\Phi,w)}(P,A)=0.$

In particular, the natural functor

$\displaystyle\operatorname{K}(\mathcal{Q},\Phi,w)\to\operatorname{D^{co}}(\mathcal{A},\Phi,w)$

is fully faithful.

  * $2.$

If every object in $\mathcal{C}$ has a finite projective resolution in
$\mathcal{C}$, the natural functor

$\displaystyle\operatorname{K}(\mathcal{Q},\Phi,w)\to\operatorname{D}(\mathcal{C},\Phi,w)$

is essentially surjective.

###### Proof.

(1) The latter statement follows from the former one by [12, Proposition B.2].
The vanishing
$\operatorname{Hom}_{\operatorname{K}(\mathcal{A},\Phi,w)}(P,A)=0$ reduces to
the case when $A\in\operatorname{A}(\mathcal{A},\Phi,w)$, since the components
of $P$ are compact by our assumption. If
$A\in\operatorname{A}(\mathcal{A},\Phi,w)$, the vanishing follows from [2,
Lemma 2.4].

(2) This follows from [2, Proposition 2.22]. ∎

######  Definition 2.8.

Let $F\colon\mathcal{A}\to\mathcal{B}$ be a factored functor with respect to
$(\Phi,w)$ and $(\Psi,v)$.

  * 1.

Assume that $\mathcal{A}$ has enough injectives and that the coproducts of
injectives are injective. If $F$ is left exact, we define the right derived
functor

$\displaystyle\mathbb{R}F\colon\
\operatorname{D^{co}}(\mathcal{A},\Phi,w)\to\operatorname{D^{co}}(\mathcal{B},\Psi,v)$

of
$F\colon\operatorname{K}(\mathcal{A},\Phi,w)\to\operatorname{K}(\mathcal{B},\Psi,v)$
to be the composition

${\operatorname{D^{co}}(\mathcal{A},\Phi,w)}$${\operatorname{K}(\mathcal{I},\Phi,w)}$${\operatorname{K}(\mathcal{B},\Psi,v)}$${\operatorname{D^{co}}(\mathcal{B},\Psi,v),}$$\scriptstyle{\sim}$$\scriptstyle{F}$$\scriptstyle{Q}$

where the first functor is the equivalence in Proposition 2.6, and $Q$ is the
natural quotient functor.

  * 2.

Let $\mathcal{C}\subset\mathcal{A}$ and $\mathcal{D}\subset\mathcal{B}$ be
abelian subcategories that are preserved by $\Phi$ and $\Psi$ respectively,
and assume that the factored functor $F$ restricts to
$F\colon\mathcal{C}\to\mathcal{D}$. Assume that every object in $\mathcal{C}$
is compact in $\mathcal{A}$ and has a finite projective resolution in
$\mathcal{C}$, and that the natural functor
$\operatorname{D}(\mathcal{C},\Phi,w)\to\operatorname{D^{co}}(\mathcal{A},\Phi,w)$
is fully faithful. If $F$ is right exact, we define the left derived functor

$\displaystyle\mathbb{L}F\colon\
\operatorname{D}(\mathcal{C},\Phi,w)\to\operatorname{D}(\mathcal{D},\Psi,v)$

of
$F\colon\operatorname{K}(\mathcal{C},\Phi,w)\to\operatorname{K}(\mathcal{D},\Psi,v)$
to be the composition

${\operatorname{D}(\mathcal{C},\Phi,w)}$${\operatorname{K}(\mathcal{Q},\Phi,w)}$${\operatorname{K}(\mathcal{D},\Psi,v)}$${\operatorname{D}(\mathcal{D},\Psi,v),}$$\scriptstyle{\sim}$$\scriptstyle{F}$$\scriptstyle{Q}$

where the first functor is the equivalence in Proposition 2.7.

###  2.2 Quasi-coherent modules over sheaves of algebras

In this subsection, we recall fundamental properties of quasi-coherent modules
over sheaves of algebras over schemes.

Let $X$ be a scheme. An $\mathcal{O}_{X}$-module $\mathcal{F}$ is said to be
quasi-coherent if for any point $x\in X$, there are an open neighborhood $U$
of $x$ and an exact sequence

$\displaystyle\mathcal{O}_{U}^{\oplus I}\to\mathcal{O}_{U}^{\oplus
J}\to\mathcal{F}|_{U}\to 0,$

where $I$ and $J$ are sets. A quasi-coherent $\mathcal{O}_{X}$-module
$\mathcal{F}$ is called a coherent $\mathcal{O}_{X}$-module, if for every
affine open subset $U=\operatorname{Spec}R$ of $X$, $\Gamma(U,\mathcal{F})$ is
a finitely generated $R$-module. We denote by $\operatorname{Qcoh}X$ (resp.
$\operatorname{coh}X$) the category of quasi-coherent (resp. coherent)
$\mathcal{O}_{X}$-modules.

######  Definition 2.9.

An $X$-algebra is a sheaf $\mathcal{A}$ of not necessarily commutative
algebras together with a morphism $\uprho\colon\mathcal{O}_{X}\to\mathcal{A}$
of sheaves of algebras such that the image of $\uprho$ lies in the center of
$\mathcal{A}$ and that $\mathcal{A}$ is a quasi-coherent
$\mathcal{O}_{X}$-module. An $X$-algebra
$\uprho\colon\mathcal{O}_{X}\to\mathcal{A}$ is said to be coherent if
$\mathcal{A}$ is a coherent $\mathcal{O}_{X}$-module. A morphism of
$X$-algebras is a morphism $\varphi\colon\mathcal{A}\to\mathcal{B}$ of sheaves
of algebras that is $\mathcal{O}_{X}$-linear.

For an $X$-algebra $\uprho\colon\mathcal{O}_{X}\to\mathcal{A}$, a quasi-
coherent $($resp. coherent$)$ $\mathcal{A}$-module is a right
$\mathcal{A}$-module $\mathcal{M}$ that is a quasi-coherent (resp. coherent)
$\mathcal{O}_{X}$-module. For quasi-coherent $\mathcal{A}$-modules
$\mathcal{M}$ and $\mathcal{N}$, a morphism from $\mathcal{M}$ to
$\mathcal{N}$ is a morphism of sheaves of right $\mathcal{A}$-modules. We
denote by

$\displaystyle\operatorname{Qcoh}\mathcal{A}$

the category of quasi-coherent $\mathcal{A}$-modules, and we write
$\operatorname{coh}\mathcal{A}$ for the full subcategory of coherent
$\mathcal{A}$-modules.

######  Remark 2.10.

Since the category of right $\mathcal{A}$-modules and the category
$\operatorname{Qcoh}X$ are both abelian, $\operatorname{Qcoh}\mathcal{A}$ is
also an abelian category. Furthermore, in Proposition 3.19 we will see that
$\operatorname{Qcoh}\mathcal{A}$ is a Grothendieck category.

Let $\varphi\colon\mathcal{A}\to\mathcal{B}$ be a morphism of $X$-algebras.
For an object $\mathcal{N}\in\operatorname{Qcoh}\mathcal{B}$, we define a
quasi-coherent $\mathcal{A}$-module $\mathcal{N}_{\varphi}$ by the composition

$\displaystyle\mathcal{N}\times\mathcal{A}\xrightarrow{\operatorname{id}\times\varphi}\mathcal{N}\times\mathcal{B}\to\mathcal{N},$

where the second morphism is the right $\mathcal{B}$-action of $\mathcal{N}$.
This defines the functor

$(-)_{\varphi}\colon\
\operatorname{Qcoh}\mathcal{B}\to\operatorname{Qcoh}\mathcal{A},$ (2.2)

and we call the functor $(-)_{\varphi}$ the restriction by $\varphi$.

Conversely, for an object $\mathcal{M}\in\operatorname{Qcoh}\mathcal{A}$, the
tensor product $\mathcal{M}\otimes_{\mathcal{A}}\mathcal{B}$ has a natural
right $\mathcal{B}$-action, and this defines the functor

$(-)\otimes_{\mathcal{A}}\mathcal{B}\colon\
\operatorname{Qcoh}\mathcal{A}\to\operatorname{Qcoh}\mathcal{B},$ (2.3)

and we call this functor the extension by $\varphi$. It is standard that we
have an adjunction

$\displaystyle(-)\otimes_{\mathcal{A}}\mathcal{B}\dashv(-)_{\varphi}.$

Although the following propositions might be well known, we give the proofs
for reader’s convenience.

######  Proposition 2.11.

Let $X$ be a scheme, and $\uprho\colon\mathcal{O}_{X}\to\mathcal{A}$ an
$X$-algebra.

  * $1.$

For every $\mathcal{M}\in\operatorname{Qcoh}\mathcal{A}$ and every point $x\in
X$, there are an open neighborhood $U$ of $x$ and a surjective
$\mathcal{A}|_{U}$-linear map

$\displaystyle\mathcal{A}|_{U}^{\oplus J}\twoheadrightarrow\mathcal{M}|_{U},$

where $J$ is a set.

  * $2.$

A right $\mathcal{A}$-module $\mathcal{M}$ is quasi-coherent if and only if
for every point $x\in X$ there are an open neighborhood $U$ and an exact
sequence

$\displaystyle\mathcal{A}|_{U}^{\oplus I}\to\mathcal{A}|_{U}^{\oplus
J}\to\mathcal{M}|_{U}\to 0.$

in the category of right $\mathcal{A}$-modules.

###### Proof.

(1) Let $\mathcal{M}\in\operatorname{Qcoh}\mathcal{A}$ and $x\in X$. Then
there are an open neighborhood $U$ of $x$, a set $J$ and an surjective
morphism $\pi\colon\mathcal{O}_{U}^{\oplus
J}\twoheadrightarrow\mathcal{M}_{\uprho}|_{U}$ in $\operatorname{Qcoh}X$.
Since the functor
$(-)\otimes_{\mathcal{O}_{U}}\mathcal{A}|_{U}\colon\operatorname{Qcoh}U\to\operatorname{Qcoh}\mathcal{A}|_{U}$
has a right adjoint functor, it is right exact and commutes with small direct
sums. Hence we have a natural isomorphism $\mathcal{O}_{U}^{\oplus
J}\otimes_{\mathcal{O}_{U}}\mathcal{A}_{U}\cong\mathcal{A}|_{U}^{\oplus J}$
and the morphism

$\displaystyle\pi\otimes_{\mathcal{O}_{U}}\mathcal{A}|_{U}\colon\
\mathcal{A}_{U}^{\oplus
J}\to\mathcal{M}_{\uprho}|_{U}\otimes_{\mathcal{O}_{U}}\mathcal{A}|_{U}$

is surjective. Composing this surjection with a natural surjective morphism
$\mathcal{M}_{\uprho}|_{U}\otimes_{\mathcal{O}_{U}}\mathcal{A}|_{U}\twoheadrightarrow\mathcal{M}|_{U}$;
$m\otimes a\mapsto ma$, we have a surjective morphism
$\mathcal{A}|_{U}^{\oplus J}\twoheadrightarrow\mathcal{M}|_{U}$.

(2) ($\Rightarrow$) Assume that a right $\mathcal{A}$-module $\mathcal{M}$ is
quasi-coherent and let $x\in X$ be a point. Then by (1) there are an open
neighbborhood $U$ of $x$ and a surjective $\mathcal{A}|_{U}$-linear map
$p\colon\mathcal{A}|_{U}^{\oplus J}\twoheadrightarrow\mathcal{M}|_{U}$. Denote
by $\mathcal{K}$ the Kernel of $p$ and by
$i\colon\mathcal{K}\hookrightarrow\mathcal{A}|_{U}^{\oplus J}$ the natural
inclusion. Then $\mathcal{K}$ is also a quasi-coherent $\mathcal{A}$-module,
and thus, by shrinking $U$ if necessary, we have a surjective
$\mathcal{A}|_{U}$-linear map $q\colon\mathcal{A}|_{U}^{\oplus
I}\twoheadrightarrow\mathcal{K}|_{U}$. Then the sequence

$\displaystyle\mathcal{A}|_{U}^{\oplus I}\xrightarrow{i\circ
q}\mathcal{A}|_{U}^{\oplus J}\xrightarrow{p}\mathcal{M}|_{U}\to 0$

is an exact.

($\Leftarrow$) Assume that a right $\mathcal{A}$-module $\mathcal{M}$ is
locally isomorphic to the cokernel of a morphism between free modules. Then,
since $\mathcal{A}$ is a quasi-coherent $\mathcal{O}_{X}$-module and
$\mathcal{M}$ is locally isomorphic to the cokernel of a morphism of quasi-
coherent modules, $\mathcal{M}$ is a locally quasi-coherent module. Thus
$\mathcal{M}$ is a quasi-coherent $\mathcal{O}_{X}$-module. ∎

###  2.3 Equivariant sheaves

We briefly recall the basics of equivariant quasi-coherent sheaves. For more
details, see, for example, [3, Section 2].

Let $G$ be an algebraic group, and denote by
$\varepsilon\colon\operatorname{Spec}k\to G$, $\mu\colon G\times G\to G$ and
$\iota\colon G\to G$ the morphisms defining the identity, the multiplication
and the inversion of $G$. Let $X$ be a scheme with an algebraic left
$G$-action $\sigma\colon G\times X\to X$. We denote by $\pi\colon G\times X\to
X$ the natural projection, and we write

$\displaystyle\varepsilon_{X}\colon\ X\to G\times X$

for the composition
$X\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\operatorname{Spec}k\times
X\xrightarrow{\varepsilon\times\operatorname{id}_{X}}G\times X$.

######  Definition 2.12.

A $G$-equivariant structure on a quasi-coherent sheaf
$\mathcal{F}\in\operatorname{Qcoh}X$ is an isomorphism

$\displaystyle\uptheta\colon\
\pi^{*}\mathcal{F}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\sigma^{*}\mathcal{F}$

of $\mathcal{O}_{G\times X}$-modules such that the equations

$\varphi^{*}\uptheta\circ(\operatorname{id}_{G}\times\pi)^{*}\uptheta=(\mu\times\operatorname{id}_{X})^{*}\uptheta\qquad\text{and}\qquad\varepsilon_{X}^{*}\uptheta=\operatorname{id}_{\mathcal{F}}$
(2.4)

hold, where $\varphi\colon G\times G\times X\to G\times X$ is the morphism
defined by $\varphi(g,h,x)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=(h,gx)$.

######  Remark 2.13.

For any closed point $g\in G$, there is the corresponding morphism
$g\colon\operatorname{Spec}k\to G$, and this induces the morphism
$g\times\operatorname{id}_{X}\colon X\to G\times X$. The pull-back of a
$G$-equivariant structure
$\uptheta\colon\pi^{*}\mathcal{F}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\sigma^{*}\mathcal{F}$
by $g\times\operatorname{id}_{X}$ defines the isomorphism

$\displaystyle\uptheta_{g}\colon\
\mathcal{F}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\sigma_{g}^{*}\mathcal{F},$

where $\sigma_{g}\colon X\to X$ is the isomorphism defined by
$\sigma_{g}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\sigma\circ(g\times\operatorname{id}_{X})$.
The pull-back of the first equation in (2.4) by the morphism $g\times
h\times\operatorname{id}_{X}\colon X\to G\times G\times X$ implies the
equation

$\displaystyle\sigma_{g}^{*}\uptheta_{h}\circ\uptheta_{g}=\uptheta_{gh},$

and the pull-back of the second equation is nothing but the equation
$\uptheta_{1_{G}}=\operatorname{id}_{\mathcal{F}}$.

A $G$-equivariant quasi-coherent sheaf is a pair $(\mathcal{F},\uptheta)$ of a
quasi-coherent sheaf $\mathcal{F}\in\operatorname{Qcoh}X$ and a
$G$-equivariant structure $\uptheta$ on $\mathcal{F}$. We say that
$(\mathcal{F},\uptheta)$ is coherent (resp. locally free) if $\mathcal{F}$ is
coherent (resp. locally free). If there is no risk of confusion, a
$G$-equivariant quasi-coherent sheaf $(\mathcal{F},\uptheta)$ is denoted
simply by $\mathcal{F}$.

Let $\big{(}\mathcal{F},\uptheta^{\mathcal{F}}\big{)}$ and
$\big{(}\mathcal{G},\uptheta^{\mathcal{G}}\big{)}$ be $G$-equivariant quasi-
coherent sheaves on $X$. A morphism $\varphi$: $\mathcal{F}\to\mathcal{G}$ of
$\mathcal{O}_{X}$-modules is said to be $G$-equivariant if the diagram

${\pi^{*}\mathcal{F}}$${\pi^{*}\mathcal{G}}$${\sigma^{*}\mathcal{F}}$${\sigma^{*}\mathcal{G}}$$\scriptstyle{\pi^{*}\varphi}$$\scriptstyle{\uptheta^{\mathcal{F}}}$$\scriptstyle{\uptheta^{\mathcal{G}}}$$\scriptstyle{\sigma^{*}\varphi}$
(2.5)

is commutative.

