# Resolution of the diagonal on the Root Stacks

Yu Zhao

###### Abstract.

In this paper, we give a new constructive proof of the semi-orthogonal
decomposition of the derived category of (quasi)-coherent sheaves of root
stacks, through an explicit resolution of the diagonal.

> To laugh is to live profoundly.
>
> Milan Kundera

##  1\. Introduction

###  1.1. The main result of this paper

The main result of this paper is a constructive proof on the semi-orthogonal
decomposition of derived category of (quasi)-coherent sheaves on root stacks,
which has been intensively studied by Ishii-Ueda [11], Bergh-Lunts-Schnürer
[4], Kuznetsov-Perry [15], Bergh-Schnürer [5] and recently by Bodzenta-Donovan
[6]:

######  Theorem 1.1.

Let $\mathcal{D}$ be an effective Cartier divisor of an algebraic stack
$\mathcal{X}$. Given an integer $l>1$, let $\mathcal{X}_{\mathcal{D},l}$ as
the $l$-th root stack of $\mathcal{X}$ along $\mathcal{D}$ and
$\mathcal{D}_{l}:=\mathcal{D}\times_{B\mathbb{G}_{m}}B\mathbb{G}_{m}$, where
the map from $\mathcal{D}\to B\mathbb{G}_{m}$ is decided by the line bundle
$\mathcal{L}_{\mathcal{D},l}:=\mathcal{O}_{\mathcal{X}}(-\mathcal{D})|_{\mathcal{D}}$
and the map $B\mathbb{G}_{m}\to B\mathbb{G}_{m}$ is the $l$-th power map. Then
under the following natural commutative diagram

${\mathcal{D}_{l}}$${\mathcal{X}_{\mathcal{D},l}}$${\mathcal{D}}$${\mathcal{X}.}$$\scriptstyle{i_{\mathcal{X}_{\mathcal{D},l}}}$$\scriptstyle{Bt_{\mathcal{D}}^{l}}$$\scriptstyle{\theta_{\mathcal{X}}^{l}}$$\scriptstyle{i_{\mathcal{X}}}$

the following functors

$\displaystyle\triangleright_{i,\mathcal{X}}:=(-\otimes\mathcal{L}_{D,l}^{i})\circ
Ri_{\mathcal{X}_{\mathcal{D},l}*}\circ
LBt_{\mathcal{D}}^{l*}:D^{+}_{qcoh}(\mathcal{D})\to
D_{qcoh}^{+}(\mathcal{X}_{\mathcal{D},l}),$ $\displaystyle
L\theta_{\mathcal{X}}^{l*}:D_{qcoh}^{+}(\mathcal{X})\to
D_{qcoh}^{+}(\mathcal{X}_{\mathcal{D},l})$

are fully faithful. Moreover, let $D_{l,\mathcal{D}}^{i}$ be the image of
$\triangleright_{i,\mathcal{X}}$, then for any $0\leq i\leq l-1$, we have the
semi-orthogonal decomposition

(1.1)
$D^{+}_{qcoh}(\mathcal{X}_{\mathcal{D},l}):=<D_{l,\mathcal{D}}^{i-l+1},\cdots,D_{l,\mathcal{D}}^{-1},L\theta^{*}_{l}D^{+}_{qcoh}(\mathcal{X}),D_{l,\mathcal{D}}^{0},\cdots,D_{l,\mathcal{D}}^{i-1}>.$

Similar arguments also hold for complexes with bounded or coherent
cohomologies.

######  Remark 1.2.

Our setting is similar to Bergh-Lunts-Schnürer [4], which only assumes that
$\mathcal{X}$ is an algebraic stack over $\mathbb{Z}$. Our strategy could also
be generalized to the derived category of perfect complexes with very mild
modifications. We will leave it as an exercise for readers with interest.

######  Remark 1.3.

Bodzenta-Donovan [6] pointed out that the above semi-orthogonal decomposition
is $2l$-periodic, and thus induces higher spherical functors. We will try to
sketch a potential relation of those higher spherical functors with the
categorical representation theory in Section 1.4.

######  Remark 1.4.

The root stack construction by Cadman [7] or Abramovich-Graber-Vistoli [1]
only requires a global section of a line bundle $\mathcal{L}$ over
$\mathcal{X}$. In this paper, we only consider the case that the global
section is injective i.e. induced by an effective Cartier divisor. We make
this restriction so we can work in a more classical framework rather than the
derived algebraic geometry, and this issue can be solved by introducing the
definition “virtual effective Cartier divisor” in Khan-Rydh [12].

###  1.2. Resolutions of the diagonal and the semi-orthogonal decomposition

Given an integer $n>0$, Beilinson [3] gave a resolution of the diagonal of
$\mathbb{P}^{n}$ by the long-exact sequence:

$0\to\wedge^{n}\Omega_{\mathbb{P}^{n}}(n)\boxtimes\mathcal{O}_{\mathbb{P}^{n}}(-n)\to\cdots\to\Omega_{\mathbb{P}^{n}}\boxtimes\mathcal{O}_{\mathbb{P}^{n}}(-1)\to\mathcal{O}_{\mathbb{P}^{n}\times\mathbb{P}^{n}}\to\Delta_{*}\mathcal{O}_{\mathbb{P}^{n}}\to
0,$

which induces the semi-orthogonal decomposition

$D^{b}_{coh}(\mathbb{P}^{n})=<\mathcal{O}_{\mathbb{P}_{n}},\mathcal{O}_{\mathbb{P}_{n}}(1),\cdots,\mathcal{O}_{\mathbb{P}_{m}}(n)>=<\mathcal{O}_{\mathbb{P}_{n}},\Omega_{\mathbb{P}_{n}}(1),\cdots,\wedge^{n}\Omega_{\mathbb{P}_{n}}(n)>.$