######  Remark 2.14.

Let $\big{(}\mathcal{F},\uptheta^{\mathcal{F}}\big{)}$ and
$\big{(}\mathcal{G},\uptheta^{\mathcal{G}}\big{)}$ be $G$-equivariant quasi-
coherent sheaves on $X$. Assume that $X$ is of finite type over $k$. Then the
set of closed points of $G\times X$ is equal to the set of pairs $(x,g)$ of
closed points $x\in X$ and $g\in G$. This implies that a morphism
$\varphi\colon\mathcal{F}\to\mathcal{G}$ of the $\mathcal{O}_{X}$-modules is
$G$-equivariant if and only if for any closed point $g\in G$ the pull-back

${\mathcal{F}}$${\mathcal{G}}$${\sigma_{g}^{*}\mathcal{F}}$${\sigma_{g}^{*}\mathcal{G}}$$\scriptstyle{\varphi}$$\scriptstyle{\uptheta^{\mathcal{F}}_{g}}$$\scriptstyle{\uptheta^{\mathcal{G}}_{g}}$$\scriptstyle{\sigma^{*}_{g}\varphi}$

of the diagram (2.5) by $g\times\operatorname{id}_{X}$ is commutative.

$G$-equivariant quasi-coherent sheaves on $X$ and $G$-equivariant morphisms
define the abelian category

$\displaystyle\operatorname{Qcoh}_{G}X.$

It is standard that the category $\operatorname{Qcoh}_{G}X$ is equivalent to
the category $\operatorname{Qcoh}[X/G]$ of quasi-coherent sheaves on the
quotient stack $[X/G]$;

$\operatorname{Qcoh}_{G}X\cong\operatorname{Qcoh}[X/G],$ (2.6)

and for $\mathcal{F}\in\operatorname{Qcoh}_{G}X$ we denote by
$[\mathcal{F}/G]\in\operatorname{Qcoh}[X/G]$ the image of $\mathcal{F}$ by the
equivalence (2.6). We denote by $\operatorname{coh}_{G}X$ the subcategory of
$G$-equivariant coherent sheaves on $X$. This category is an exact
subcategory, and if $X$ is noetherian, it is an abelian subcategory.
Furthermore, if $X$ is of finite type over $k$, by Remark 2.14, we have

$\operatorname{Hom}_{\operatorname{Qcoh}_{G}X}\big{(}\big{(}\mathcal{F},\uptheta^{\mathcal{F}}\big{)},\big{(}\mathcal{G},\uptheta^{\mathcal{G}}\big{)}\big{)}=\operatorname{Hom}_{\operatorname{Qcoh}X}(\mathcal{F},\mathcal{G})^{G},$
(2.7)

where the right hand side
$\operatorname{Hom}_{\operatorname{Qcoh}X}(\mathcal{F},\mathcal{G})^{G}$ is
the $G$-invariant subset with respect to the $G$-action on
$\operatorname{Hom}_{\operatorname{Qcoh}X}(\mathcal{F},\mathcal{G})$ defined
by $g\cdot\varphi\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=(\uptheta^{\mathcal{G}}_{g})^{-1}\circ\sigma_{g}^{*}\varphi\circ\uptheta^{\mathcal{F}}_{g}$.

Let $Y$ be another $G$-scheme, and $f\colon X\to Y$ a $G$-equivariant morphism
that is quasi-compact and quasi-separated. Then $f$ induces the direct image

$\displaystyle f_{*}\colon\
\operatorname{Qcoh}_{G}X\to\operatorname{Qcoh}_{G}Y$

and the inverse image

$\displaystyle f^{*}\colon\
\operatorname{Qcoh}_{G}Y\to\operatorname{Qcoh}_{G}X.$

It is standard that $f^{*}$ is a left adjoint functor of $f_{*}$.

Tensor products and sheaf Hom define bi-functors

$\displaystyle(-)\otimes_{X}(-)\colon\
\operatorname{Qcoh}_{G}X\times\operatorname{Qcoh}_{G}X\to\operatorname{Qcoh}_{G}X,$
$\displaystyle\operatorname{{\mathcal{H}}om}(-,-)\colon\
(\operatorname{coh}_{G}X)^{\rm
op}\times\operatorname{Qcoh}_{G}X\to\operatorname{Qcoh}_{G}X,$

and it is also standard that for any $\mathcal{F}\in\operatorname{coh}_{G}X$,
the functor
$\operatorname{{\mathcal{H}}om}(\mathcal{F},-)\colon\operatorname{Qcoh}_{G}X\to\operatorname{Qcoh}_{G}X$
is right adjoint to the functor
$(-)\otimes_{X}\mathcal{F}\colon\operatorname{Qcoh}_{G}X\to\operatorname{Qcoh}_{G}X$.

###  2.4 Tilting objects and derived equivalences

We recall that tilting objects induce derived equivalences. Throughout this
subsection, $\mathcal{A}$ is an abelian category with small coproducts and
enough injectives, and we assume that small coproducts of families of short
exact sequences in $\mathcal{A}$ are exact. These properties of $\mathcal{A}$
are satisfied if $\mathcal{A}$ is a Grothendieck category.

######  Definition 2.15.

Let $T\in\mathcal{A}$ be an object. The object $T$ is called a tilting object
if the following conditions are satisfied:

  * $1.$

$\operatorname{Ext}_{\mathcal{A}}^{i}(T,T)=0$ for all $i>0$.

  * $2.$

$T$ is compact in $\operatorname{D}(\mathcal{A})$, i.e., the natural map

$\displaystyle\bigoplus_{i\in\mathcal{I}}\operatorname{Hom}_{\operatorname{D}(\mathcal{A})}(T,A_{i}^{\bullet})\to\operatorname{Hom}_{\operatorname{D}(\mathcal{A})}\Big{(}T,\bigoplus_{i\in\mathcal{I}}A_{i}^{\bullet}\Big{)}$

is an isomorphism of abelian groups for any set $\mathcal{I}$ and any family
$\\{A_{i}^{\bullet}\\}_{i\in I}$.

  * $3.$

$T$ is a generator of $\operatorname{D}(\mathcal{A})$, i.e., for an object
$A^{\bullet}\in\operatorname{D}(\mathcal{A})$,

$\operatorname{{\mathbb{R}}Hom}(T,A^{\bullet})\cong
0in\operatorname{D}\big{(}\operatorname{Mod}\operatorname{End}_{\mathcal{A}}(T)\big{)}$

implies $A^{\bullet}\cong 0$ in $\operatorname{D}(\mathcal{A})$.

We call $T$ a partial tilting object if the conditions (1) and (2) are
satisfied.

For an object $T\in\mathcal{A}$, we denote by

$\displaystyle\operatorname{Add}T$

the smallest additive subcategory containing all direct summands of small
coproducts of $T$. We write $\operatorname{add}T\subset\operatorname{Add}T$
for the subcategory consisting of direct summands of finite coproducts of $T$.
If $T$ is a partial tilting object, then for arbitrary sets $\mathcal{I}$ and
$\mathcal{J}$, we have isomorphisms

$\displaystyle\operatorname{Ext}^{n}_{\mathcal{A}}\Big{(}\bigoplus_{i\in\mathcal{I}}T,\bigoplus_{j\in\mathcal{J}}T\Big{)}$
$\displaystyle\cong\operatorname{Hom}_{\operatorname{D}(\mathcal{A})}\Big{(}\bigoplus_{i\in\mathcal{I}}T,\bigoplus_{j\in\mathcal{J}}(T[n])\Big{)}$
$\displaystyle\cong\prod_{i\in\mathcal{I}}\operatorname{Hom}_{\operatorname{D}(\mathcal{A})}\Big{(}T,\bigoplus_{j\in\mathcal{J}}(T[n])\Big{)}$
$\displaystyle\cong\prod_{i\in\mathcal{I}}\Big{(}\bigoplus_{j\in\mathcal{J}}\operatorname{Hom}_{\operatorname{D}(\mathcal{A})}(T,T[n])\Big{)}.$

This implies that for any $X,Y\in\operatorname{Add}T$, we have
$\operatorname{Ext}_{\mathcal{A}}^{n}(X,Y)=0$ for any $n>0$. Thus any short
exact sequence $0\to X\to Z\to Y\to 0$ in $\mathcal{A}$ with
$X,Y\in\operatorname{Add}T$ splits, and in particular $\operatorname{Add}T$ is
an exact subcategory of $\mathcal{A}$.

In the remaining of this subsection, $T\in\mathcal{A}$ is a partial tilting
object, and we assume that every right module over the endomorphism ring
$\Lambda\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{End}_{\mathcal{A}}(T)$
is of finite projective dimension. Consider the functor

$\displaystyle{\operatorname{\sf{F}}}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{Hom}_{\mathcal{A}}(T,-)\colon\
\mathcal{A}\to\operatorname{Mod}\Lambda$

and assume that

  * $(i)$

$\operatorname{\sf{F}}$ has a left adjoint functor
$\operatorname{\sf{G}}\colon\operatorname{Mod}\Lambda\to\mathcal{A}$ such that
the adjunction morphisms
$\sigma({\Lambda})\colon\Lambda\to\operatorname{\sf{F}}\operatorname{\sf{G}}(\Lambda)$
and $\tau(T)\colon\operatorname{\sf{G}}\operatorname{\sf{F}}(T)\to T$ are
isomorphisms.

  * $(ii)$

The right derived functor
$\mathbb{R}\operatorname{\sf{F}}\colon\operatorname{D}^{+}(\mathcal{A})\to\operatorname{D}^{+}(\operatorname{Mod}\Lambda)$
restricts to the functor

$\displaystyle\mathbb{R}\operatorname{\sf{F}}\colon\
\operatorname{D^{b}}(\mathcal{A})\to\operatorname{D^{b}}(\operatorname{Mod}\Lambda)$

between bounded derived categories.

Note that both of the functors $\operatorname{\sf{F}}$ and
$\operatorname{\sf{G}}$ commute with small coproducts since $T$ is compact and
$\operatorname{\sf{G}}$ admits a right adjoint functor. In particular,
$\operatorname{\sf{F}}$ and $\operatorname{\sf{G}}$ restrict to the following
functors

$\displaystyle\operatorname{\sf{F}}\colon\
\operatorname{Add}T\to\operatorname{Proj}\Lambda,$
$\displaystyle\operatorname{\sf{G}}\colon\
\operatorname{Proj}\Lambda\to\operatorname{Add}T,$

where $\operatorname{Proj}\Lambda$ is the category of projective right
$\Lambda$-modules.

######  Lemma 2.16.

The functor
$\operatorname{\sf{F}}\colon\operatorname{Add}T\to\operatorname{Proj}\Lambda$
is an equivalence, and $\operatorname{\sf{G}}\cong\operatorname{\sf{F}}^{-1}$.

###### Proof.

The adjunction morphisms $\sigma({P})\colon
P\to\operatorname{\sf{F}}\operatorname{\sf{G}}(P)$ and
$\tau(S)\colon\operatorname{\sf{G}}\operatorname{\sf{F}}(S)\to S$ are
bijective for any $P\in\operatorname{Mod}\Lambda$ and
$S\in\operatorname{Add}T$, since $\sigma({\Lambda})$ and $\tau({T})$ are
bijective and $\operatorname{Proj}\Lambda=\operatorname{Add}\Lambda$. Hence,
the functors $\operatorname{\sf{F}}$ and $\operatorname{\sf{G}}$ are
equivalences, and $\operatorname{\sf{G}}\cong\operatorname{\sf{F}}^{-1}$. ∎

By our assumptions, the functor $\operatorname{\sf{F}}$ defines the right
derived functor

$\displaystyle\mathbb{R}\operatorname{\sf{F}}\colon\
\operatorname{D^{b}}(\mathcal{A})\to\operatorname{D^{b}}(\operatorname{Mod}\Lambda)$

between bounded derived categories, and the functor $\operatorname{\sf{G}}$
also defines the left derived functor

$\displaystyle\mathbb{L}\operatorname{\sf{G}}\colon\
\operatorname{D^{b}}(\operatorname{Mod}\Lambda)\to\operatorname{D^{b}}(\mathcal{A})$

between bounded derived categories. The following is well known to experts.

######  Theorem 2.17.

The functor
$\mathbb{L}\operatorname{\sf{G}}\colon\operatorname{D^{b}}(\operatorname{Mod}\Lambda)\to\operatorname{D^{b}}(\mathcal{A})$
is fully faithful. Furthermore, if $T$ is a tilting object, then
$\mathbb{L}\operatorname{\sf{G}}$ is an equivalence and
$\mathbb{R}\operatorname{\sf{F}}\cong(\mathbb{L}\operatorname{\sf{G}})^{-1}$.

###### Proof.

We have the following commutative diagram

${\operatorname{K^{b}}(\operatorname{Proj}\Lambda)}$${\operatorname{K^{b}}(\operatorname{Add}T)}$${\operatorname{D^{b}}(\operatorname{Mod}\Lambda)}$${\operatorname{D^{b}}(\mathcal{A}),}$$\scriptstyle{\operatorname{\sf{G}}}$$\scriptstyle{\mathbb{L}\operatorname{\sf{G}}\,\,\,\,\,\,}$

where the vertical arrows are the natural quotient functors. The left vertical
functor is an equivalence, since every right $\Lambda$-module has a finite
projective resolution. Moreover, by Lemma 2.16, the top horizontal functor is
also an equivalence. Hence, for the former assertion, it is enough to show
that the natural functor

$\displaystyle\operatorname{K^{b}}(\operatorname{Add}T)\to\operatorname{D^{b}}(\mathcal{A})$

is fully faithful, and this follows from an identical argument as in [6,
Chapter III, Lemma 2.1].

For the latter assertion, assume that $T$ is a tilting object. It is enough to
prove that $\mathbb{R}\operatorname{\sf{F}}$ is also fully faithful. For any
$A^{\bullet}\in\operatorname{D}(\mathcal{A})$, consider the following triangle

$\displaystyle\mathbb{L}\operatorname{\sf{G}}\mathbb{R}\operatorname{\sf{F}}(A^{\bullet})\xrightarrow{\tau}A^{\bullet}\to
C(\tau)\to\mathbb{L}\operatorname{\sf{G}}\mathbb{R}\operatorname{\sf{F}}(A^{\bullet})[1],$

where $\tau$ is the adjunction morphism. Applying the functor
$\mathbb{R}\operatorname{\sf{F}}$ to the above triangle and using the natural
isomorphism
$\mathbb{R}\operatorname{\sf{F}}\circ\,\mathbb{L}\operatorname{\sf{G}}\cong\operatorname{id}$,
we see that $\mathbb{R}\operatorname{\sf{F}}(C(\tau))\cong 0$ in
$\operatorname{D^{b}}(\operatorname{Mod}\Lambda)$. Since $T$ is a generator of
$\operatorname{D}(\mathcal{A})$, this implies that $C(\tau)\cong 0$ in
$\operatorname{D}(\mathcal{A})$. Hence the adjunction morphism $\tau$ is an
isomorphism. This completes the proof. ∎

######  Remark 2.18.

Our assumption that every $\Lambda$-module has a finite projective resolution
might be weakened. However, the result under our assumption is enough for our
purpose.

##  3 Equivariant algebras and equivariant tilting modules

In this section, we introduce equivariant algebras and equivariant modules,
and define several functors between equivariant modules that are
generalizations of functors in [3, Section 2]. We also show that equivariant
tilting modules induce derived equivalences of equivariant modules, which are
simultaneous generalizations of tilting equivalences induced by tilting
bundles on schemes and tilting modules over noncommutative algebras.

###  3.1 Equivariant modules over equivariant algebras

In this subsection, we give the definitions of equivariant algebras and
equivariant modules, and discuss basic properties. Notation is the same as in
Section 2.3. Let $\uprho\colon\mathcal{O}_{X}\to\mathcal{A}$ be an
$X$-algebra.

######  Definition 3.1.

A $G$-equivariant structure on the $X$-algebra
$\uprho\colon\mathcal{O}_{X}\to\mathcal{A}$ is an isomorphism

$\displaystyle\uptheta^{\mathcal{A}}\colon\
\pi^{*}\mathcal{A}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\sigma^{*}\mathcal{A}$

of sheaves of algebras on $G\times X$ such that it is $\mathcal{O}_{G\times
X}$-linear and it gives a $G$-equivariant structure on the
$\mathcal{O}_{X}$-module $\mathcal{A}$. The pair
$\big{(}\mathcal{A},\uptheta^{\mathcal{A}}\big{)}$ is called a $G$-equivariant
algebra over $X$, or $G$-equivariant $X$-algebra. We say that
$\big{(}\mathcal{A},\uptheta^{\mathcal{A}}\big{)}$ is coherent if the
underlying $X$-algebra $\mathcal{A}$ is coherent $\mathcal{O}_{X}$-module. We
write simply $\mathcal{A}$ for
$\big{(}\mathcal{A},\uptheta^{\mathcal{A}}\big{)}$ if no confusion seems
likely to occur.