We refer to Kuznetsov [13][14] for a general introduction to the semi-
orthogonal decomposition of the derived category of coherent sheaves on
algebraic varieties. One of the main purposes of this paper is to give an
explicit resolution of the diagonal which induced the semi-orthogonal
decomposition, following Appendix A of our work [18] for Orlov’s semi-
orthogonal decomposition theorem for blow-up of smooth varieties [16]. We
prove that

######  Theorem 1.5 (Theorem 4.6).

Let

$\triangleleft_{i,\mathcal{X}}:D_{qcoh}^{+}(\mathcal{X}_{\mathcal{D},l})\to
D_{qcoh}^{+}(\mathcal{D})$

be the cohomological degree $1$ shift of the right adjoint functor of
$\triangleright_{i,\mathcal{X}}$. Given $0\leq n\leq m\leq n-1$, there exists

$\tau_{n,m,\mathcal{X}}:=D^{b}_{coh}(\mathcal{X}_{\mathcal{D},l}\times_{\theta_{\mathcal{X}}^{l},\mathcal{X},\theta_{\mathcal{X}}^{l}}\mathcal{X}_{\mathcal{D},l}).$

such that the Fourier-Mukai functors generated by kernels
$\tau_{n,m,\mathcal{X}}$ satisfy:

$\overline{\tau}_{n,m,\mathcal{X}}:D^{+}_{qcoh}(\mathcal{X}_{\mathcal{D},l})\to
D^{+}_{qcoh}(\mathcal{X}_{\mathcal{D},l})$

  1. (1)

we have

$\overline{\tau_{n,n,\mathcal{X}}}\cong\otimes\mathcal{O}_{\mathcal{X}_{\mathcal{D},l}}(-\mathcal{D}_{l})^{n},\quad\overline{\tau_{0,l-1,\mathcal{X}}}\cong
L\theta_{\mathcal{X}}^{l*}R\theta_{\mathcal{X}*}^{l}$

  2. (2)

for any $0\leq n<m\leq l-1$, we have canonical triangles:

$\displaystyle\overline{\tau_{n,m-1,\mathcal{X}}}\to\overline{\tau_{n,m,\mathcal{X}}}\to\bigoplus_{i=0}^{m}\triangleright_{m-i,\mathcal{X}}\triangleleft_{i,\mathcal{X}}$
$\displaystyle\overline{\tau_{n,m,\mathcal{X}}}\to\overline{\tau_{n+1,m,\mathcal{X}}}\to\bigoplus_{i=n+1}^{l-1}\triangleright_{i-1-n,\mathcal{X}}\triangleleft_{i,\mathcal{X}}.$

.

  3. (3)

All those functors map complexes with bounded cohomologies (resp. coherent
cohomologies) to complexes with bounded cohomologies (resp. coherent
cohomologies).

###  1.3. Motivation and derived birational geometry

In [18], we found a surprising relation between the resolution of the diagonal
with the birational geometry of derived schemes. Now we try to briefly
introduce this insight without introducing the machinery of
$\infty$-categories.

For the simplicity, we assume that
$\mathcal{X}=[\mathbb{A}^{1}_{\mathbb{C}}/\mathbb{C}^{*}]$ and the Cartier
divisor is $B\mathbb{C}^{*}$. In this situation $\mathcal{X}_{\mathcal{D},l}$
is also $[\mathbb{A}^{1}_{\mathbb{C}^{1}}/\mathbb{C}^{*}]$ and

$\mathcal{X}_{\mathcal{D},l}\times_{\mathcal{X}}\mathcal{X}_{\mathcal{D},l}\cong[\\{(x,y)\in\mathbb{A}^{2}_{\mathbb{C}}|\prod_{j\in\mathbb{Z}/l\mathbb{Z}}(x-e^{\frac{2\pi
i}{l}\cdot j}y)=0\\}/(\mathbb{C}^{*}\times\mathbb{Z}/l\mathbb{Z})]$

which is $l$-lines that intersect at the original point, quotient the action
of $\mathbb{C}^{*}$ by the scalar action on both $x$ and $y$ and
$\mathbb{Z}/l\mathbb{Z}$ by multiplying the roots of unity on $y$.

Now we blow up the origin point $(0,0)$ in

$\alpha_{l}:=\\{(x,y)\in\mathbb{A}^{2}_{\mathbb{C}}|\prod_{j\in\mathbb{Z}/l\mathbb{Z}}(x-e^{\frac{2\pi
i}{l}\cdot j}y)=0\\}.$

In the classical setting, we will get $l$ lines which do not intersect with
each other, which is exactly $\mathcal{X}_{\mathcal{D},l}$ after quotient
$\mathbb{C}^{*}\times\mathbb{Z}/l\mathbb{Z}$, and the natural projection is
the diagonal morphism. However, in the sense of derived setting of Hekking
[10], we should consider a family of varieties

$\mathcal{M}_{k}:=\\{((x_{1},y_{1}),[x_{2},y_{2}])\in\mathbb{A}^{2}_{\mathbb{C}}\times\mathbb{P}^{1}_{\mathbb{C}}|x_{1}y_{2}=y_{1}x_{2},x_{1}^{l-k}x_{2}^{l}=y_{1}^{l-k}y_{2}^{l}\\},\quad
0\leq k\leq l.$

Then
$\mathcal{M}_{l}\cong\mathbb{A}^{1}_{\mathbb{C}}\times\mathbb{Z}/l\mathbb{Z}$
and for other $k<l$, $\mathcal{M}_{k}$ contains $\mathbb{P}^{1}_{\mathbb{C}}$
which contains points such that $x_{1}=y_{1}=0$. In the sense of derived blow-
up of Hekking [10], we will have

$\mathbb{R}Bl_{\mathcal{M}_{k}}\mathbb{P}^{1}\cong\mathcal{M}_{k+1},\quad\mathbb{R}Bl_{\alpha_{k}}\\{(0,0)\\}\cong\mathcal{M}_{1}$

Moreover, in [18] we systematically studied the variation of certain quasi-
coherent sheaves after the derived blow-up. Thus Theorem 1.1 actually is a
direct corollary of the generalized vanishing theorem in [18].

For algebraic stacks over $\mathbb{Z}$, we should be careful as the group of
$l$-th of unity $\mu_{l}$ and the constant group $\mathbb{Z}/l\mathbb{Z}$ are
not isomorphic in this situation. A detailed computation in this case is given
in Section 2.5.

###  1.4. Relations with the categorical representation theory

In this subsection, we describe a potential relation between the semi-
orthogonal decomposition of root stacks with the categorical representation,
which we will study the detail in future work.

We denote $\Theta:=[\mathbb{A}^{1}/\mathbb{G}_{m}]$ and the morphism of $l$-th
power on $\mathbb{A}^{1}$ induces a morphism $\theta_{l}:\Theta\to\Theta$.
Then the composition of relative Fourier-Mukai transforms induce a monoidal
structure on

$D^{b}_{coh}(\Theta\times_{\theta_{l},\Theta,\theta_{l}}\Theta).$

Then for any $l$-the root stack $\mathcal{X}_{\mathcal{D},l}$, the pull-back
morphisms induce that $D^{b}_{Coh}(\mathcal{X}_{\mathcal{D},l})$ is a
representation of the monoidal category
$D^{b}_{coh}(\Theta\times_{\theta_{l},\Theta,\theta_{l}}\Theta)$. Moreover, we
notice that $Coh(\Theta\times_{\theta_{l},\Theta,\theta_{l}}\Theta)$ consists
of $\mathbb{Z}[x]\otimes_{\mathbb{Z}[x^{l}]}\mathbb{Z}[x]$ bimodules with
certain grading. We expect that the semi-orthogonal decomposition of the root
stacks follows directly from the categorical representation theory of those
bimodules.

###  1.5. Organization of this paper

As we will work on stacks over $\mathrm{Spec}(\mathbb{Z})$, we review the
background of affine groups over commutative rings and the derived category of
(quasi)-coherent sheaves on algebraic stacks in Section 2 and Section 3
respectively. Experts or readers who only care about schemes over $\mathbb{C}$
should feel free to just read Section 2.5, Section 3.4 and Section 3.5. The
proof of Theorem 1.1 and Theorem 1.5 is given in Section 4.

###  1.6. Acknowledgments

The paper is directly inspired by Bodzenta-Donovan [6], where the author
learned and discussed with them in the conference “Current Trends in
Categorically Approach to Algebraic and Symplectic Geometry II” in June 2023
at Kavli IPMU. The author would like to appreciate Agnieszka Bodzenta and Will
Donovan for their interest and many helpful discussions in this topic.

Part of this paper was given in my lecture series in the Shanghai Tech
University and Chinese Academy of Sciences. The author would like to thank the
above institutes and Zhiyuan Ding, Mingliang Cai, Ziyu Zhang, Siqi He and
Baohua Fu for their invitations and support.

The author is supported by World Premier International Research Center
Initiative (WPI initiative), MEXT, Japan, and Grant-in-Aid for Scientific
Research grant (No. 22K13889) from JSPS Kakenhi, Japan.

##  2\. Affine group schemes and actions

In this section, we always assume that $R$ is a commutative ring. We identify
the category of affine schemes over $\mathrm{Spec}(R)$ with the opposite
category of commutative $R$-algebras. Given an affine $R$-scheme $X:=R[X]$, we
identify the abelian category of quasi-coherent (resp. coherent if $R[X]$ is
Noetherian) sheaves $\mathrm{QCoh}(X)$ (resp. $\mathrm{Coh}(X)$) with the
abelian category of (resp. finite generated) $R[X]$-modules
$R[X]-\mathrm{Mod}$ (resp. $R[X]-\mathrm{Mod}^{fg}$).