######  Remark 3.2.

Note that for an isomorphism
$\upphi\colon\pi^{*}\mathcal{A}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\sigma^{*}\mathcal{A}$
of sheaves of algebras which defines a $G$-equivariant structure on the quasi-
coherent sheaf $\mathcal{A}$, $\upphi$ is $\mathcal{O}_{G\times X}$-linear if
and only if the morphism $\uprho\colon\mathcal{O}_{X}\to\mathcal{A}$ is a
$G$-equivariant morphism of $G$-equivariant quasi-coherent sheaves
$(\mathcal{O}_{X},\uptheta^{\rm can})$ and $(\mathcal{A},\upphi)$, where
$\uptheta^{\rm can}$ is the canonical equivariant structure on the structure
sheaf $\mathcal{O}_{X}$ induced by the $G$-action on $X$.

Let $X=\operatorname{Spec}R$ be an affine scheme with an action from an affine
algebraic group $G$, and set $R_{G}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=k[G]\otimes_{k}R$,
where $k[G]$ denotes the coordinate ring of $G$. We denote by

$\displaystyle R_{G}^{\pi}\qquad\big{(}\text{resp.}~{}R_{G}^{\sigma}\big{)}$

the $R$-algebra $R\to R_{G}$ induced by the morphism $\pi\colon G\times X\to
X$ (resp. $\sigma\colon G\times X\to X$). If
$\big{(}\mathcal{A},\uptheta^{\mathcal{A}}\big{)}$ is a $G$-equivariant
$X$-algebra, taking global sections induces an $R$-algebra $A\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\mathcal{A}(X)$
and an isomorphism

$\displaystyle\uptheta^{A}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\uptheta^{\mathcal{A}}(X)\colon\
A\otimes_{R}R_{G}^{\pi}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,A\otimes_{R}R_{G}^{\sigma}$

of $R_{G}$-algebras such that $\uptheta^{A}$ gives a $G$-equivariant structure
on the $R$-module $A$. This yields the following definition.

######  Definition 3.3.

We call a pair $(\Lambda,\uptheta^{\Lambda})$ a $G$-equivariant $R$-algebra,
if $\Lambda$ is an $R$-algebra and
$\uptheta^{\Lambda}\colon\Lambda\otimes_{R}R_{G}^{\pi}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\Lambda\otimes_{R}R_{G}^{\sigma}$
is an isomorphism of $R_{G}$-algebras such that it defines a $G$-equivariant
structure on the $R$-module $\Lambda$.

######  Remark 3.4.

For any commutative ring $S$, via the natural equivalence
$\operatorname{Mod}S\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\operatorname{Qcoh}\operatorname{Spec}S$,
to give a $G$-equivariant algebra over $\operatorname{Spec}S$ is equivalent to
give a $G$-equivariant $S$-algebra.

######  Example 3.5.

Notation is the same as above.

  1. 1.

Let $\mathcal{F}\in\operatorname{coh}_{G}X$ be a $G$-equivariant coherent
sheaf on $X$. Then the algebra

$\displaystyle\mathcal{A}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{{\mathcal{E}}nd}_{X}(\mathcal{F})$

over $X$ has a natural $G$-equivariant structure. Indeed, the $G$-equivariant
structure

$\displaystyle\uptheta\colon\
\pi^{*}\operatorname{{\mathcal{E}}nd}_{X}(\mathcal{F})\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\sigma^{*}\operatorname{{\mathcal{E}}nd}_{X}(\mathcal{F})$

on the $G$-equivariant coherent sheaf
$\operatorname{{\mathcal{E}}nd}_{X}(\mathcal{F})\in\operatorname{coh}_{G}X$ is
a morphism of algebras, and so the pair $(\mathcal{A},\uptheta$) is a
$G$-equivariant algebra over $X$.

  2. 2.

Let $\big{(}\mathcal{A},\uptheta^{\mathcal{A}}\big{)}$ be a $G$-equivariant
$X$-algebra. If $Y$ is another $G$-scheme and $f\colon X\to Y$ is a
$G$-equivariant morphism that is quasi-separated and quasi-compact, then the
direct image $\big{(}f_{*}\mathcal{A},(\operatorname{id}_{G}\times
f)_{*}\uptheta^{\mathcal{A}}\big{)}$ is a $G$-equivariant $Y$-algebra. If
$\mathcal{A}$ is coherent and $f$ is proper, the algebra $f_{*}\mathcal{A}$ is
also coherent. Similarly, if $g\colon Y\to X$ is a $G$-equivariant morphism,
then the pull-back $\big{(}g^{*}\mathcal{A},(\operatorname{id}_{G}\times
g)^{*}\uptheta^{\mathcal{A}}\big{)}$ is a $G$-equivariant $Y$-algebra, and it
is coherent if $\mathcal{A}$ is coherent.

We define equivariant modules over equivariant algebras.

######  Definition 3.6.

Let $\big{(}\mathcal{A},\uptheta^{\mathcal{A}}\big{)}$ be a $G$-equivariant
$X$-algebra.

  * 1.

A $G$-equivariant structure on a quasi-coherent right $\mathcal{A}$-module
$\mathcal{M}$ is an isomorphism

$\displaystyle\uptheta^{\mathcal{M}}\colon\
\pi^{*}\mathcal{M}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\sigma^{*}\mathcal{M}$

of $\pi^{*}\mathcal{A}$-modules such that $\uptheta^{\mathcal{M}}$ satisfies
the condition (2.4), where the $\pi^{*}\mathcal{A}$-module structure on the
$\sigma^{*}\mathcal{A}$-module $\sigma^{*}\mathcal{M}$ is given by
$\uptheta^{\mathcal{A}}$. We call the pair
$\big{(}\mathcal{M},\uptheta^{\mathcal{M}}\big{)}$ a $G$-equivariant quasi-
coherent right $\mathcal{A}$-module, or simply $G$-equivariant
$\mathcal{A}$-module. If the underlying sheaf $\mathcal{M}$ of modules is
coherent, we call $\big{(}\mathcal{M},\uptheta^{\mathcal{M}}\big{)}$ a
$G$-equivariant coherent $\mathcal{A}$-module. We sometimes write just
$\mathcal{M}$ for $\big{(}\mathcal{M},\uptheta^{\mathcal{M}}\big{)}$.

  * 2.

Let $\big{(}\mathcal{M},\uptheta^{\mathcal{M}}\big{)}$ and
$\big{(}\mathcal{N},\uptheta^{\mathcal{N}}\big{)}$ be $G$-equivariant
$\mathcal{A}$-modules. A morphism $\varphi\colon\mathcal{M}\to\mathcal{N}$ of
$\mathcal{A}$-modules is $G$-equivariant if the following diagram

${\pi^{*}\mathcal{M}}$${\pi^{*}\mathcal{N}}$${\sigma^{*}\mathcal{M}}$${\sigma^{*}\mathcal{N}}$$\scriptstyle{\pi^{*}\varphi}$$\scriptstyle{\uptheta^{\mathcal{M}}}$$\scriptstyle{\uptheta^{\mathcal{N}}}$$\scriptstyle{\sigma^{*}\varphi}$

is commutative.

  * 3.

We denote by

$\displaystyle\operatorname{Qcoh}_{G}\mathcal{A}$

the category of $G$-equivariant $\mathcal{A}$-modules whose morphisms are
$G$-equivariant, and write $\operatorname{coh}_{G}\mathcal{A}$ for the full
subcategory of $G$-equivariant coherent $\mathcal{A}$-modules.

######  Remark 3.7.

(1) If $X$ is a $G$-scheme, then $\mathcal{O}_{X}$ is a natural
$G$-equivariant algebra over $X$. For any
$(\mathcal{F},\uptheta)\in\operatorname{Qcoh}_{G}X$, the $G$-equivariant
structure $\uptheta$ is automatically $\pi^{*}\mathcal{O}_{X}$-linear.
Therefore, we have a natural identification

$\displaystyle\operatorname{Qcoh}_{G}\mathcal{O}_{X}=\operatorname{Qcoh}_{G}X.$

Hence $G$-equivariant modules are generalizations of $G$-equivariant quasi-
coherent sheaves.

######  Definition 3.8.

Let $X=\operatorname{Spec}R$ be an affine scheme with an action from an affine
algebraic group $G$, and $(\Lambda,\uptheta^{\Lambda})$ a $G$-equivariant
$R$-algebra. A $G$-equivariant structure on a right $\Lambda$-module $M$ is an
isomorphism

$\displaystyle\uptheta^{M}\colon\
M\otimes_{R}R_{G}^{\pi}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,M\otimes_{R}R_{G}^{\sigma}$

of $\big{(}\Lambda\otimes_{R}R_{G}^{\pi}\big{)}$-modules, where
$\big{(}\Lambda\otimes_{R}R_{G}^{\sigma}\big{)}$-module
$M\otimes_{R}R_{G}^{\sigma}$ is considered as a
$\big{(}\Lambda\otimes_{R}R_{G}^{\pi}\big{)}$-module via the ring isomorphism
$\uptheta^{\Lambda}\colon\Lambda\otimes_{R}R_{G}^{\pi}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\Lambda\otimes_{R}R_{G}^{\sigma}$.
We call such a pair $\big{(}M,\uptheta^{M}\big{)}$ a $G$-equivariant
$\Lambda$-module. We denote by

$\displaystyle\operatorname{Mod}_{G}\Lambda$

the category of $G$-equivariant $\Lambda$-modules whose morphisms are defined
similarly to Definition 3.6, and we denote by

$\displaystyle\operatorname{mod}_{G}\Lambda\subset\operatorname{Mod}_{G}\Lambda$

the full subcategory consisting of equivariant modules that are finitely
generated over $\Lambda$.

######  Remark 3.9.

Let $X=\operatorname{Spec}R$ be an affine scheme with an action from an affine
algebraic group $G$, and $\big{(}\mathcal{A},\uptheta^{\mathcal{A}}\big{)}$ a
$G$-equivariant $X$-algebra. Then it induces a $G$-equivariant $R$-algebra
$A\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\mathcal{A}(X)$,
and we have a natural equivalence

$\displaystyle\operatorname{Qcoh}_{G}\mathcal{A}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\operatorname{Mod}_{G}A$

of abelian categories, which restricts to an equivalence
$\operatorname{coh}_{G}\mathcal{A}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\operatorname{mod}_{G}A$.

Let $\big{(}\mathcal{M},\uptheta^{\mathcal{M}}\big{)}$ be a $G$-equivariant
$\mathcal{A}$-module. Recall from Remark 2.13 that for each closed point $g\in
G$, we have the induced isomorphism

$\displaystyle\uptheta_{g}^{\mathcal{M}}\colon\
\mathcal{M}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\sigma_{g}^{*}\mathcal{M}.$

If $\big{(}\mathcal{N},\uptheta^{\mathcal{N}}\big{)}$ is another
$G$-equivariant $\mathcal{A}$-module, we have the $G$-action on
$\operatorname{Hom}_{\operatorname{Qcoh}\mathcal{A}}(\mathcal{M},\mathcal{N})$
defined by

$\displaystyle g\cdot\varphi\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\big{(}\uptheta^{\mathcal{N}}_{g}\big{)}^{-1}\circ\sigma_{g}^{*}\varphi\circ\uptheta^{\mathcal{M}}_{g}.$

The following is a generalization of the equality in (2.7).

######  Proposition 3.10.

Assume that $X$ is of finite type over $k$. We have

$\displaystyle\operatorname{Hom}_{\operatorname{Qcoh}_{G}\mathcal{A}}\big{(}\big{(}\mathcal{M},\uptheta^{\mathcal{M}}\big{)},\big{(}\mathcal{N},\uptheta^{\mathcal{N}}\big{)}\big{)}=\operatorname{Hom}_{\operatorname{Qcoh}\mathcal{A}}(\mathcal{M},\mathcal{N})^{G}.$

Moreover, if $G$ is reductive, we have

$\displaystyle\operatorname{Ext}_{\operatorname{Qcoh}_{G}\mathcal{A}}^{i}\big{(}\big{(}\mathcal{M},\uptheta^{\mathcal{M}}\big{)},\big{(}\mathcal{N},\uptheta^{\mathcal{N}}\big{)}\big{)}=\operatorname{Ext}_{\operatorname{Qcoh}\mathcal{A}}^{i}(\mathcal{M},\mathcal{N})^{G}.$

for any $i\geq 0$.

###### Proof.

The first equality follows from an identical argument as in Remark 2.14. Since
$G$ is reductive, the functor of taking $G$-invariant parts is exact, and thus
the second equality follows from the first one. ∎

######  Example 3.11.

Notation is the same as in Example 3.5(1).

  * $1.$

For any $\mathcal{G}\in\operatorname{Qcoh}_{G}X$, the quasi-coherent
$\mathcal{A}$-module

$\displaystyle\operatorname{{\mathcal{H}}om}_{X}(\mathcal{F},\mathcal{G})$

has a natural $G$-equivariant structure induced by the $G$-equivariant
structure on

$\displaystyle\operatorname{{\mathcal{H}}om}_{X}(\mathcal{F},\mathcal{G})\in\operatorname{Qcoh}_{G}X.$

  * $2.$

For $\mathcal{M}\in\operatorname{Mod}_{G}\mathcal{A}$ and
$\mathcal{F}\in\operatorname{Qcoh}_{G}X$, the tensor product

$\displaystyle\mathcal{F}\otimes_{X}\mathcal{M}$

of $\mathcal{O}_{X}$-modules is an $\mathcal{A}$-module with a natural
$G$-equivariant structure induced by the $G$-equivariant structure on
$\mathcal{F}\otimes_{X}\mathcal{M}\in\operatorname{Qcoh}_{G}X$.

If $\big{(}\mathcal{A},\uptheta^{\mathcal{A}}\big{)}$ is a $G$-equivariant
$X$-algebra, the ring homomorphism $\uprho\colon\mathcal{O}_{X}\to\mathcal{A}$
is $G$-equivariant by Remark 3.2. We define the sheaf of rings

$\displaystyle[\mathcal{A}/G]$

on the quotient stack $[X/G]$ to be the image of
$\mathcal{A}\in\operatorname{Qcoh}_{G}X$ by the equivalence

$\displaystyle[-/G]\colon\
\operatorname{Qcoh}_{G}X\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\operatorname{Qcoh}[X/G]$

in (2.6). Since $[\mathcal{O}_{X}/G]$ is canonically isomorphic to the
structure sheaf $\mathcal{O}_{[X/G]}$, we have the following morphism

$\displaystyle[\uprho/G]\colon\ \mathcal{O}_{[X/G]}\to[\mathcal{A}/G]$

of sheaf of rings and this makes the sheaf $[\mathcal{A}/G]$ an algebra over
$[X/G]$. The following is a generalization of (2.6), and it follows from
Proposition A.11.

######  Proposition 3.12.

We have an equivalence

$\displaystyle\operatorname{Qcoh}_{G}\mathcal{A}\cong\operatorname{Qcoh}[\mathcal{A}/G]$

of abelian categories.

######  Remark 3.13.

If an affine algebraic group $G$ is abelian, by a similar argument as in [4,
Section 2.1] $G$-equivariant algebras correspond to $\widehat{G}$-graded
algebras, where $\widehat{G}$ is the character group of $G$. Since we do not
need this correspondence in the present paper, we do not give a formulation of
the correspondence and its proof.

###  3.2 Functors of equivariant modules

In this subsection, we define fundamental functors between equivariant
modules. Let $X$ be a quasi-compact and quasi-separated scheme, and $G$ an
algebraic group acting on $X$ by $\sigma\colon G\times X\to X$. Denote by
$\pi\colon G\times X\to X$ the natural projection.