###  2.1. Affine group schemes

######  Definition 2.1.

We define an $R$-affine group $G$ as a flat, finite type and commutative
$R$-algebra $\pi_{G}:R\to R[G]$ with $R$-algebra morphisms

$m_{G}:R[G]\to R[G]\otimes_{R}R[G],\quad i_{G}:R[G]\to R[G],\quad
e_{G}:R[G]\to R$

which we call multiplication, inversion, and identification respectively, such
that they satisfy the group laws:

$\displaystyle(m_{G}\otimes_{R}id_{R[G]})\circ
m_{G}=(id_{R[G]}\otimes_{R}m_{G})\circ m_{G},$ $\displaystyle R[G]\to
R[G]\otimes_{R}R[G]\otimes_{R}R[G],$ $\displaystyle
id_{R[G]}=(id_{R[G]}\otimes_{R}e_{G})\circ
m_{G}=(e_{G}\otimes_{R}id_{R[G]})\circ m_{G},$ $\displaystyle R[G]\to R[G],$
$\displaystyle\pi_{G}\circ e_{G}=(i_{G}\otimes_{R}id_{R[G]})\circ
m_{G}=(id_{R[G]}\otimes_{R}i_{G})\circ m_{G},$ $\displaystyle R[G]\to R[G].$

We define an $R$-affine group morphism from an $R$-affine group $G$ to another
$R$-affine group $H$ as a morphism of $R$-algebras $f:R[H]\to R[G]$ such that

$\displaystyle(f\otimes_{R}f)\circ m_{H}=m_{G}\circ f:R[H]\to
R[G]\otimes_{R}R[G]$

The $R$-affine group has the following properties by definition:

  * •

Given a morphism of commutative ring $S\to T$ and an $R$-affine group $G$,
$G_{T}:=\mathrm{Spec}(R[G]\otimes_{S}T)$ is also an $T$-affine group by the
induced morphisms. Similarly, affine group morphisms are also stable under
base change.

  * •

Given two morphisms of $R$-affine groups

$f_{1}:G_{1}\to G,\quad f_{2}:G_{2}\to G,$

$G_{1}\times_{f_{1},G,f_{2}}G_{2}$ is also an affine $R$-group with the
multiplication induced by $m_{G_{1}}$ and $m_{G_{2}}$ (if it is fppf over
$\mathrm{Spec}(R)$). Particularly, given a morphism of $R$-affine group
$f:G\to H$, we denote $ker(f):=G\times_{f,H,\pi_{H}}\mathrm{Spec}(R)$.

######  Example 2.2.

The scheme $\mathbb{G}_{m}:=\mathrm{Spec}(\mathbb{Z}[t,t^{-1}])$ has a
$\mathbb{Z}$-affine group structure

$\displaystyle
m_{\mathbb{G}_{m}}:\mathbb{Z}[t,t^{-1}]\to\mathbb{Z}[t_{1},t^{-1}_{1},t_{2},t_{2}^{-1}],$
$\displaystyle\quad t\to t_{1}t_{2},$ $\displaystyle
i_{\mathbb{G}_{m}}:\mathbb{Z}[t,t^{-1}]\to\mathbb{Z}[t,t^{-1}],$
$\displaystyle\quad t\to t^{-1},$ $\displaystyle
e_{\mathbb{G}_{m}}:\mathbb{Z}[t,t^{-1}]\to\mathbb{Z},$ $\displaystyle\quad
t\to 1.$

Given any integer $l$, we have a $\mathbb{Z}$-group
$t^{l}:\mathbb{G}_{m}\to\mathbb{G}_{m}$ by

$t^{l}:\mathbb{Z}[t,t^{-1}]\to\mathbb{Z}[t,t^{-1}],\quad t\to t^{l}.$

When $l>0$, we denote

$\mu_{l}:=ker(t^{l})=\mathrm{Spec}(\mathbb{Z}[t]/(t^{l-1})).$

We denote $\mathbb{G}_{m,R}:=\mathbb{G}_{m}\times_{Z}R$ and make similar
notations for $\mu_{l,R}$ and $t^{l}_{R}$.

###  2.2. Representations of an affine group

######  Definition 2.3.

Let $G$ be an $R$-affine group. We define a $G$-representation $M$ as an
$R$-module with an $R$-module morphism

$\sigma_{G,M}:M\to R[G]\otimes_{R}M$

such that

$\displaystyle(m_{G}\otimes_{R}id_{M})\circ\sigma_{G,M}=(id_{R[G]}\otimes_{R}\sigma_{G,M})\circ\sigma_{G,M},$
$\displaystyle M\to R[G]\otimes_{R}R[G]\otimes_{R}M$ $\displaystyle
id_{M}=(e_{G}\otimes_{R}id_{M})\circ\sigma_{G,M},$ $\displaystyle M\to M.$

We say that $M$ is a finitely generated $G$-representation if $M$ is finitely
generated as an $R$-module. Given two $G$-representations $M$ and $N$, we
define a $G$-equivariant morphism from $M$ to $N$ as an $R$-module morphism
$f:M\to N$ such that

$\sigma_{G,N}\circ f=(id_{R[G]}\otimes_{R}f)\circ\sigma_{G,M},\quad M\to
R[G]\otimes_{R}N.$

The category of $G$-representations with $G$-equivariant morphisms forms an
abelian category, which we denote as $\mathrm{Rep}(G)$. Moreover, if $R$ is
Noetherian, the category of finitely generated $G$-representations form an
abelian subcategory of $\mathrm{Rep}(G)$, which we denote as
$\mathrm{Rep}(G)^{fg}$.

The category of $G$-representations has the following properties by
definition:

  1. (1)

The tensor product of $R$-modules induces a canonical monoidal structure for
$\mathrm{Rep}(G)$.

  2. (2)

Given a $R$-affine group morphism $\psi:G\to H$, let $M$ be an
$H$-representation. Then

$\sigma_{G,M}:=\psi\otimes_{R}\sigma_{H,M}:M\to R[G]\otimes_{R}M$

induces a $G$-representation structure on $M$. Hence it induces an exact and
monoidal pull-back functor:

$\psi^{*}:\mathrm{Rep}(H)\to\mathrm{Rep}(G).$

######  Example 2.4.

There exists a canonical monoidal equivalence between
$\mathrm{Rep}(\mathbb{G}_{m,R})$ with the category of $\mathbb{Z}$-graded
(resp. $\mathbb{Z}/l\mathbb{Z}$) graded $R$-modules
$R-\mathrm{Mod}_{\mathbb{Z}}$ in the following way: given a
$\mathbb{Z}$-graded $R$-module $M=\oplus_{d\in\mathbb{Z}}M_{d}$, we denote
$M|_{d}:M\to M_{d}$ as the projection morphism. Then

$\sigma_{\mathbb{G}_{m,R},M}:=\bigoplus_{d\in\mathbb{Z}}t^{d}\otimes_{R}M|_{d}:M\to
M[t,t^{-1}]$

is a $\mathbb{G}_{m,R}$-representation structure on $M$. Conversely, given
$\mathbb{G}_{m,R}$ representation $M$, the $R$-module morphism

$\sigma_{G,M}:M\to M[t,t^{-1}]$

could be written as

$\sum_{d\in\mathbb{Z}}t^{d}\otimes\sigma_{d}$

where $\sigma_{d}$ is an $R$-module endmorphism of $M$ for each $d$. The law
of group actions requires that

$\sigma_{d}\sigma_{e}=\delta_{de}\sigma_{e},\quad\sum_{d\in\mathbb{Z}}\sigma_{d}=1,$

where $\delta_{de}$ is the Kronecker symbol. Hence we have
$M\cong\oplus_{d\in\mathbb{Z}}\sigma_{d}M$.

Under the above equivalences, we compute the pull-back and push-forward
functor of $t^{l}_{R}:\mathbb{G}_{m,R}\to\mathbb{G}_{m,R}$, which is
represented by

$\displaystyle t^{l*}_{R}:R-\mathrm{Mod}_{\mathbb{Z}}\to
R-\mathrm{Mod}_{\mathbb{Z}},\quad\bigoplus_{d\in\mathbb{Z}}M_{d}\to\bigoplus_{d\in
l\mathbb{Z}}M_{d/l},$ $\displaystyle t^{l}_{R*}:R-\mathrm{Mod}_{\mathbb{Z}}\to
R-\mathrm{Mod}_{\mathbb{Z}},\quad\bigoplus_{d\in\mathbb{Z}}M_{d}\to\bigoplus_{d\in\mathbb{Z}}M_{dl}.$

Both $t_{R}^{l*}$ and $t^{l}_{R*}$ are exact functors and maps finite
generated $R$-modules to finite generated $R$-modules.

Given a rational number $i$, we denote

$\mathcal{L}^{i}_{R}:=\begin{cases}\bigoplus_{d=i}R,&i\in\mathbb{Z},\\\
0,&i\not\in\mathbb{Z}.\end{cases}$

Then we have

$t_{R}^{l*}(\mathcal{L}_{R}^{i})\cong\mathcal{L}_{R}^{li},\quad
t_{R*}^{l}(\mathcal{L}_{R}^{i})\cong\mathcal{L}_{R}^{i/l}.$

Like the case of $\mathbb{G}_{m,R}$, we have a canonical equivalence between
the category of $\mu_{n,R}$ representations and the category of
$\mathbb{Z}/l\mathbb{Z}$-graded $R$-modules.

$\mathrm{Rep}(\mu_{n,R})\cong R-\mathrm{Mod}_{\mathbb{Z}/l\mathbb{Z}}.$

###  2.3. Equivariant affine schemes and morphisms

######  Definition 2.5.

Given an $R$-affine group $G$, we define a $G$-equivariant affine scheme as a
$G$-representation $R[X]$ with a $G$-equivariant morphism

$*_{X}:R[X]\otimes_{R}R[X]\to R[X].$

such that $R[X]$ is an commutative $R$-algebra under the multiplication
$*_{X}$. We denote

$\sigma_{G,X}:=\sigma_{G,R[X]},\quad\pi_{G,X}:=\pi_{G,R[X]}.$

We define an $R$-affine equivariant scheme as a pair $(G,X)$ where $G$ is an
$R$-affine group and $X$ is a $G$-equivariant affine scheme. Given two affine
equivariant schemes $(G,X)$ and $(H,Y)$, an equivariant morphism from $X$ to
$Y$ is defined as a pair $(\psi,f)$, where $\psi:G\to H$ is an $R$-affine
group morphism and $f:R[Y]\to R[X]$ is an $R$-algebra morphism such that

$(\psi\otimes_{R}f)\circ\sigma_{H,Y}=\sigma_{G,X}\circ g,\quad R[Y]\to
R[G]\otimes_{R}R[X]$

The category of equivariant affine schemes and morphisms has the following
properties by definitions:

  1. (1)

Every $R$-affine group morphism $f:G\to H$ induces a $G$-equivariant structure
on $H$. Particularly, there is a canonical $G$-equivariant structure on
$\mathrm{Spec}(R)$.