####  3.2.1 Restrictions and extensions

Let $\mathcal{A}$ and $\mathcal{B}$ be $G$-equivariant $X$-algebras, and let
$\varphi\colon\mathcal{A}\to\mathcal{B}$ be a $G$-equivariant morphism of
$X$-algebras. The functors in (2.2) and (2.3) define the restriction by
$\varphi$

$\displaystyle(-)_{\varphi}\colon\
\operatorname{Qcoh}_{G}\mathcal{B}\to\operatorname{Qcoh}_{G}\mathcal{A}$

and the extension by $\varphi$

$\displaystyle(-)\otimes_{\mathcal{A}}\mathcal{B}\colon\
\operatorname{Qcoh}_{G}\mathcal{A}\to\operatorname{Qcoh}_{G}\mathcal{B},$

and we have an adjunction

$(-)\otimes_{\mathcal{A}}\mathcal{B}\dashv(-)_{\varphi}.$ (3.1)

####  3.2.2 Direct image functors and pull-back functors

Let $Y$ be another quasi-compact and quasi-separated $G$-scheme, and $f\colon
X\to Y$ a $G$-equivariant morphism. Let $\mathcal{A}$ be a $G$-equivariant
$X$-algebra, and $\mathcal{B}$ a $G$-equivariant $Y$-algebra. Recall from
Example 3.5(2) that the push-forward $f_{*}\mathcal{A}$ is a $G$-equivariant
$Y$-algebra and that the pull-back $f^{*}\mathcal{B}$ is a $G$-equivariant
$X$-algebra. The push-forwards and the pull-backs of $G$-equivariant modules
define the following additive functors

$\displaystyle f_{\star}\colon\
\operatorname{Qcoh}_{G}\mathcal{A}\to\operatorname{Qcoh}_{G}f_{*}\mathcal{A},$
$\displaystyle f^{*}\colon\
\operatorname{Qcoh}_{G}\mathcal{B}\to\operatorname{Qcoh}_{G}f^{*}\mathcal{B}.$

Using natural morphisms of equivariant algebras $\varphi\colon
f^{-1}f_{*}\mathcal{A}\to\mathcal{A}$ and $\psi\colon\mathcal{B}\to
f_{*}f^{*}\mathcal{B}$, we define additive functors

$\displaystyle f^{\star}\colon\
\operatorname{Qcoh}_{G}f_{*}\mathcal{A}\to\operatorname{Qcoh}_{G}\mathcal{A},$
$\displaystyle f_{*}\colon\
\operatorname{Qcoh}_{G}f^{*}\mathcal{B}\to\operatorname{Qcoh}_{G}\mathcal{B}$

by $f^{\star}(-)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=f^{-1}(-)\otimes_{f^{-1}f_{*}\mathcal{A}}\mathcal{A}$
and $f_{*}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=(-)_{\psi}\circ
f_{\star}$. By standard arguments, we see that the above functors induce
adjunctions;

$\displaystyle f^{*}\dashv f_{*}\qquad\text{and}\qquad f^{\star}\dashv
f_{\star}.$

Let $G^{\prime}$ be another algebraic group acting on another scheme $Z$, and
let $\alpha\colon G\to G^{\prime}$ be a morphism of algebraic groups. Let
$h\colon X\to Z$ be a morphism of schemes, and $\mathcal{C}$ an
$G^{\prime}$-equivariant $Z$-algebra. We say that $h$ is $\alpha$-equivariant
if we have $h(gx)=\alpha(g)h(x)$ for all $(g,x)\in G\times X$. If $h$ is
$\alpha$-equivariant, then $h$ induces the pull-back functor

$\displaystyle h^{*}_{\alpha}\colon\
\operatorname{Qcoh}_{G^{\prime}}Z\to\operatorname{Qcoh}_{G}X$

defined by $h^{*}(\mathcal{F},\uptheta)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=(h^{*}\mathcal{F},(\alpha\times
h)^{*}\uptheta)$, and we have the associated $G$-equivariant $X$-algebra
$h^{*}\mathcal{C}$. This functor extends to the functor

$h^{*}_{\alpha}\colon\
\operatorname{Qcoh}_{G^{\prime}}\mathcal{C}\to\operatorname{Qcoh}_{G}h^{*}\mathcal{C}.$

####  3.2.3 Tensor products and sheaf Homs

Let $\uprho\colon\mathcal{O}_{X}\to\mathcal{A}$ and
$\uprho^{\prime}\colon\mathcal{O}_{X}\to\mathcal{A}^{\prime}$ be
$G$-equivariant $X$-algebras. A $G$-equivariant quasi-coherent $($resp.
coherent$)$ $(\mathcal{A}^{\prime},\mathcal{A})$-bimodule is a pair
$\big{(}\mathcal{M},\uptheta^{\mathcal{M}}\big{)}$ of a
$(\mathcal{A}^{\prime},\mathcal{A})$-bimodule $\mathcal{M}$ such that the left
action of $\mathcal{O}_{X}$ via $\uprho^{\prime}$ coincides with the right
action of $\mathcal{O}_{X}$ via $\uprho$ and that $\mathcal{M}$ is a quasi-
coherent (resp. coherent) sheaf on $X$ and an isomorphism
$\uptheta\colon\pi^{*}\mathcal{M}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\sigma^{*}\mathcal{M}$
of $(\pi^{*}\mathcal{A}^{\prime},\pi^{*}\mathcal{A})$-bimodules such that
$\uptheta^{\mathcal{M}}$ satisfies the condition (2.4). We denote by
$\operatorname{Qcoh}_{G}(\mathcal{A}^{\prime},\mathcal{A})$ the category of
$G$-equivariant quasi-coherent $(\mathcal{A}^{\prime},\mathcal{A})$-bimodules,
and $\operatorname{coh}_{G}(\mathcal{A}^{\prime},\mathcal{A})$ the full
subcategory of $G$-equivariant coherent
$(\mathcal{A}^{\prime},\mathcal{A})$-bimodules.

Let $\mathcal{F}\in\operatorname{Qcoh}_{G}(\mathcal{A}^{\prime},\mathcal{A})$,
$\mathcal{M}\in\operatorname{Qcoh}_{G}\mathcal{A}^{\prime}$ and
$\mathcal{N}\in\operatorname{Qcoh}_{G}\mathcal{A}$. Then we have the tensor
product

$\displaystyle\mathcal{M}\otimes_{\mathcal{A}^{\prime}}\mathcal{F}\in\operatorname{Qcoh}_{G}\mathcal{A}.$

If $\mathcal{F}\in\operatorname{coh}_{G}(\mathcal{A}^{\prime},\mathcal{A})$,
we also have the sheaf Hom

$\displaystyle\operatorname{{\mathcal{H}}om}_{\mathcal{A}}(\mathcal{F},\mathcal{N})\in\operatorname{Qcoh}_{G}\mathcal{A}^{\prime},$

and there is a functorial isomorphism

$\operatorname{Hom}_{\operatorname{Qcoh}_{G}\mathcal{A}}\big{(}\mathcal{M}\otimes_{\mathcal{A}^{\prime}}\mathcal{F},\mathcal{N}\big{)}\cong\operatorname{Hom}_{\operatorname{Qcoh}_{G}\mathcal{A}^{\prime}}\big{(}\mathcal{M},\operatorname{{\mathcal{H}}om}_{\mathcal{A}}(\mathcal{F},\mathcal{N})\big{)}.$

In other words, if
$\mathcal{F}\in\operatorname{coh}_{G}(\mathcal{A}^{\prime},\mathcal{A})$, the
functor

$\displaystyle(-)\otimes_{\mathcal{A}^{\prime}}\mathcal{F}\colon\
\operatorname{Qcoh}_{G}\mathcal{A}^{\prime}\to\operatorname{Qcoh}_{G}\mathcal{A}$

is left adjoint to the functor

$\displaystyle\operatorname{{\mathcal{H}}om}_{\mathcal{A}}(\mathcal{F},-)\colon\
\operatorname{Qcoh}_{G}\mathcal{A}\to\operatorname{Qcoh}_{G}\mathcal{A}^{\prime}.$

The following is a version of projection formula for equivariant
$\mathcal{A}$-modules.

######  Lemma 3.14 (Projection formula).

Let $Y$ be a quasi-compact and quasi-separated scheme with a $G$-action, and
$g\colon Y\to X$ a $G$-equivariant morphism. If $g$ is flat and affine, for
any $\mathcal{E}\in\operatorname{Qcoh}_{G}Y$ and
$\mathcal{F}\in\operatorname{Qcoh}_{G}\mathcal{A}$, we have an isomorphism of
$G$-equivariant $\mathcal{A}$-modules

$\displaystyle
g_{*}(\mathcal{E})\otimes_{\mathcal{O}_{X}}\mathcal{F}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,g_{*}\\!\big{(}\mathcal{E}\otimes_{\mathcal{O}_{Y}}g^{*}\mathcal{F}\big{)}.$

###### Proof.

Since $\mathbb{R}g_{*}\cong g_{*}$ and $\mathbb{L}\,g^{*}\cong g^{*}$, by the
projection formula [11, Proposition 3.9.4], we have a quasi-isomorphism

$\displaystyle
g_{*}\mathcal{E}\otimes^{\mathbb{L}}_{\mathcal{O}_{X}}\mathcal{F}\cong
g_{*}\big{(}\mathcal{E}\otimes^{\mathbb{L}}_{\mathcal{O}_{Y}}g^{*}\mathcal{F}\big{)}$

of complexes of quasi-coherent $\mathcal{O}_{X}$-modules. Hence we have
isomorphisms

$\displaystyle g_{*}(\mathcal{E})\otimes_{\mathcal{O}_{X}}\mathcal{F}$
$\displaystyle\cong
H^{0}\big{(}g_{*}(\mathcal{E})\otimes^{\mathbb{L}}_{\mathcal{O}_{X}}\mathcal{F}\big{)}$
$\displaystyle\cong
H^{0}\big{(}g_{*}\big{(}\mathcal{E}\otimes^{\mathbb{L}}_{\mathcal{O}_{Y}}g^{*}\mathcal{F}\big{)}\big{)}$
$\displaystyle\cong
g_{*}\big{(}H^{0}\big{(}\mathcal{E}\otimes^{\mathbb{L}}_{\mathcal{O}_{Y}}g^{*}\mathcal{F}\big{)}\big{)}$
$\displaystyle\cong
g_{*}\big{(}\mathcal{E}\otimes_{\mathcal{O}_{Y}}g^{*}\mathcal{F}\big{)},$

where the third isomorphism follows since $g_{*}$ is an exact functor. This
isomorphism

$\displaystyle\varphi\colon\
g_{*}(\mathcal{E})\otimes_{\mathcal{O}_{X}}\mathcal{F}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,g_{*}\\!\big{(}\mathcal{E}\otimes_{\mathcal{O}_{Y}}g^{*}\mathcal{F}\big{)}$

of $\mathcal{O}_{X}$-modules is nothing but the composition

$\displaystyle
g_{*}(\mathcal{E})\otimes_{\mathcal{O}_{X}}\mathcal{F}\longrightarrow
g_{*}(\mathcal{E})\otimes_{\mathcal{O}_{X}}g_{*}g^{*}\mathcal{F}\longrightarrow
g_{*}\\!\big{(}\mathcal{E}\otimes_{\mathcal{O}_{Y}}g^{*}\mathcal{F}\big{)},$

where the first morphism is the morphism induced by the adjunction
$\mathcal{F}\to g_{*}g^{*}\mathcal{F}$ and the second morphism is the
canonical morphism defined by $x\otimes y\mapsto x\otimes y$. Since these
morphisms are $\mathcal{A}$-linear and $G$-equivariant, so is the composition
$\varphi$. This finishes the proof. ∎

####  3.2.4 Taking invariant sections.

Let $H$ be a closed normal subgroup of $G$. Assume that the restriction
$\sigma_{H}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\sigma|_{H\times
X}\colon H\times X\to X$ is the trivial action, so that $\sigma_{H}=\pi_{H}$,
where $\pi_{H}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\pi|_{H\times
X}\colon H\times X\to X$ is the natural projection. Then we have the induced
$G/H$-action on $X$ denoted by

$\displaystyle\overline{\sigma}\colon\ G/H\times X\to X.$

We write $\overline{\pi}\colon G/H\times X\to X$ for the natural projection.

For
$\big{(}\mathcal{F},\uptheta^{\mathcal{F}}\big{)}\in\operatorname{Qcoh}_{G}X$,
we define the subsheaf $\mathcal{F}^{H}\subseteq\mathcal{F}$ to be the kernel
of the composition

${\mathcal{F}}$${(\pi_{H})_{*}(\pi_{H})^{*}\mathcal{F}}$${(\pi_{H})_{*}(\pi_{H})^{*}\mathcal{F},}$$\scriptstyle{\mu}$$\scriptstyle{\operatorname{id}-(\pi_{H})_{*}\uptheta_{H}^{\mathcal{F}}}$

where $\mu$ is the adjunction morphism and
$\uptheta^{\mathcal{F}}_{H}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\uptheta^{\mathcal{F}}|_{H\times
X}\colon(\pi_{H})^{*}\mathcal{F}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,(\sigma_{H})^{*}\mathcal{F}=(\pi_{H})^{*}\mathcal{F}$.
Then the pair
$\big{(}\mathcal{F}^{H},\uptheta^{\mathcal{F}}|_{\pi^{*}\mathcal{F}^{H}}\big{)}$
is a $G$-equivariant quasi-coherent sheaf. Since the restriction of the
isomorphism
$\uptheta^{\mathcal{F}}|_{\pi^{*}\mathcal{F}^{H}}\colon{\pi^{*}\mathcal{F}^{H}}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,{\sigma^{*}\mathcal{F}^{H}}$
to $H\times X$ is the identity, there is a unique $G/H$-equivariant structure
$\uptheta^{\mathcal{F}^{H}}\colon\overline{\pi}^{*}\mathcal{F}^{H}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\overline{\sigma}^{*}\mathcal{F}^{H}$
on $\mathcal{F}^{H}$ such that
$\uptheta^{\mathcal{F}}|_{\pi^{*}\mathcal{F}^{H}}={(p\times\operatorname{id}_{X})}^{*}\uptheta^{\mathcal{F}^{H}}$,
where $p\colon G\to G/H$ is the natural projection. This defines the functor

$(-)^{H}\colon\ \operatorname{Qcoh}_{G}X\to\operatorname{Qcoh}_{G/H}X.$ (3.2)

If $\uprho\colon\mathcal{O}_{X}\to\mathcal{A}$ is a $G$-equivariant
$X$-algebra, the induced morphism

$\displaystyle\mathcal{O}_{X}=\mathcal{O}_{X}^{H}\xrightarrow{\uprho^{H}}\mathcal{A}^{H}$

defines a $G/H$-equivariant $X$-algebra. If
$\big{(}\mathcal{M},\uptheta^{\mathcal{M}}\big{)}$ is a $G$-equivariant
$\mathcal{A}$-module, the morphism
$\alpha\colon\mathcal{A}\times\mathcal{M}\to\mathcal{M}$ defining the
$\mathcal{A}$-module structure on $\mathcal{M}$ is $G$-equivariant since
$\uptheta^{\mathcal{M}}$ is $\pi^{*}\mathcal{A}$-linear. The induced morphism

$\displaystyle\alpha^{H}\colon\
\mathcal{A}^{H}\times\mathcal{M}^{H}\to\mathcal{M}^{H}$

defines a $\mathcal{A}^{H}$-module structure on $\mathcal{M}^{H}$, and so the
pair $\big{(}\mathcal{M}^{H},\uptheta^{\mathcal{M}^{H}}\big{)}$ is a
$G/H$-equivariant $\mathcal{A}^{H}$-module. Thus the functor (3.2) extends to
the functor

$\displaystyle(-)^{H}\colon\
\operatorname{Qcoh}_{G}\mathcal{A}\to\operatorname{Qcoh}_{G/H}\mathcal{A}^{H}.$

The following is a generalization of [3, Lemma 2.22].

######  Proposition 3.15.

The composition

${\operatorname{Qcoh}_{G/H}\mathcal{A}^{H}}$${\operatorname{Qcoh}_{G}\mathcal{A}^{H}}$${\operatorname{Qcoh}_{G}\mathcal{A}}$$\scriptstyle{\operatorname{id}_{p}^{*}}$$\scriptstyle{(-)\otimes_{\mathcal{A}^{H}}\mathcal{A}}$

is left adjoint to the functor $(-)^{H}$.

###### Proof.

Let $\mathcal{M}\in\operatorname{Qcoh}_{G/H}\mathcal{A}^{H}$ and
$\mathcal{N}\in\operatorname{Qcoh}_{G}\mathcal{A}$. The result follows from
the following sequences of isomorphisms

$\displaystyle\operatorname{Hom}_{\operatorname{Qcoh}_{G}\mathcal{A}}\big{(}\operatorname{id}_{p}^{*}\mathcal{M}\otimes_{\mathcal{A}^{H}}\mathcal{A},\mathcal{N}\big{)}$
$\displaystyle\cong\operatorname{Hom}_{\operatorname{Qcoh}_{G}\mathcal{A}^{H}}\big{(}\operatorname{id}_{p}^{*}\mathcal{M},\mathcal{N}_{\iota}\big{)}$
$\displaystyle\cong\operatorname{Hom}_{\operatorname{Qcoh}_{G/H}\mathcal{A}^{H}}\big{(}\mathcal{M},(\mathcal{N}_{\iota})^{H}\big{)}$
$\displaystyle\cong\operatorname{Hom}_{\operatorname{Qcoh}_{G/H}\mathcal{A}^{H}}\big{(}\mathcal{M},\mathcal{N}^{H}\big{)},$

where $\mathcal{N}_{\iota}$ is the restriction of $\mathcal{N}$ by the natural
inclusion $\iota\colon\mathcal{A}^{H}\hookrightarrow\mathcal{A}$, the first
isomorphism follows from (3.1), the second isomorphism follows from an
identical argument as in the proof of [3, Lemma 2.22], and the last
isomorphism follows from a natural isomorphism
$(\mathcal{N}_{\iota})^{H}\cong\mathcal{N}^{H}$. ∎

####  3.2.5 Restriction functors and induction functors

Let $H$ be a closed subgroup of $G$, and

$\displaystyle\alpha\colon\ H\hookrightarrow G$

the natural inclusion morphism. Let $\mathcal{A}$ be a $G$-equivariant
$X$-algebra. We define the restriction functor

$\displaystyle\operatorname{\sf Res}^{G}_{H}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=(\operatorname{id}_{X})_{\alpha}^{*}\colon\
\operatorname{Qcoh}_{G}\mathcal{A}\to\operatorname{Qcoh}_{H}\mathcal{A}$

to be the pull-back by the identity morphism $\operatorname{id}_{X}\colon X\to
X$ that is $\alpha$-equivariant. If $H$ is trivial, we write
$\operatorname{\sf Res}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{\sf
Res}^{G}_{\\{1\\}}$, which is nothing but the forgetful functor
$\big{(}\mathcal{M},\uptheta^{\mathcal{M}}\big{)}\mapsto\mathcal{M}$.