  2. (2)

Let $\psi:G\to H$ be an $R$-affine group morphism and $Y$ be a $H$-equivariant
scheme, then

$\sigma_{G,X}:=(\psi\otimes_{R}id_{R[X]})\circ\sigma_{H,X}:R[X]\to
R[G]\otimes_{R}R[X]$

induces a $G$-equivariant structure on $X$. Moreover, any equivariant morphism
of equivariant schemes:

$(G,X)\xrightarrow{(\psi,f)}(H,Y)$

factors through

$(G,X)\xrightarrow{(id,f)}(G,Y)\xrightarrow{(\psi,id)}(H,Y).$

  3. (3)

Given two equivariant morphisms

$\displaystyle(\psi_{1},f_{1}):G_{1}\times_{\mathrm{Spec}(R)}X_{1}\to
G\times_{\mathrm{Spec}(R)}X,$
$\displaystyle(\psi_{2},f_{2}):G_{2}\times_{\mathrm{Spec}(R)}X_{2}\to
G\times_{\mathrm{Spec}(R)}X,$

it induces a $G_{1}\times_{G}G_{2}$-equivariant structure on
$X_{1}\times_{X}X_{2}$ by the actions $\sigma_{G_{1},X_{1}}$ and
$\sigma_{G_{2},X_{2}}$.

######  Example 2.6 (GIT quotients).

By Example 2.4, there is a canonical equivalence between the opposite category
of $\mathbb{Z}$-graded (resp. $\mathbb{Z}/l\mathbb{Z}$) commutative
$R$-algebras with the category of $\mathbb{G}_{m,R}$ (resp. $\mu_{l,R}$)
equivariant affine schemes. Moreover, given an $\mathbb{Z}$-graded (resp.
$\mathbb{Z}/l\mathbb{Z}$-graded) algebra

$R[X]=\bigoplus_{d\in\mathbb{Z}}R[X]_{d}\textit{ (resp.
}\bigoplus_{d\in\mathbb{Z}/l\mathbb{Z}}R[X]_{d}\textit{)},$

$R[X]_{0}$ is also an $R$-algebra and all $R[X]_{i}$ is an $R[X]_{0}$-module.
The inclusion of $R[X]_{0}$ into $R[X]$ induces a canonical equivariant
morphism

$\phi_{0,X}:(\mathbb{G}_{m},X)\to(\mathrm{Spec}(R),\mathrm{Spec}(R[X]_{0}))\textit{
or }\phi_{l,X}:(\mu_{l},X)\to(\mathrm{Spec}(R),R[X]_{0}).$

###  2.4. Equivariant modules

######  Definition 2.7.

Given an $R$-affine equivariant scheme $(G,X)$, we define a $G$-equivariant
$R[X]$-module as a $G$-representation $M$ with a $G$-equivariant morphism

$*_{M}:R[X]\otimes_{R}M\to M$

such that $*_{m}$ induces an $R[X]$-module structure on $M$. Given two
$G$-equivariant $R[X]$-modules $M,N$, we define a $G$-equivariant
$R[X]$-module morphism from $M$ to $N$ as a $G$-equivariant morphism $f:M\to
N$ such that $f\circ*_{M}=*_{N}\circ id_{R[X]}\otimes_{R}f$. We say that $M$
is finitely generated if it is finitely generated as $R[X]$-module.

We denote $\mathrm{QCoh}_{G}(X)$ as the abelian category of $G$-equivariant
$R[X]$-modules. If $R[X]$ is Noetherian, we denote $\mathrm{Coh}_{G}(X)$ as
the abelian category of finite generated $G$-equivariant $R[X]$-modules.

The category of equivariant modules has the following property by definition

  1. (1)

We have $\mathrm{QCoh}_{G}(\mathrm{Spec}(R))=\mathrm{Rep}(G)$ and
$\mathrm{Coh}_{G}(\mathrm{Spec}(R))=\mathrm{Rep}(G)^{fg}$.

  2. (2)

The category of $G$-equivariant $R[X]$-modules has a canonical monoidal
structure: let $M$ and $N$ be two $G$-equivariant $R[X]$-modules, the
$R[X]$-module $M\otimes_{R[X]}N$, regarding as an $R$-module, is the cokernel
of

$R[X]\otimes_{R}M\otimes_{R}N\to M\otimes_{R}N,\quad x\otimes m\otimes
n\to*_{M}(xm)\otimes n-x\otimes*_{N}(xn)$

which has a canonical $G$-equivariant structure.

  3. (3)

Given an equivariant morphism $(\psi,f):(G,X)\to(H,Y)$, there is a canonical
pull-back functor

$(\psi,f)^{*}:\mathrm{QCoh}_{H}(Y)\to\mathrm{QCoh}_{G}(X)$

defined in the following way:

     * •

when $\psi=id$, we define $(f,id)^{*}M:=M\otimes_{R[X]}R[Y]$;

     * •

when $f=id$, we define $(f,id)^{*}M:=f^{*}M$ as a $G$-representation, and the
$R[X]$-module structure is still induced by $*_{M}$.

     * •

in general, we define $(\psi,f)^{*}:=(\psi,id)^{*}(id,f)^{*}$.

The pull-back functor $(\psi,f)^{*}$ maps finite generated $H$-equivariant
$R[Y]$-modules to finite generated $G$-equivariant $R[X]$-modules. We denote

$(\psi,f)_{*}:\mathrm{QCoh}_{G}(X)\to\mathrm{QCoh}_{H}(X)$

as the right adjoint functor of $(\psi,f)^{*}$ if it exists.

######  Example 2.8.

Let $R[X]$ be a $\mathbb{Z}$-graded $R$-algebra, with the $\mathbb{G}_{m,R}$
action induced by the grading. By Example 2.4 and Example 2.6, there is a
canonical equivalence between $\mathrm{QCoh}_{\mathbb{G}_{m,R}}(X)$ (resp.
$\mathrm{Coh}_{\mathbb{G}_{m,,R}}(X)$ if $R[X]$ is Noetherian) with the
category of $\mathbb{Z}$-graded (resp. finite generated) $R[X]$-modules. The
similar arguments also work for $\mu_{l,R}$-equivariant schemes.

Particularly, given a $\mathbb{Z}/l\mathbb{Z}$-graded $R$-algebra $R[X]$,
under the GIT quotient map

$\phi_{l,X}:(\mu_{l},X)\to(\mathrm{Spec}(R),\mathrm{Spec}(R[X]_{0}))$

the pull-back and push-forward maps are represented by

$\displaystyle\phi_{l,X}^{*}:\mathrm{QCoh}(\mathrm{Spec}(R[X]_{0}))\to\mathrm{QCoh}_{\mu_{l,R}}(X):$
$\displaystyle\quad
M\to\bigoplus_{d\in\mathbb{Z}/l\mathbb{Z}}M\otimes_{R[X]_{0}}R[X]_{i},$
$\displaystyle\phi_{l,X*}:\mathrm{QCoh}_{\mu_{l,R}}(X)\to\mathrm{QCoh}(\mathrm{Spec}(R[X]_{0})):$
$\displaystyle\quad\bigoplus_{d\in\mathbb{Z}/l\mathbb{Z}}M_{i}\to M_{0}$

Particularly, $\phi_{0,X*}$ is always exact, and maps finite generated
$R[X]$-graded modules to finite generated $R[X]_{0}$-modules if $R[X]$ is
finite generated as a $R[X]_{0}$-module and $R[X]_{0}$ is Noetherian.

###  2.5. The $\mathbb{G}_{m,R}$-equivariant scheme $\mathbb{A}_{R}^{1}$

The scheme $\mathbb{A}^{1}_{R}:=\mathrm{Spec}(R[x])$ has a universal
$\mathbb{G}_{m,R}$-equivariant structure by associating the degree of $x$ as
$1$. Given a positive integer $l$, we denote
$x^{l}:\mathbb{A}^{1}\to\mathbb{A}^{1}$ by

$\mathbb{Z}[x]\to\mathbb{Z}[x],\quad x\to x^{l}.$
$(t^{l},x^{l}):(\mathbb{G}_{m},\mathbb{A}^{1})\to(\mathbb{G}_{m},\mathbb{A}^{1})$

is an equivariant action. Moreover, we have

$(\mathbb{G}_{m},\mathbb{A}^{1})_{(t^{l},x^{l})}\times_{(\mathbb{G}_{m},\mathbb{A}^{1}),(t^{l},x^{l})}(\mathbb{G}_{m},\mathbb{A}^{1})\cong(\mathbb{G}_{m}\times\mu_{l},\alpha_{l})$

where $\alpha_{l}:=\mathrm{Spec}(\mathbb{Z}[x,y]/(x^{l}-y^{l}))$ and the group
action is induced by

$(t,\mu)(x,y)\to(tx,t\mu y)$

There are two equivariant morphisms:

$\alpha_{l,1},\alpha_{l,2}:(\mathbb{G}_{m}\times\mu_{l},\alpha_{l})\to(\mathbb{G}_{m},\mathbb{A}^{1})$

such that

$\alpha_{l,1}((t,\mu),(x,y))=(t,x),\quad\alpha_{l,2}((t,\mu),(x,y))=(t\mu,y).$

The affine group $\mathbb{G}_{m}\times\mu_{l}$ also acts on
$\mathbb{A}^{1}\times\mu_{l}$ by
$(t,\mu_{1})\circ(x,\mu_{2})\to(tx,\mu_{1}\mu_{2})$. We have an
$\mathbb{G}_{m}\times\mu_{l}$-equivariant morphism of affine schemes from
$\mathbb{A}^{1}\times\mu_{l}$ to $\alpha_{l}$ by

$\Delta_{t^{l},x^{l}}:\mathbb{Z}[x,y]/(x^{l}-y^{l})\to\mathbb{Z}[x,t]/(t^{l}-1),\textit{
by }x\to x,y\to ty$

Here $x,y,t$ had homogeneous degree $(1,0)$, $(1,1)$ and $(0,1)$ with respect
to the $\mathbb{G}_{m}\times\mu_{l}$ action.