Next, we will construct the adjoint functor of this restriction functor. We
define an $H$-action on $G\times X$ by

$\displaystyle h\cdot(g,x)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\big{(}g\big{(}h^{-1}\big{)},hx\big{)}$

for any $h\in H$, $g\in G$ and $x\in X$, and we write

$\displaystyle G\times^{H\hskip 0.56905pt}\\!\\!X\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=[G\times
X/H]$

for the associated quotient stack. By [3, Lemma 2.16(a)], the quotient stack
$G\times^{H\hskip 0.56905pt}\\!\\!X$ is representable by the scheme $G/H\times
X$. Since the morphism $\sigma\colon G\times X\to X$ defining the $G$-action
on $X$ is $H$-invariant, we have the induced morphism

$\displaystyle\sigma^{H}\colon\ G\times^{H\hskip 0.56905pt}\\!\\!X\to X.$

We define the morphism

$\displaystyle\varepsilon_{X}^{H}\colon\ X\to G\times^{H\hskip
0.56905pt}\\!\\!X$

to be the composition $X\xrightarrow{\varepsilon_{X}}G\times
X\xrightarrow{\,\,q\,\,}G\times^{H\hskip 0.56905pt}\\!\\!X$, where $q$ is the
canonical quotient map. Then we have
$\sigma^{H}\circ\varepsilon_{X}^{H}=\operatorname{id}_{X}$. The $G$-action on
$G\times X$ given by

$\displaystyle\widetilde{\sigma}\colon\ G\times(G\times X)\to G\times
X,\qquad(g,g^{\prime},x)\mapsto(gg^{\prime},x)$ (3.3)

induces the $G$-action on $G\times^{H\hskip 0.56905pt}\\!\\!X$. With respect
to this $G$-action on $G\times^{H\hskip 0.56905pt}\\!\\!X$, the morphism
$\varepsilon_{X}^{H}$ is $\alpha$-equivariant, and $\sigma^{H}$ is
$G$-equivariant. Thus $\big{(}\sigma^{H}\big{)}^{*}\mathcal{A}$ is a
$G$-equivariant $G\times^{H\hskip 0.56905pt}\\!\\!X$-algebra, and we have the
following functor

$\displaystyle\Phi\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\big{(}\varepsilon_{X}^{H}\big{)}^{*}_{\alpha}\colon\
\operatorname{Qcoh}_{G}\big{(}\sigma^{H}\big{)}^{*}\mathcal{A}\to\operatorname{Qcoh}_{H}\mathcal{A}.$

The following is a generalization of [3, Lemma 2.13]

######  Lemma 3.16.

The functor
$\Phi\colon\operatorname{Qcoh}_{G}\big{(}\sigma^{H}\big{)}^{*}\mathcal{A}\to\operatorname{Qcoh}_{H}\mathcal{A}$
is an equivalence.

###### Proof.

This can be proved by a similar argument as in [20, Lemma 1.3]. Since
$\big{(}\sigma^{H}\big{)}^{*}\mathcal{A}$ is isomorphic to the restriction of
the big fppf sheaf $[\sigma^{*}\mathcal{A}/H]$ to the small Zariski site of
$G\times^{H\hskip 0.56905pt}\\!\\!X$, by Proposition 3.12 we have an
equivalence
$\operatorname{Qcoh}\big{(}\sigma^{H}\big{)}^{*}\mathcal{A}\cong\operatorname{Qcoh}_{H}\sigma^{*}\mathcal{A}$.
Thus objects in
$\operatorname{Qcoh}_{G}\big{(}\sigma^{H}\big{)}^{*}\mathcal{A}$ are
identified with pairs

$\Big{(}\big{(}\mathcal{M},\uptheta^{\mathcal{M}}\big{)},\uptheta^{\left(\mathcal{M},\uptheta^{\mathcal{M}}\right)}\Big{)}$
(3.4)

of
$\big{(}\mathcal{M},\uptheta^{\mathcal{M}}\big{)}\in\operatorname{Qcoh}_{H}\sigma^{*}\mathcal{A}$
and an isomorphism
$\uptheta^{\left(\mathcal{M},\uptheta^{\mathcal{M}}\right)}\colon\widetilde{\pi}^{*}\big{(}\mathcal{M},\uptheta^{\mathcal{M}}\big{)}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\widetilde{\sigma}^{*}\big{(}\mathcal{M},\uptheta^{\mathcal{M}}\big{)}$
in $\operatorname{Qcoh}_{H}(\sigma\circ\widetilde{\sigma})^{*}\mathcal{A}$
satisfying the conditions as in (2.4), where $\widetilde{\pi}\colon G\times
G\times X\to G\times X$ and $\widetilde{\sigma}\colon G\times G\times X\to
G\times X$ are the projection and the group action in (3.3) respectively. If
we write
$\widetilde{\uptheta}^{\mathcal{M}}\colon\widetilde{\pi}^{*}\mathcal{M}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\widetilde{\sigma}^{*}\mathcal{M}$
for the isomorphism in
$\operatorname{Qcoh}(\sigma\circ\widetilde{\sigma})^{*}\mathcal{A}$ defining
the isomorphism $\uptheta^{\left(\mathcal{M},\uptheta^{\mathcal{M}}\right)}$,
then the pair $(\mathcal{M},\widetilde{\uptheta}^{\mathcal{M}})$ is an object
in $\operatorname{Qcoh}_{G}\sigma^{*}\mathcal{A}$, and
$\uptheta^{\mathcal{M}}$ defines a $H$-equivariant structure on
$\big{(}\mathcal{M},\widetilde{\uptheta}^{\mathcal{M}}\big{)}$. Thus the
object (3.4) uniquely corresponds to an object in
$\operatorname{Qcoh}_{H}[\sigma^{*}\mathcal{A}/G]$. Since $G$ trivially acts
on $X$ in the $G$-action (3.3), the composition of $\varepsilon_{X}\colon X\to
G\times X$ and the natural projection $G\times X\to[G\times X/G]$ gives an
isomorphism
$X\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,[G\times
X/G]$. Via this isomorphism, the sheaf $[\sigma^{*}\mathcal{A}/G]$ on
$[G\times X/G]$ corresponds to the sheaf $\mathcal{A}$ on $X$. Hence the
assignment

$\displaystyle\Bigl{(}\big{(}\mathcal{M},\uptheta^{\mathcal{M}}\big{)},\uptheta^{\left(\mathcal{M},\uptheta^{\mathcal{M}}\right)}\Bigr{)}\mapsto\big{(}\varepsilon_{X}^{*}\mathcal{M},\uptheta^{\mathcal{M}}|_{H\times\\{1\\}\times
X}\big{)}$

defines an equivalence
$\operatorname{Qcoh}_{G}[\sigma^{*}\mathcal{A}/H]\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\operatorname{Qcoh}_{H}\mathcal{A}$,
and $\Phi$ is isomorphic to this equivalence via
$\operatorname{Qcoh}_{G}\big{(}\sigma^{H}\big{)}^{*}\mathcal{A}\cong\operatorname{Qcoh}_{G}[\sigma^{*}\mathcal{A}/H]$.
∎

Since the morphism $\sigma^{H}\colon G\times^{H\hskip 0.56905pt}\\!\\!X\to X$
is $G$-equivariant, we have the direct image functor
$\sigma^{H}_{*}\colon\operatorname{Qcoh}_{G}\big{(}\sigma^{H}\big{)}^{*}\mathcal{A}\to\operatorname{Qcoh}_{G}\mathcal{A}$,
and we define the induction functor

$\displaystyle\operatorname{\sf Ind}^{G}_{H}\colon\
\operatorname{Qcoh}_{H}\mathcal{A}\to\operatorname{Qcoh}_{G}\mathcal{A}$

to be the composition $\big{(}\sigma^{H}\big{)}_{*}\circ\Phi^{-1}$. If $H$ is
trivial, we write $\operatorname{\sf Ind}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{\sf
Ind}^{G}_{\\{1\\}}$.

######  Lemma 3.17.

Notation is the same as above.

  * $1.$

The functor $\operatorname{\sf Res}^{G}_{H}$ is exact, and if $G/H$ is affine,
the functor $\operatorname{\sf Ind}^{G}_{H}$ is also exact.

  * $2.$

The restriction functor $\operatorname{\sf Res}^{G}_{H}$ is left adjoint to
the induction functor $\operatorname{\sf Ind}^{G}_{H}$.

  * $3.$

If $H$ is a normal subgroup of $G$ and $G/H$ is reductive affine algebraic
group, the adjunction morphism

$\displaystyle\mu\colon\ \operatorname{id}\to\operatorname{\sf
Ind}^{G}_{H}\circ\operatorname{\sf Res}^{G}_{H}$

is a split mono, i.e., there exists a functor morphism
$\nu\colon\operatorname{\sf Ind}^{G}_{H}\circ\operatorname{\sf
Res}^{G}_{H}\to\operatorname{id}$ such that the composition $\nu\circ\mu$ is
the identity morphism of functors.

###### Proof.

(1) Since the morphism $\sigma^{H}$ is flat, the pull-back
$\big{(}\sigma^{H}\big{)}^{*}$ is exact. Since
$\sigma^{H}\circ\varepsilon_{X}^{H}=\operatorname{id}_{X}$, the functor
$\operatorname{\sf Res}^{G}_{H}$ is isomorphic to the composition
$\Phi\circ\big{(}\sigma^{H}\big{)}^{*}$, and thus $\operatorname{\sf
Res}^{G}_{H}$ is exact. If $G/H$ is affine, the morphism $\sigma^{H}$ is an
affine morphism since $G\times^{H\hskip 0.56905pt}\\!\\!X$ is isomorphic to
$G/H\times X$. Then the direct image $\big{(}\sigma^{H}\big{)}_{*}$ is exact,
and so is $\operatorname{\sf Ind}^{G}_{H}$.

(2) This follows from the adjunction
$\big{(}\sigma^{H}\big{)}^{*}\dashv\big{(}\sigma^{H}\big{)}_{*}$ and the
isomorphism $\operatorname{\sf
Res}^{G}_{H}\cong\Phi\circ\big{(}\sigma^{H}\big{)}^{*}$.

(3) It is enough to show that the adjunction morphism

$\displaystyle\widetilde{\mu}\colon\
\operatorname{id}\to\big{(}\sigma^{H}\big{)}_{*}\big{(}\sigma^{H}\big{)}^{*}$

is a split mono. Note that we have the following cartesian square

${G\times^{H\hskip
0.56905pt}\\!\\!X}$${X}$${G/H}$${\operatorname{Spec}k,}$$\scriptstyle{\sigma^{H}}$$\scriptstyle{q}$$\scriptstyle{p}$

where $p$ and $q$ are the morphisms defining the base field $k$. If $G/H$ is a
reductive affine algebraic group, it is linearly reductive. Then the natural
morphism $k\to p_{*}p^{*}k=k[G/H]$ is a split mono, and so is the adjunction
morphism

$\displaystyle\mathcal{O}_{X}\to\big{(}\sigma^{H}\big{)}_{*}\big{(}\sigma^{H}\big{)}^{*}\mathcal{O}_{X}$

by the base change formula for the above cartesian square. For any
$\mathcal{M}\in\operatorname{Qcoh}_{G}\mathcal{A}$, by Lemma 3.14 the
adjunction
$\widetilde{\mu}(\mathcal{M})\colon\mathcal{M}\to\big{(}\sigma^{H}\big{)}_{*}\big{(}\sigma^{H}\big{)}^{*}\mathcal{M}$
is isomorphic to the tensor product

$\displaystyle\Bigr{(}\mathcal{O}_{X}\to\big{(}\sigma^{H}\big{)}_{*}\big{(}\sigma^{H}\big{)}^{*}\mathcal{O}_{X}\Bigr{)}\otimes_{\mathcal{O}_{X}}\mathcal{M},$

and therefore $\widetilde{\mu}(\mathcal{M})$ is also a split mono. ∎

We will apply the above result to the following:

######  Lemma 3.18.

Let $\mathcal{C}$ and $\mathcal{D}$ be additive categories, and
$F\colon\mathcal{C}\to\mathcal{D}$ and $G\colon\mathcal{D}\to\mathcal{C}$
additive functors. If $F$ is left adjoint to $G$, and if the adjunction
morphism $\mu\colon\operatorname{id}_{\mathcal{C}}\to G\circ F$ is a split
mono, then the functor $F$ is faithful.

###### Proof.

For a morphism $f\colon A\to B$ in $\mathcal{C}$, assume that $F(f)=0$. It is
enough to show that $f=0$. We have a commutative diagram

${A}$${GF(A)}$${A}$${B}$${GF(B)}$${B.}$$\scriptstyle{\mu(A)}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{f}$$\scriptstyle{GF(f)}$$\scriptstyle{\nu(A)}$$\scriptstyle{f}$$\scriptstyle{\mu(B)}$$\scriptstyle{\nu(B)}$

Therefore, $f=\nu(B)\circ GF(f)\circ\mu(A)=\nu(B)\circ G(0)\circ\mu(A)=0$. ∎

###  3.3 Fundamental properties of equivariant modules

In this subsection, we prove that the category of equivariant modules over a
certain equivariant algebra has enough injectives and enough locally free
modules.

Let $X$ be a quasi-compact and quasi-separated scheme with an action from an
affine algebraic group $G$, and $\uprho\colon\mathcal{O}_{X}\to\mathcal{A}$ a
$G$-equivariant $X$-algebra.

######  Proposition 3.19.

The category $\operatorname{Qcoh}_{G}\mathcal{A}$ is a Grothendieck category.
In particular, it has enough injectives.

###### Proof.

The category $\operatorname{Qcoh}_{G}\mathcal{A}$ is an abelian category with
small direct sums, and so it suffices to prove that (1) filtered colimits are
exact in $\operatorname{Qcoh}_{G}\mathcal{A}$ and that (2)
$\operatorname{Qcoh}_{G}\mathcal{A}$ has a generator.

(1) For a point $x\in X$, denote by
$F_{x}\colon\operatorname{Qcoh}_{G}\mathcal{A}\to\operatorname{Mod}\mathcal{A}_{x}$
the composition

$\displaystyle\operatorname{Qcoh}_{G}\mathcal{A}\xrightarrow{\operatorname{\sf
Res}}\operatorname{Qcoh}\mathcal{A}\xrightarrow{(-)_{x}}\operatorname{Mod}\mathcal{A}_{x},$

where the latter is taking the stalk at $x$, and $\mathcal{A}_{x}$ is the
stalk of $\mathcal{A}$ at $x$. Then a sequence in
$\operatorname{Qcoh}_{G}\mathcal{A}$ is exact if and only if for every $x\in
X$ the sequence in $\operatorname{Mod}\mathcal{A}_{x}$ induced by $F_{x}$ is
exact. Therefore, since $F_{x}$ commutes with filtered colimits and
$\operatorname{Mod}\mathcal{A}_{x}$ is a Grothendieck category, filtered
colimits in $\operatorname{Qcoh}_{G}\mathcal{A}$ are also exact.

(2) It is well known that $\operatorname{Qcoh}_{G}X$ is a Grothendieck
category (more generally, the category of quasi-coherent sheaves on an
algebraic stack is a Grothendieck category [19, Proposition 14.2; 0781], [1,
Corollary 5.10]). In particular, $\operatorname{Qcoh}_{G}X$ has a generator
$\mathcal{G}\in\operatorname{Qcoh}_{G}X$. We show that

$\displaystyle\mathcal{G}_{\mathcal{A}}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\mathcal{G}\otimes_{\mathcal{O}_{X}}\mathcal{A}\in\operatorname{Qcoh}_{G}\mathcal{A}$

is a generator. Let $\mathcal{M}\in\operatorname{Qcoh}_{G}\mathcal{A}$ be an
object. Then, there is a set $I$ and a surjective morphism

$\displaystyle p\colon\ \mathcal{G}^{\oplus
I}\twoheadrightarrow\mathcal{M}_{\uprho}$

in $\operatorname{Qcoh}_{G}X$. Since the functor
$(-)\otimes_{\mathcal{O}_{X}}\mathcal{A}\colon\operatorname{Qcoh}_{G}X\to\operatorname{Qcoh}_{G}\mathcal{A}$
is left adjoint to the functor $(-)_{\uprho}$, it is right exact and commutes
with small direct sums. Hence the extension of $p$ by
$\uprho\colon\mathcal{O}_{X}\to\mathcal{A}$ defines a surjective morphism

$\displaystyle p_{\mathcal{A}}\colon\ \mathcal{G}_{\mathcal{A}}^{\oplus
I}\twoheadrightarrow\mathcal{M}_{\uprho}\otimes_{\mathcal{O}_{X}}\mathcal{A}.$

Composing this with a natural surjection
$\mathcal{M}_{\uprho}\otimes_{\mathcal{O}_{X}}\mathcal{A}\to\mathcal{M}$, we
have a surjective morphism $\mathcal{G}_{\mathcal{A}}^{\oplus
I}\twoheadrightarrow\mathcal{M}$. This shows that $\mathcal{G}_{\mathcal{A}}$
is a generator in $\operatorname{Qcoh}_{G}\mathcal{A}$. ∎

######  Definition 3.20.

Notation is the same as above.

  * 1.

A $G$-equivariant $\mathcal{A}$-module
$\big{(}\mathcal{M},\uptheta^{\mathcal{M}}\big{)}$ is said to be locally free,
if there is an open covering $\\{U_{i}\hookrightarrow X\\}_{i\in I}$ of $X$
such that for every $U_{i}$ the restriction $\mathcal{M}|_{U_{i}}$ of the
underlying $\mathcal{A}$-module $\mathcal{M}$ is a free
$\mathcal{A}|_{U_{i}}$-module.

  * 2.

We say that $\big{(}\mathcal{A},\uptheta^{\mathcal{A}}\big{)}$ satisfies the
resolution property, if for any $G$-equivariant coherent $\mathcal{A}$-module
$\mathcal{N}\in\operatorname{coh}_{G}\mathcal{A}$, there exists a surjective
morphism
$\big{(}\mathcal{E},\uptheta^{\mathcal{E}}\big{)}\twoheadrightarrow\big{(}\mathcal{N},\uptheta^{\mathcal{N}}\big{)}$
in $\operatorname{coh}_{G}\mathcal{A}$ from a $G$-equivariant locally free
coherent $\mathcal{A}$-module
$\big{(}\mathcal{E},\uptheta^{\mathcal{E}}\big{)}$.

  * 3.

A $G$-scheme $X$ satisfies the $G$-equivariant resolution property if the
$G$-equivariant $X$-algebra $(\mathcal{O}_{X},\uptheta^{\operatorname{can}})$
has the resolution property.

######  Remark 3.21.

(1) If $X=\operatorname{Spec}R$ is an affine noetherian scheme, a
$G$-equivariant coherent $R$-algebra $A$ satisfies the resolution property if
and only if the category $\operatorname{Mod}_{G}A$ has enough projectives.

(2) Assume that $X$ is noetherian and that $\mathcal{A}$ is coherent. If
$\big{(}\mathcal{A},\uptheta^{\mathcal{A}}\big{)}$ satisfies the resolution
property, for every $G$-equivariant $\mathcal{A}$-module $\mathcal{M}$, there
is a surjection $\mathcal{E}\twoheadrightarrow\mathcal{M}$ from a
$G$-equivariant locally free $\mathcal{A}$-module $\mathcal{E}$. This follows
since there is a set $\\{\mathcal{M}_{i}\\}_{i\in I}$ of $G$-equivariant
coherent submodules $\mathcal{M}_{i}$ of $\mathcal{M}$ such that the natural
map $\bigoplus_{i\in I}\mathcal{M}_{i}\to\mathcal{M}$ is surjective.

######  Proposition 3.22.

Assume that $\big{(}\mathcal{A},\uptheta^{\mathcal{A}}\big{)}$ is coherent and
that $X$ has the $G$-equivariant resolution property. Then
$\big{(}\mathcal{A},\uptheta^{\mathcal{A}}\big{)}$ satisfies the resolution
property.

###### Proof.

Let $\mathcal{M}\in\operatorname{coh}_{G}\mathcal{A}$ be a $G$-equivariant
coherent $\mathcal{A}$-module. By the assumption, there is a surjective
morphism $p\colon\mathcal{E}\twoheadrightarrow\mathcal{M}_{\uprho}$ from a
$G$-equivariant locally free coherent $\mathcal{O}_{X}$-module $\mathcal{E}$.
Then the extension

$\displaystyle p\otimes_{\mathcal{O}_{X}}\mathcal{A}\colon\
\mathcal{E}\otimes_{\mathcal{O}_{X}}\mathcal{A}\to\mathcal{M}_{\uprho}\otimes_{\mathcal{O}_{X}}\mathcal{A}$

of $p$ by $\uprho\colon\mathcal{O}_{X}\to\mathcal{A}$ is also a surjection in
$\operatorname{Qcoh}_{G}\mathcal{A}$. Composing with the natural surjection
$\mathcal{M}_{\uprho}\otimes_{\mathcal{O}_{X}}\mathcal{A}\twoheadrightarrow\mathcal{M}$
gives a surjective morphism
$\mathcal{E}\otimes_{\mathcal{O}_{X}}\mathcal{A}\twoheadrightarrow\mathcal{M}$
from a $G$-equivariant locally free $\mathcal{A}$-module. ∎

######  Definition 3.23.

A $G$-equivariant (resp. coherent) $\mathcal{A}$-module
$\big{(}\mathcal{M},\uptheta^{\mathcal{M}}\big{)}$ has a finite locally free
resolution in $\operatorname{Qcoh}_{G}\mathcal{A}$ (resp. in
$\operatorname{coh}_{G}\mathcal{A}$) if there exists an exact sequence

$\displaystyle
0\to\mathcal{E}^{-n}\to\cdots\to\mathcal{E}^{0}\to\mathcal{M}\to 0$

in $\operatorname{Qcoh}_{G}\mathcal{A}$ (resp. in
$\operatorname{coh}_{G}\mathcal{A}$) such that each $\mathcal{E}^{i}$ is
locally free.

######  Lemma 3.24.

Assume that $X$ is a normal scheme of finite type over $k$ and that $G$ is
affine. If $X$ has an ample family of line bundles, then
$\big{(}\mathcal{A},\uptheta^{\mathcal{A}}\big{)}$ satisfies the resolution
property.

###### Proof.

This follows from [3, Theorem 2.29] and Proposition 3.22. ∎

######  Lemma 3.25.

Let $R$ be a normal ring of finite type over $k$, and $G$ an affine reductive
algebraic group acting on $\operatorname{Spec}R$. Let
$\big{(}A,\uptheta^{A}\big{)}$ be a $G$-equivariant coherent $R$-algebra, and
$\big{(}M,\uptheta^{M}\big{)}\in\operatorname{Mod}_{G}A$ a $G$-equivariant
$A$-module. If the underlying $A$-module $M\in\operatorname{Mod}A$ has a
finite projective resolution, so does $\big{(}M,\uptheta^{M}\big{)}$. If $M$
is finitely generated and has a finite projective resolution in
$\operatorname{mod}A$, then $\big{(}M,\uptheta^{M}\big{)}$ has a finite
projective resolution in $\operatorname{mod}_{G}A$.

###### Proof.

By Lemma 3.24, the categories $\operatorname{Mod}_{G}A$ and
$\operatorname{mod}_{G}A$ have enough projectives. Since $A$ is coherent
$R$-algebra, $\operatorname{mod}_{G}A$ is an abelian subcategory of
$\operatorname{Mod}_{G}A$. It is standard that an object $x$ in an abelian
category $\mathcal{A}$ has a finite projective resolution if and only if there
exists $n>0$ such that $\operatorname{Ext}_{\mathcal{A}}^{i}(x,y)=0$ for any
$i>n$ and any $y\in\mathcal{A}$. Thus the statements follow from Proposition
3.10. ∎

###  3.4 Equivariant tilting modules and derived equivalences

In this subsection, we prove that an equivariant tilting module induces a
derived equivalence of equivariant algebras, which is considered as an
equivariant Morita theory.

Let $X$ be a separated scheme of finite type over $k$, and $G$ an affine
algebraic group acting on $X$. Let $H\subseteq G$ be a closed normal subgroup
of $G$, and $\mathcal{A}$ a $G$-equivariant coherent $X$-algebra.

######  Definition 3.26.

Let $\mathcal{T}\in\operatorname{coh}_{G}\mathcal{A}$ be a $G$-equivariant
coherent $\mathcal{A}$-module.

  * 1.

$\mathcal{T}$ is called a $(G,H)$-tilting module (resp. partial
$(G,H)$-tilting module), if the restriction $\operatorname{\sf
Res}^{G}_{H}(\mathcal{T})$ is a tilting object (resp. partial tilting object)
in $\operatorname{Qcoh}_{H}\mathcal{A}$.

  * 2.

$\mathcal{T}$ is called a $G$-tilting module (resp. partial $G$-tilting
module) if it is $(G,\\{1\\})$-tilting (resp. partial $(G,\\{1\\})$-tilting).

Let $R$ be a normal ring of finite type over $k$, and suppose that there is a
$G$-action $\sigma\colon G\times\operatorname{Spec}R\to\operatorname{Spec}R$
on $\operatorname{Spec}R$ such that the $H$-action on $\operatorname{Spec}R$,
which is given by the restriction of $\sigma$, is trivial. Let $f\colon
X\to\operatorname{Spec}R$ be a $G$-equivariant morphism such that
$f_{*}\mathcal{O}_{X}=R$ and the associated morphism
$\overline{f}\colon[X/H]\to\operatorname{Spec}R$ is proper. Let
$\big{(}\mathcal{T},\uptheta^{\mathcal{T}}\big{)}\in\operatorname{coh}_{G}\mathcal{A}$
be a $G$-equivariant coherent $\mathcal{A}$-module. Then the endomorphism ring

$\displaystyle\Lambda\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{End}_{\mathcal{A}}(\mathcal{T})=f_{*}\operatorname{{\mathcal{E}}nd}_{\mathcal{A}}(\mathcal{T})$

of the underlying coherent $\mathcal{A}$-module $\mathcal{T}$ is a
$G$-equivariant $R$-algebra. We define the functor

$\displaystyle\operatorname{\sf{F}}\colon\
\operatorname{Qcoh}_{G}\mathcal{A}\to\operatorname{Mod}_{G/H}\Lambda^{H}$

to be the composition

${\operatorname{Qcoh}_{G}\mathcal{A}}$${\operatorname{Qcoh}_{G}\operatorname{{\mathcal{E}}nd}_{\mathcal{A}}(\mathcal{T})}$${\operatorname{Mod}_{G}\Lambda}$${\operatorname{Mod}_{G/H}\Lambda^{H}.}$$\scriptstyle{\operatorname{{\mathcal{H}}om}_{\mathcal{A}}(\mathcal{T}\mbox{\tiny,}-)}$$\scriptstyle{f_{\star}}$$\scriptstyle{(-)^{H}}$

Then the composition

$\displaystyle\operatorname{\sf{G}}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\left(-\otimes\mathcal{T}\right)\circ
f^{\star}\circ\left(\operatorname{id}_{p}^{*}(-)\otimes_{\mathcal{A}^{H}}\mathcal{A}\right)\colon\
\operatorname{Mod}_{G/H}\Lambda^{H}\to\operatorname{Qcoh}_{G}\mathcal{A}$

is left adjoint to $\operatorname{\sf{F}}$, and it preserves equivariant
coherent modules. Note that since the morphism
$\overline{f}\colon[X/H]\to\operatorname{Spec}R$ is proper, the functor
$\operatorname{\sf{F}}$ also preserves equivariant coherent modules. The
following is one of our motivations of considering equivariant algebras and
equivariant modules.

######  Theorem 3.27.

Assume that $G/H$ is reductive and that every $\Lambda^{H}$-module
$M\in\operatorname{Mod}\Lambda^{H}$ is of finite projective dimension.

  * $1.$

If $\big{(}\mathcal{T},\uptheta^{\mathcal{T}}\big{)}$ is partial
$(G,H)$-tilting, then the functor

$\displaystyle\mathbb{L}\operatorname{\sf{G}}\colon\
\operatorname{D^{b}}\big{(}\operatorname{Mod}_{G/H}\Lambda^{H}\big{)}\hookrightarrow\operatorname{D^{b}}\big{(}\operatorname{Qcoh}_{G}{\mathcal{A}}\big{)}$

is fully faithful, and it restricts to the fully faithful functor

$\displaystyle\mathbb{L}\operatorname{\sf{G}}\colon\
\operatorname{D^{b}}\big{(}\operatorname{mod}_{G/H}\Lambda^{H}\big{)}\hookrightarrow\operatorname{D^{b}}\big{(}\operatorname{coh}_{G}{\mathcal{A}}\big{)}.$

  * $2.$

If $\big{(}\mathcal{T},\uptheta^{\mathcal{T}}\big{)}$ is $(G,H)$-tilting, then
the functor

$\displaystyle\mathbb{L}\operatorname{\sf{G}}\colon\
\operatorname{D^{b}}\big{(}\operatorname{Mod}_{G/H}\Lambda^{H}\big{)}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\operatorname{D^{b}}\big{(}\operatorname{Qcoh}_{G}{\mathcal{A}}\big{)}$

is an equivalence, and it restricts to the equivalence

$\displaystyle\mathbb{L}\operatorname{\sf{G}}\colon\
\operatorname{D^{b}}\big{(}\operatorname{mod}_{G/H}\Lambda^{H}\big{)}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\operatorname{D^{b}}\big{(}\operatorname{coh}_{G}{\mathcal{A}}\big{)}.$

Before we prove this theorem, we prepare the following lemmas.

######  Lemma 3.28.

Notation is the same as above, and assume that $G/H$ is reductive. Then the
restrictions

$\displaystyle\operatorname{\sf Res}^{G}_{H}\colon\
\operatorname{D^{b}}\big{(}\operatorname{Qcoh}_{G}\mathcal{A}\big{)}\to\operatorname{D^{b}}\big{(}\operatorname{Qcoh}_{H}\mathcal{A}\big{)},$
$\displaystyle\operatorname{\sf Res}\colon\
\operatorname{D^{b}}\big{(}\operatorname{Mod}_{G/H}\Lambda^{H}\big{)}\to\operatorname{D^{b}}\big{(}\operatorname{Mod}\Lambda^{H}\big{)}$

are faithful functors.

###### Proof.

We only prove $\operatorname{\sf Res}^{G}_{H}$ is faithful. Since $G/H$ is
affine, $\operatorname{\sf
Ind}^{G}_{H}\colon\operatorname{Qcoh}_{H}\mathcal{A}\to\operatorname{Qcoh}_{G}\mathcal{A}$
is an exact functor, and so it extends to the functor

$\displaystyle\operatorname{\sf Ind}^{G}_{H}\colon\
\operatorname{D^{b}}\big{(}\operatorname{Qcoh}_{H}\mathcal{A}\big{)}\to\operatorname{D^{b}}\big{(}\operatorname{Qcoh}_{G}\mathcal{A}\big{)}$

which is right adjoint to $\operatorname{\sf Res}^{G}_{H}$. Since $G/H$ is
reductive, the adjunction
$\mu\colon\operatorname{id}_{\operatorname{Qcoh}_{G}\mathcal{A}}\to\operatorname{\sf
Ind}^{G}_{H}\operatorname{\sf Res}^{G}_{H}$ is a split mono by Lemma 3.17, and
this splitting naturally extends to the splitting of the adjunction
$\mu\colon\operatorname{id}_{\operatorname{D^{b}}\left(\operatorname{Qcoh}_{G}\mathcal{A}\right)}\to\operatorname{\sf
Ind}^{G}_{H}\operatorname{\sf Res}^{G}_{H}$ since $\operatorname{\sf
Ind}^{G}_{H}$ and $\operatorname{\sf Res}^{G}_{H}$ are exact functors. Hence
the functor $\operatorname{\sf Res}^{G}_{H}$ is faithful by Lemma 3.18. ∎

######  Lemma 3.29.

For each $i=1,2$, let $F_{i}\colon\mathcal{C}_{i}\to\mathcal{D}_{i}$ and
$G_{i}\colon\mathcal{D}_{i}\to\mathcal{C}_{i}$ be exact functors between
triangulated categories such that $G_{i}\dashv F_{i}$. Denote by
$\sigma_{i}\colon G_{i}\circ F_{i}\to\operatorname{id}$ and
$\tau_{i}\colon\operatorname{id}\to F_{i}\circ G_{i}$ the adjunction
morphisms, and let $P\colon\mathcal{C}_{1}\to\mathcal{C}_{2}$ and
$Q\colon\mathcal{D}_{1}\to\mathcal{D}_{2}$ be faithful exact functors. Assume
that we have functor isomorphisms $\varphi\colon Q\circ
F_{1}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,F_{2}\circ
P$ and $\psi\colon P\circ
G_{1}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,G_{2}\circ
Q$ and that the following diagrams are commutative

${PG_{1}F_{1}}$${G_{2}F_{2}P}$${QF_{1}G_{1}}$${F_{2}G_{2}Q}$${P}$${Q}$$\scriptstyle{\sim}$$\scriptstyle{P\sigma_{1}}$$\scriptstyle{\sigma_{2}P}$$\scriptstyle{\sim}$$\scriptstyle{Q\tau_{1}}$$\scriptstyle{\tau_{2}Q}$

where the top horizontal arrows are the isomorphisms induced by $\varphi$ and
$\psi$.

  * $1.$

If $F_{2}$ $($resp. $G_{2})$ is fully faithful, so is $F_{1}$ $($resp.
$G_{1})$.

  * $2.$

If $F_{2}$ $($resp. $G_{2})$ is an equivalence, so is $F_{1}$ $($resp.
$G_{1})$.

###### Proof.

(2) follows from (1). We only prove (1) for $F_{i}$, since the statement for
$G_{i}$ follows by a similar argument. For an object $C\in\mathcal{C}_{1}$
consider the following triangle

$\displaystyle G_{1}F_{1}(C)\xrightarrow{\sigma_{1}(C)}C\to{\rm
Cone}(\sigma_{1}(C))\to G_{1}F_{1}(C)[1].$

It suffices to prove that ${\rm Cone}(\tau_{1}(C))$ is the zero object. By the
assumption, $P({\rm Cone}(\sigma_{1}(C)))$ is isomorphic to the cone of the
morphism $\sigma_{2}(P(C))$, which is the zero object since $F_{2}$ is fully
faithful. Hence ${\rm Cone}(\sigma_{1}(C))$ is also the zero object since $P$
is faithful. ∎

Proof of Theorem 3.27. We prove (1) and (2) simultaneously. We define a
functor

$\displaystyle\operatorname{\sf{F}}_{H}\colon\
\operatorname{Qcoh}_{H}\mathcal{A}\to\operatorname{Mod}\Lambda^{H}$

to be the composition

${\operatorname{Qcoh}_{H}\mathcal{A}}$${\operatorname{Qcoh}_{H}\operatorname{{\mathcal{E}}nd}_{\mathcal{A}}(\mathcal{T})}$${\operatorname{Mod}_{H}\Lambda}$${\operatorname{Mod}\Lambda^{H}.}$$\scriptstyle{\operatorname{{\mathcal{H}}om}_{\mathcal{A}}(\mathcal{T}\mbox{\tiny,}-)}$$\scriptstyle{f_{\star}}$$\scriptstyle{(-)^{H}}$

If we set $\mathcal{T}_{H}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{\sf
Res}^{G}_{H}\big{(}\mathcal{T},\uptheta^{\mathcal{T}}\big{)}\in\operatorname{Qcoh}_{H}\mathcal{A}$,
since
$\Lambda^{H}\cong\operatorname{End}_{\mathcal{A}}(\mathcal{T})^{H}=\operatorname{End}_{\operatorname{Qcoh}_{H}\mathcal{A}}(\mathcal{T}_{H})$,
the functor $\operatorname{\sf{F}}_{H}$ is isomorphic to the functor

$\displaystyle\operatorname{Hom}_{\operatorname{Qcoh}_{H}\mathcal{A}}(\mathcal{T}_{H},-)\colon\
\operatorname{Qcoh}_{H}\mathcal{A}\to\operatorname{Mod}\Lambda^{H}.$

Note that $\operatorname{\sf{F}}_{H}$ has a left adjoint functor
$\operatorname{\sf{G}}_{H}\colon\operatorname{Mod}\Lambda^{H}\to\operatorname{Qcoh}_{H}\mathcal{A}$
which is given by

$\displaystyle\operatorname{\sf{G}}_{H}(-)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=f^{\star}\big{(}\operatorname{id}_{p}^{*}(-)\otimes_{\Lambda^{H}}\Lambda\big{)}\otimes_{\operatorname{{\mathcal{E}}nd}(\mathcal{T})}\mathcal{T}\cong
f^{-1}\big{(}\operatorname{id}_{p}^{*}(-)\otimes_{\Lambda^{H}}\Lambda\big{)}\otimes_{f^{-1}\Lambda}\mathcal{T},$

where $p\colon H\to H/H=\\{1\\}$ is the natural projection. By construction,
we have natural isomorphisms
$\operatorname{\sf{F}}_{H}(\mathcal{T}_{H})\cong\Lambda^{H}$ and
$\operatorname{\sf{G}}_{H}\big{(}\Lambda^{H}\big{)}\cong\mathcal{T}_{H}$, and
this implies that the adjunction morphisms

$\displaystyle\Lambda^{H}\to\operatorname{\sf{F}}_{H}\operatorname{\sf{G}}_{H}\big{(}\Lambda^{H}\big{)}\qquad\text{and}\qquad\operatorname{\sf{G}}_{H}\operatorname{\sf{F}}_{H}(\mathcal{T}_{H})\to\mathcal{T}_{H}$

are isomorphisms. Since $\mathcal{T}_{H}$ is a partial tilting object in
$\operatorname{Qcoh}_{H}\mathcal{A}$, by Theorem 2.17, the functor

$\displaystyle\mathbb{L}\operatorname{\sf{G}}_{H}\colon\
\operatorname{D^{b}}\big{(}\operatorname{Mod}\Lambda^{H}\big{)}\to\operatorname{D^{b}}\big{(}\operatorname{Qcoh}_{H}\mathcal{A}\big{)}$

is fully faithful, and if $\mathcal{T}_{H}$ is tilting, it is an equivalence.
Now we have the following commutative diagram:

${\operatorname{D^{b}}\big{(}\operatorname{Mod}_{G/H}\Lambda^{H}\big{)}}$${\operatorname{D^{b}}\big{(}\operatorname{Qcoh}_{G}\mathcal{A}\big{)}}$${\operatorname{D^{b}}\big{(}\operatorname{Mod}_{G/H}\Lambda^{H}\big{)}}$${\operatorname{D^{b}}\big{(}\operatorname{Mod}\Lambda^{H}\big{)}}$${\operatorname{D^{b}}\big{(}\operatorname{Qcoh}_{H}\mathcal{A}\big{)}}$${\operatorname{D^{b}}\big{(}\operatorname{Mod}\Lambda^{H}\big{)}.}$$\scriptstyle{\mathbb{L}\operatorname{\sf{G}}}$$\scriptstyle{\operatorname{\sf
Res}}$$\scriptstyle{\operatorname{\sf
Res}^{G}_{H}}$$\scriptstyle{\mathbb{R}\operatorname{\sf{F}}}$$\scriptstyle{\operatorname{\sf
Res}}$$\scriptstyle{\mathbb{L}\operatorname{\sf{G}}_{H}}$$\scriptstyle{\mathbb{R}\operatorname{\sf{F}}_{H}}$

Hence (1) and (2) follows from Lemmas 3.28 and 3.29.∎

The following is a special case of Theorem 3.27, which is an equivariant
version of derived equivalences induced by tilting bundles and tilting
modules.

######  Corollary 3.30.

Notation is the same as above. Assume that $G$ is reductive.

  * $1.$

Let $\mathcal{T}$ be a $G$-equivariant vector bundle on $X$ and set
$\Lambda\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{End}_{X}(\mathcal{T})$.
If $\mathcal{T}$ is $G$-tilting and $\Lambda$ is of finite global dimension,
we have an equivalence

$\displaystyle\operatorname{D^{b}}\big{(}\operatorname{coh}_{G}X\big{)}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\operatorname{D^{b}}\big{(}\operatorname{mod}_{G}\Lambda\big{)}.$

  * $2.$

Let $\Gamma$ be a $G$-equivariant $R$-algebra, and $T$ a $G$-equivariant
$\Gamma$-module. Set $\Lambda\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{End}_{\Gamma}(T)$.
If $T$ is $G$-tilting and $\Lambda$ is of finite global dimension, we have an
equivalence

$\displaystyle\operatorname{D^{b}}\big{(}\operatorname{mod}_{G}\Gamma\big{)}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\operatorname{D^{b}}\big{(}\operatorname{mod}_{G}\Lambda\big{)}.$

##  4 Equivariant tilting objects and factorizations

In this section, we prove that equivariant tilting modules induce equivalences
of derived factorization categories.

###  4.1 Tilting objects and factorizations

In this subsection, we prove that a tilting object in a Gorthendieck category
induces an equivalence of derived factorization categories. To prove this, we
need the following lemmas:

Let $\mathcal{A}$ be a Grothendieck category and $(\mathcal{E},\Phi,w)$ a
triple as in (2.1).

######  Lemma 4.1.

Assume that we have $\operatorname{Ext}^{i}_{\mathcal{A}}(A,B)=0$ for
arbitrary objects $A,B\in\mathcal{E}$ and all $i>0$. Then for objects
$E,F\in\operatorname{Fact}(\mathcal{E},\Phi,w)$ and an injective resolution
$\varepsilon\colon F\to J^{\bullet}$ in
$\operatorname{Ch}(\operatorname{Fact}(\mathcal{A},\Phi,w))$, the map

$\displaystyle\varepsilon_{*}\colon\
\operatorname{Hom}_{\operatorname{\sf{Fact}}(\mathcal{A},\Phi,w)}(E,F)^{p}\to\operatorname{Hom}_{\operatorname{\sf{Fact}}(\mathcal{A},\Phi,w)}(E,J^{\bullet})^{p}$

of cochain complexes of abelian groups is a quasi-isomorphism for every
$p\in\mathbb{Z}$. Here injective resolution means that $\varepsilon$ is a
quasi-isomorphism given by an injection $\varepsilon_{0}\colon
F\hookrightarrow J^{0}$, and all components of the factorizations $J^{n}$ are
injective objects in $\mathcal{A}$.

###### Proof.

Since
$\operatorname{Hom}_{\operatorname{\sf{Fact}}(\mathcal{A},\Phi,w)}(E,F)^{p}\cong\operatorname{Hom}_{\operatorname{\sf{Fact}}(\mathcal{A},\Phi,w)}(E[p],F)^{0}$,
we may assume that $p=0$. If $\varepsilon\colon F\to J^{\bullet}$ is an
injective resolution, then we have the induced injective resolutions
$\varepsilon_{i}\colon F_{i}\to J^{\bullet}_{i}$ for $i=0,1$. Then the map
$\varepsilon_{*}\colon\operatorname{Hom}_{\operatorname{\sf{Fact}}(\mathcal{A},\Phi,w)}(E,F)^{0}\to\operatorname{Hom}_{\operatorname{\sf{Fact}}(\mathcal{A},\Phi,w)}(E,J^{\bullet})^{0}$
of cochain complexes is the direct sum of two maps

$\displaystyle(\varepsilon_{*})_{i}\colon\
\operatorname{Hom}_{\mathcal{A}}(E_{i},F_{i})\to\operatorname{Hom}_{\mathcal{A}}(E_{i},J^{\bullet}_{i})\qquad(i=0,1)$

of cochain complexes. The $i$-th cohomology of the cochain complex
$\operatorname{Hom}_{\mathcal{A}}(E_{i},J^{\bullet}_{i})$ is nothing but
$\operatorname{Ext}^{i}_{\mathcal{A}}(E_{i},F_{i})$. Hence, by the assumption,
the map $(\varepsilon_{*})_{i}$ is a quasi-isomorphism, and so is
$\varepsilon_{*}$. ∎

######  Lemma 4.2.

Assume that we have $\operatorname{Ext}^{i}_{\mathcal{A}}(A,B)=0$ for
arbitrary objects $A,B\in\mathcal{E}$ and all $i>0$. Then the natural functor
$\operatorname{K}(\mathcal{E},\Phi,w)\to\operatorname{D^{co}}(\mathcal{A},\Phi,w)$
is fully faithful.

###### Proof.

Let $E,F\in\operatorname{Fact}(\mathcal{E},\Phi,w)$ be objects. We need to
show that the map

$\displaystyle Q\colon\operatorname{Hom}_{{\rm
K}(\mathcal{A},\Phi,w)}(E,F)\to\operatorname{Hom}_{\operatorname{D^{co}}(\mathcal{A},\Phi,w)}(E,F)$

defined by the Verdier quotient $Q\colon{\rm
K}(\mathcal{A},\Phi,w)\to\operatorname{D^{co}}(\mathcal{A},\Phi,w)$ is an
isomorphism. By [2, Proposition 2.19] we can take an injective resolution
$\iota\colon F\to(I^{\bullet},d_{I}^{\bullet})=\big{(}\cdots\to 0\to
I^{0}\xrightarrow{d_{I}^{0}}I^{1}\xrightarrow{d_{I}^{1}}\cdots\big{)}$ of $F$
in $\operatorname{Ch}(\operatorname{Fact}(\mathcal{A},\Phi,w))$ in the sense
that all $I^{i}$ has injective components. Then, we have the induced map

$\displaystyle\operatorname{Tot}(\iota)\colon\
F\to\operatorname{Tot}(I^{\bullet})$

in $\operatorname{Fact}(\mathcal{A},\Phi,w)$ which becomes an isomorphism in
$\operatorname{D^{co}}(\mathcal{A},\Phi,w)$. Consider the following
commutative diagram:

${\operatorname{Hom}_{{\rm
K}(\mathcal{A},\Phi,w)}(E,F)}$${\operatorname{Hom}_{\operatorname{D^{co}}(\mathcal{A},\Phi,w)}(E,F)}$${\operatorname{Hom}_{{\rm
K}(\mathcal{A},\Phi,w)}(E,\operatorname{Tot}(I^{\bullet}))}$${\operatorname{Hom}_{\operatorname{D^{co}}(\mathcal{A},\Phi,w)}(E,\operatorname{Tot}(I^{\bullet})).}$$\scriptstyle{Q}$$\scriptstyle{\operatorname{Tot}(\iota)_{*}}$$\scriptstyle{\operatorname{Tot}(\iota)_{*}}$$\scriptstyle{Q}$

In the above diagram, the vertical arrow on the right hand side is an
isomorphism, and the horizontal arrow on the bottom is also an isomorphism by
[2, Lemma 2.24] and [12, Proposition B.2(I)] (see also [12, Remark 2.14]).
Hence it suffices to show that the vertical arrow on the left hand side is an
isomorphism, and for this we prove that the map

$\operatorname{Tot}(\iota)_{*}\colon\
\operatorname{Hom}_{\operatorname{\sf{Fact}}(\mathcal{A},\Phi,w)}(E,F)\to\operatorname{Hom}_{\operatorname{\sf{Fact}}(\mathcal{A},\Phi,w)}(E,\operatorname{Tot}(I^{\bullet}))$
(4.1)

of cochain complexes is a quasi-isomorphism. The complex
$\operatorname{Hom}_{\operatorname{\sf{Fact}}(\mathcal{A},\Phi,w)}(E,F)$ can
be seen as the double complex $X^{\bullet,\bullet}$ defined by

$X^{p,q}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\begin{cases}\operatorname{Hom}_{\operatorname{\sf{Fact}}(\mathcal{A},\Phi,w)}(E,F)^{p},&q=0,\\\
0,&q\neq 0.\end{cases}$

On the other hand, we can consider the double complex $Y^{\bullet,\bullet}$
defined by

$Y^{p,q}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{Hom}_{\operatorname{\sf{Fact}}(\mathcal{A},\Phi,w)}(E,I^{q})^{p},$

where the differentials $d_{Y}^{p,\bullet}\colon Y^{p,\bullet}\to
Y^{p,\bullet+1}$ and $d_{Y}^{\bullet,q}\colon Y^{\bullet,q}\to
Y^{\bullet+1,q}$ are given by $d_{Y}^{p,\bullet}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=(d_{I}^{\bullet})_{*}$
and $d_{Y}^{\bullet,q}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=d_{(E,I^{q})}^{\bullet}$,
where $d_{(E,I^{q})}^{\bullet}$ is the differential of the complex
$\operatorname{Hom}_{\operatorname{\sf{Fact}}(\mathcal{A},\Phi,w)}(E,I^{q})^{\bullet}$.
Then the injective map $\iota\colon F\to I^{0}$ defining the resolution
$\iota\colon F\to I^{\bullet}$ gives the morphism

$\iota_{*}\colon\ X^{\bullet,\bullet}\to Y^{\bullet,\bullet}$ (4.2)

of double complexes. By definition, we have isomorphisms
$\operatorname{Tot}(X^{\bullet,\bullet})=\operatorname{Hom}_{\operatorname{\sf{Fact}}(\mathcal{A},\Phi,w)}(E,F)$
and
$\operatorname{Tot}(Y^{\bullet,\bullet})\cong\operatorname{Hom}_{\operatorname{\sf{Fact}}(\mathcal{A},\Phi,w)}(E,\operatorname{Tot}(I^{\bullet}))$,
and the map $\operatorname{Tot}(\iota)_{*}$ in (4.1) is the totalization
$\operatorname{Tot}(\iota_{*})$ of the map $\iota_{*}$ in (4.2). By Lemma 4.1
the map $\iota_{*}\colon X^{p,\bullet}\to Y^{p,\bullet}$ of cochain complexes
is a quasi-isomorphism for any $p\in\mathbb{Z}$, and therefore the map in
(4.1) is also a quasi-isomorphism by [9, Theorem 1.9.3] (see also [12, Lemma
2.46]). This completes the proof. ∎

######  Lemma 4.3.

Let $F^{\bullet}\in\operatorname{D}^{+}(\mathcal{A})$ be an object, and let
$m\in\mathbb{Z}$ be an integer such that $F^{k}=0$ for any $k<m$. Then there
are a family $\\{F^{\bullet}_{i}\\}_{i\in\mathbb{Z}}$ of objects in
$\operatorname{D^{b}}(\mathcal{A})$ with $F^{k}_{i}=0$ for any $k<m$ and $i$
and an exact triangle

$\displaystyle\bigoplus_{i\in\mathbb{Z}}F^{\bullet}_{i}\to\bigoplus_{i\in\mathbb{Z}}F^{\bullet}_{i}\to
F^{\bullet}\to\bigoplus_{i\in\mathbb{Z}}F^{\bullet}_{i}[1].$

###### Proof.

Denote by $\tau_{\leq i}(F^{\bullet})$ the canonical truncation of
$F^{\bullet}$ defined by $\tau_{\leq i}(F^{\bullet})^{j}=F^{j}$ if $j<i$,
$\tau_{\leq i}(F^{\bullet})^{i}=\operatorname{Ker}(d^{i}\colon F^{i}\to
F^{i+1})$ and $\tau_{\leq i}(F^{\bullet})^{j}=0$ if $j>i$. If we set
$F_{i}^{\bullet}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\tau_{\leq
i}(F^{\bullet})$, $F_{i}^{\bullet}$ lies in
$\operatorname{D^{b}}(\mathcal{A})$ and $F^{k}_{i}=0$ for any $k<m$ and $i$.
The object $F^{\bullet}$ is the cokernel of an injective morphism

$\displaystyle\bigoplus_{i\in\mathbb{Z}}F^{\bullet}_{i}\to\bigoplus_{i\in\mathbb{Z}}F^{\bullet}_{i}$

in the abelian category $\operatorname{Ch}(\mathcal{A})$ given by
$\operatorname{id}\oplus(-\iota_{i})\colon F^{\bullet}_{i}\to
F^{\bullet}_{i}\oplus F^{\bullet}_{i+1}$, where $\iota_{i}\colon
F^{\bullet}_{i}\to F^{\bullet}_{i+1}$ is a natural injection. Hence we have a
triangle

$\displaystyle\bigoplus_{i\in\mathbb{Z}}F^{\bullet}_{i}\to\bigoplus_{i\in\mathbb{Z}}F^{\bullet}_{i}\to
F^{\bullet}\to\bigoplus_{i\in\mathbb{Z}}F^{\bullet}_{i}[1]$

in $\operatorname{D}^{+}(\mathcal{A})$. ∎

Let $\mathcal{B}$ be a Grothendieck category with enough projectives, and let
$F\colon\mathcal{A}\to\mathcal{B}$ be an additive functor such that it
commutes with small direct sums and has a left adjoint functor
$G\colon\mathcal{B}\to\mathcal{A}$. The adjunction $G\dashv F$ implies that
$F$ is left exact and $G$ is right exact. By [10, Proposition 14.3.4], $F$
admits a right derived functor

$\displaystyle\mathbb{R}F\colon\
\operatorname{D}(\mathcal{A})\to\operatorname{D}(\mathcal{B}),$

which commutes with small direct sums, and $\mathbb{R}F$ restricts to a
functor
$\mathbb{R}^{+}\\!F\colon\operatorname{D}^{+}(\mathcal{A})\to\operatorname{D}^{+}(\mathcal{B})$.
On the other hand, by [10, Theorem 14.4.3], $G$ admits a left derived functor

$\displaystyle\mathbb{L}G\colon\
\operatorname{D}(\mathcal{B})\to\operatorname{D}(\mathcal{A}),$

and by [10, Theorem 14.4.5] the adjunction $G\dashv F$ induces an adjunction

$\displaystyle\mathbb{L}G\dashv\mathbb{R}F.$

######  Lemma 4.4.

Assume that
$\mathbb{R}^{+}\\!F\colon\operatorname{D}^{+}(\mathcal{A})\to\operatorname{D}^{+}(\mathcal{B})$
restricts to the functor
$\mathbb{R}F\colon\operatorname{D^{b}}(\mathcal{A})\to\operatorname{D^{b}}(\mathcal{B})$
and that there exists a positive integer $d>0$ such that for every object
$B\in\mathcal{B}$ the projective dimension of $B$ is less than $d$. Then, we
have the following:

  * $1.$

The functor $\mathbb{L}G$ restricts to a functor
$\mathbb{L}^{+}G\colon\operatorname{D}^{+}(\mathcal{B})\to\operatorname{D}^{+}(\mathcal{A})$.

  * $2.$

If
$\mathbb{R}F\colon\operatorname{D^{b}}(\mathcal{A})\to\operatorname{D^{b}}(\mathcal{B})$
is fully faithful, so is
$\mathbb{R}^{+}\\!F\colon\operatorname{D}^{+}(\mathcal{A})\to\operatorname{D}^{+}(\mathcal{B})$.

###### Proof.

(1) Let $B^{\bullet}\in\operatorname{D}^{+}(\mathcal{B})$ be an object, and
let $m\in\mathbb{Z}$ be an integer such that $B^{k}=0$ for any $k<m$. Then by
Lemma 4.3 there are a family $\\{B^{\bullet}_{i}\\}_{i\in\mathbb{Z}}$ of
objects in $\operatorname{D^{b}}(\mathcal{B})$ with $B_{i}^{k}=0$ for any
$k<m$ and $i$ and an exact sequence

$\bigoplus_{i\in\mathbb{Z}}B^{\bullet}_{i}\to\bigoplus_{i\in\mathbb{Z}}B^{\bullet}_{i}\to
B^{\bullet}\to\bigoplus_{i\in\mathbb{Z}}B^{\bullet}_{i}[1].$ (4.3)

We set $C_{i}^{\bullet}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\mathbb{L}G(B^{\bullet}_{i})$.
Then, by the assumption, $C_{i}^{k}=0$ for any $k<m-d$ and $i$, and so
$\bigoplus_{i\in\mathbb{Z}}C_{i}^{\bullet}$ lies in
$\operatorname{D}^{+}(\mathcal{A})$. Since $\mathbb{L}G$ admits a right
adjoint functor, it commutes with small direct sums. Applying the functor
$\mathbb{L}G$ to the triangle (4.3), we obtain a triangle

$\displaystyle\bigoplus_{i\in\mathbb{Z}}C^{\bullet}_{i}\to\bigoplus_{i\in\mathbb{Z}}C^{\bullet}_{i}\to\mathbb{L}G(B^{\bullet})\to\bigoplus_{i\in\mathbb{Z}}C^{\bullet}_{i}[1],$

which implies that
$\mathbb{L}G(B^{\bullet})\in\operatorname{D}^{+}(\mathcal{A})$.

(2) Since $\mathbb{R}^{+}\\!F$ admits a left adjoint functor $\mathbb{L}^{+}G$
by (1), it is enough to show that the adjunction morphism

$\displaystyle\varepsilon\colon\ \Upphi\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\mathbb{L}^{+}G\circ\mathbb{R}^{+}\\!F\to\operatorname{id}_{\operatorname{D}^{+}(\mathcal{A})}$

is an isomorphism of functors. Let
$A^{\bullet}\in\operatorname{D}^{+}(\mathcal{A})$ be an object. Then by Lemma
4.3 there are a family $\\{A^{\bullet}_{i}\\}_{i\in\mathbb{Z}}$ of objects in
$\operatorname{D^{b}}(\mathcal{A})$ and an exact triangle

$\displaystyle\bigoplus_{i\in\mathbb{Z}}A^{\bullet}_{i}\xrightarrow{f}\bigoplus_{i\in\mathbb{Z}}A^{\bullet}_{i}\xrightarrow{g}A^{\bullet}\to\bigoplus_{i\in\mathbb{Z}}A^{\bullet}_{i}[1].$

Consider the following commutative diagram

${\Upphi\big{(}\bigoplus_{i\in\mathbb{Z}}A^{\bullet}_{i}\big{)}}$${\Upphi\big{(}\bigoplus_{i\in\mathbb{Z}}A^{\bullet}_{i}\big{)}}$${\Upphi(A^{\bullet})}$${\bigoplus_{i\in\mathbb{Z}}A^{\bullet}_{i}}$${\bigoplus_{i\in\mathbb{Z}}A^{\bullet}_{i}}$${A^{\bullet},}$$\scriptstyle{\Upphi(f)}$$\scriptstyle{\varepsilon}$$\scriptstyle{\Upphi(g)}$$\scriptstyle{\varepsilon}$$\scriptstyle{\varepsilon}$$\scriptstyle{f}$$\scriptstyle{g}$

where the horizontal sequences are triangles and the vertical arrows are
induced by $\varepsilon$. Since $\Upphi$ commutes with direct sums, the
vertical arrows on the left and middle are the direct sums of morphisms
$\varepsilon_{i}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\varepsilon(A_{i}^{\bullet})\colon\Upphi(A_{i}^{\bullet})\to
A_{i}^{\bullet}$, and each $\varepsilon_{i}$ is an isomorphism since
$\mathbb{R}F\colon\operatorname{D^{b}}(\mathcal{A})\to\operatorname{D^{b}}(\mathcal{B})$
is fully faithful and $A_{i}^{\bullet}\in\operatorname{D^{b}}(\mathcal{A})$.
Hence the vertical arrow on the right hand side is also an isomorphism. ∎

Now we are ready to prove that a tilting object induces an equivalence of
derived factorization categories. Let $T\in\mathcal{A}$ be a partial tilting
object such that the exact subcategory $\operatorname{Add}T\subset\mathcal{A}$
is preserved by $\Phi$. Set $\Lambda\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{End}_{\mathcal{A}}(T)$,
and let $(\Psi,v)$ be a potential on $\operatorname{Mod}\Lambda$. We define

$\displaystyle\operatorname{D^{co}Mod}(\Lambda,\Psi,v)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{D^{co}}(\operatorname{Mod}\Lambda,\Psi,v),$
$\displaystyle\operatorname{Dmod}(\Lambda,\Psi,v)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{D}(\operatorname{mod}\Lambda,\Psi,v).$

Consider the functor

$\displaystyle\operatorname{\sf{F}}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{Hom}_{\mathcal{A}}(T,-)\colon\
\mathcal{A}\to\operatorname{Mod}\Lambda,$

and assume that there is a left adjoint functor
$\operatorname{\sf{G}}\colon\operatorname{Mod}\Lambda\to\mathcal{A}$, and that
$\operatorname{\sf{F}}$ and $\operatorname{\sf{G}}$ are factored with respect
to $(\Phi,w)$ and $(\Psi,v)$. Let

$\displaystyle\mathcal{C}\subset\mathcal{A}$

be an abelian subcategory preserved by $\Phi$ such that
$\operatorname{add}T\\!\subset\mathcal{C}$,
$\operatorname{\sf{F}}(\mathcal{C})\\!\subset\operatorname{mod}\Lambda$ and
$\operatorname{\sf{G}}(\operatorname{mod}\Lambda)\subset\mathcal{C}$. Assume
that the natural functors
$\operatorname{D^{b}}(\mathcal{C})\to\operatorname{D^{b}}(\mathcal{A})$ and
$\operatorname{D}(\mathcal{C},\Phi,w)\to\operatorname{D^{co}}(\mathcal{A},\Phi,w)$
are fully faithful and that
$\mathbb{R}\operatorname{\sf{F}}\colon\operatorname{D^{b}}(\mathcal{A})\to\operatorname{D^{b}}(\operatorname{Mod}\Lambda)$
restricts to
$\mathbb{R}\operatorname{\sf{F}}\colon\operatorname{D^{b}}(\mathcal{C})\to\operatorname{D^{b}}(\operatorname{mod}\Lambda)$.
Then
$\mathbb{R}\operatorname{\sf{F}}\colon\operatorname{D^{co}}(\mathcal{A},\Phi,w)\to\operatorname{D^{co}Mod}(\Lambda,\Psi,v)$
restricts to the functor

$\displaystyle\mathbb{R}\operatorname{\sf{F}}\colon\
\operatorname{D}(\mathcal{C},\Phi,w)\to\operatorname{Dmod}(\Lambda,\Psi,v).$

Moreover we assume that $\Lambda$ is of finite global dimension. Then we have
the left derived functor

$\displaystyle\mathbb{L}\operatorname{\sf{G}}\colon\
\operatorname{Dmod}(\Lambda,\Psi,v)\to\operatorname{D}(\mathcal{C},\Phi,w)$

that is left adjoint to
$\mathbb{R}\operatorname{\sf{F}}\colon\operatorname{D}(\mathcal{C},\Phi,w)\to\operatorname{Dmod}(\Lambda,\Psi,v)$.

######  Theorem 4.5.

Suppose that the adjunction morphisms
$\Lambda\to\operatorname{\sf{F}}\operatorname{\sf{G}}(\Lambda)$ and
$\operatorname{\sf{G}}\operatorname{\sf{F}}(T)\to T$ are isomorphisms.

  * $1.$

The functor
$\mathbb{L}\operatorname{\sf{G}}\colon\operatorname{Dmod}(\Lambda,\Psi,v)\to\operatorname{D}(\mathcal{C},\Phi,w)$
is fully faithful, and it factors through an equivalence

$\displaystyle\mathbb{L}\operatorname{\sf{G}}\colon\operatorname{Dmod}(\Lambda,\Psi,v)\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\operatorname{Kadd}(T,\Phi,w).$

  * $2.$

Assume that every object in $\mathcal{A}$ has finite injective resolution and
that the right derived functor
$\mathbb{R}\operatorname{\sf{F}}\colon\operatorname{D^{b}}(\mathcal{A})\to\operatorname{D^{b}}(\operatorname{Mod}\Lambda)$
is an equivalence. Then the functor

$\displaystyle\mathbb{R}\operatorname{\sf{F}}\colon\
\operatorname{D^{co}}(\mathcal{A},\Phi,w)\to\operatorname{D^{co}Mod}(\Lambda,\Psi,v)$

is an equivalence, and it restricts to the equivalence

$\displaystyle\mathbb{R}\operatorname{\sf{F}}\colon\
\operatorname{D}(\mathcal{C},\Phi,w)\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\operatorname{Dmod}(\Lambda,\Psi,v).$

###### Proof.

(1) Recall that the functor
$\mathbb{L}\operatorname{\sf{G}}\colon\operatorname{Dmod}(\Lambda,\Psi,v)\to\operatorname{D}(\mathcal{C},\Phi,w)$
is the composition

${\operatorname{Dmod}(\Lambda,\Psi,v)}$${\operatorname{K}(\operatorname{proj}\Lambda,\Psi,v)}$${\operatorname{Kadd}(T,\Phi,w)}$${\operatorname{D}(\mathcal{C},\Phi,w),}$$\scriptstyle{\sim}$$\scriptstyle{\operatorname{\sf{G}}}$$\scriptstyle{Q}$

where the first equivalence follows from Proposition 2.7 and $Q$ is the
natural quotient functor. Since we have the equivalence
$\operatorname{\sf{G}}\colon\operatorname{proj}\Lambda\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\operatorname{add}T$,
the middle functor $\operatorname{\sf{G}}$ is also an equivalence. Thus the
statement follows from Lemma 4.2 and the assumption that the natural functor
$\operatorname{D}(\mathcal{C},\Phi,w)\to\operatorname{D^{co}}(\mathcal{A},\Phi,w)$
is fully faithful.

(2) Since
$(\mathbb{R}\operatorname{\sf{F}})^{+}\colon\operatorname{D}^{+}(\mathcal{A})\to\operatorname{D}^{+}(\operatorname{Mod}\Lambda)$
is fully faithful by Lemma 4.4, the functor $\mathbb{R}\operatorname{\sf{F}}$:
$\operatorname{D^{co}}(\mathcal{A},\Phi,w)\to\operatorname{D^{co}Mod}(\Lambda,\Psi,v)$
is also fully faithful by [2, Lemma 4.11]. Consider the following commutative
diagram:

${\operatorname{K}\operatorname{Add}(T,\Phi,w)}$${\operatorname{K}\operatorname{Proj}(\Lambda,\Psi,v)}$${\operatorname{D^{co}}(\mathcal{A},\Phi,w)}$${\operatorname{D^{co}Mod}(\Lambda,\Psi,v).}$$\scriptstyle{\operatorname{\sf{F}}}$$\scriptstyle{\mathbb{R}\operatorname{\sf{F}}}$

Since the additive functor
$\operatorname{\sf{F}}\colon\operatorname{Add}(T)\to\operatorname{Proj}\Lambda$
is an equivalence, the top horizontal arrow is an equivalence. Moreover, the
vertical arrow on the right hand side is essentially surjective by Proposition
2.7. Hence
$\mathbb{R}\operatorname{\sf{F}}\colon\operatorname{D^{co}}(\mathcal{A},\Phi,w)\to\operatorname{D^{co}Mod}(\Lambda,\Psi,v)$
is essentially surjective, and so it is an equivalence.

Since the natural embedding functors
$\operatorname{D}(\mathcal{C},\Phi,w)\to\operatorname{D^{co}}(\mathcal{A},\Phi,w)$
and
$\operatorname{Dmod}(\Lambda,\Psi,v)\to\operatorname{D^{co}Mod}(\Lambda,\Psi,v)$
are fully faithful, the functor
$\mathbb{R}\operatorname{\sf{F}}\colon\\!\operatorname{D}(\mathcal{C},\Phi,w)\\!\to\operatorname{Dmod}(\Lambda,\Psi,v)$
and its left adjoint
$\mathbb{L}\operatorname{\sf{G}}\colon\\!\operatorname{Dmod}(\Lambda,\Psi,v)\to\operatorname{D}(\mathcal{E},\Phi,w)$
are fully faithful. Hence
$\mathbb{R}\operatorname{\sf{F}}\colon\\!\operatorname{D}(\mathcal{C},\Phi,w)\to\operatorname{Dmod}(\Lambda,\Psi,v)$
is also an equivalence. ∎

###  4.2 Derived factorization categories of noncommutative gauged LG models