The morphism $\Delta_{t^{l},x^{l}}$ induces morphisms of equivariant
$\mathbb{Z}[x,y]/(x^{l}-y^{l})$ modules

$\Delta_{t^{l},x^{l}}^{n}:(x,y)^{n}\mathbb{Z}[x,y]/(x^{l}-y^{l})\to(x^{n})\mathbb{Z}[x,t]/(t^{l}-1),\quad
n\in\mathbb{Z}_{\geq 0}$

which is always injective and is an isomorphism when $n=l-1$. For integers
$0\leq m\leq n\leq l-1$, we define

$\tau_{m,n}:=\frac{(x^{n})\mathbb{Z}[x,t]/(t^{l}-1)\oplus(x,y)^{m}\mathbb{Z}[x,y]/(x^{l}-y^{l})}{(x,y)^{n}\mathbb{Z}[x,y]/(x^{l}-y^{l})}\in\mathrm{Coh}_{\mathbb{G}_{m}\times\mu_{l}}(\alpha_{l})$

The modules $\tau_{m,n}$ have the following properties:

  1. (1)

we have

$\tau_{n,n}\cong\Delta_{t^{l},x^{l}*}(x^{n}\mathbb{Z}[x,t]/(t^{l}-1)),\quad\tau_{n,l-1}\cong(x,y)^{n}\mathbb{Z}[x,y]/(x^{l}-y^{l})$

  2. (2)

When $n<m\leq l-1$, there exists a canonical injection morphism

$\tau_{n,m-1}\to\tau_{n,m},\quad\tau_{n,m}\to\tau_{n+1,m}$

such that cokernels are

$\bigoplus_{i=0}^{m}\mathbb{Z}<m,i>,\textit{ and
}\bigoplus_{i=n+1}^{l-1}\mathbb{Z}<n+1,i>,$

respectively, where $<a,b>$ is the homogeneous degree with respect to the
$\mathbb{G}_{m}\times\mu_{l}$ action.

##  3\. Derived category of (quasi)-coherent sheaves on algebraic stacks

In this section, we review the background about algebraic stacks and
(quasi)-coherent sheaves on algebraic stacks, following [17, Tag 0ELS].
Experts or readers who are only interested in the scheme cases should feel
free to read from Section 3.4.

###  3.1. Quotient stacks

Given an $R$-equivariant scheme $(G,X)$, the groupoid in schemes
$(X,G\times_{\mathrm{Spec}(R)}X,\sigma_{G,X},\pi_{G}\times_{\mathrm{Spec}(R)}id_{X},m_{G}\times_{\mathrm{Spec}(R)}id_{X})$
induces an algebraic stack which we denote as $[X/G]$, following [17, Tag
044O]. Let

$\sigma_{X}^{G}:X\to[X/G]$

be the canonical quotient map, we have the Cartesian diagram:

${G\times_{\mathrm{Spec}(R)}X}$${X}$${X}$${{[X/G]}.}$$\scriptstyle{\sigma_{G,X}}$$\scriptstyle{\pi_{G,X}}$$\scriptstyle{\sigma_{X}^{G}}$$\scriptstyle{\sigma_{X}^{G}}$

where all the morphisms are fppf coverings.

Given an $R$-equivariant morphism $(\psi,f):(G,X)\to(H,Y)$, the morphism of
groupoid in $R$-schemes $(f,f\times_{\mathrm{Spec}(R)}f)$ induces a morphism
of quotient stacks:

$[f/\psi]:[X/G]\to[Y/H],$

Particularly, given a morphism of $R$-affine groups $\psi:G\to H$, we denote

$BG:=[\mathrm{Spec}(R)/G],\quad BH:=[\mathrm{Spec}(R)/H],\quad
B\psi:=[id/\psi]:BG\to BH.$

Given two morphisms $(\psi_{1},f_{1}):(G_{1},X_{1})\to(H,Y)$ and
$(\psi_{2},f_{2}):(G_{2},X_{2})\to(H,Y)$, the fiber product of equivariant
morphisms induces the fiber product of algebraic stacks if
$G_{1}\times_{H}G_{2}$ is also fppf over $R$.

######  Example 3.1.

We follow Halpern-Leistner [9] to denote
$\Theta:=[\mathbb{A}^{1}/\mathbb{G}_{m}]$ and denote

$\theta^{l}:=[x^{l}/t^{l}]:\Theta\to\Theta.$

Then the diagonal of $\theta^{l}$, which we denote as $\Delta_{\theta^{l}}$ is
represented by

$[\Delta_{t^{l},x^{l}}/id]:[\mathbb{A}^{1}\times\mu_{l}/\mathbb{G}_{m}\times\mu_{l}]\to[\alpha_{l}/\mathbb{G}_{m}\times\mu_{l}]$

in Section 2.5.

###  3.2. Morphisms of algebraic stacks

We refer to the Stacks Project [17, Tag 04XM], for the property of being flat,
locally of finite type, quasi-compact, quasi-separated and having an affine
diagonal for morphisms of algebraic stacks. Moreover, we mention the following
facts

  1. (1)

Given a morphism $f:\mathcal{X}\to\mathcal{Y}$ such that both $\mathcal{X}$
and $\mathcal{Y}$ are represented by schemes, the definition of the above
properties coincide with the definition for morphisms of schemes.

  2. (2)

The above properties are stable under base change, i.e. given a Cartesian
diagram of algebraic stacks

(*)
${\mathcal{X}^{\prime}}$${\mathcal{X}}$${\mathcal{Y}^{\prime}}$${\mathcal{Y},}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f}$$\scriptstyle{g}$

$f^{\prime}$ is flat (resp. locally of finite type, quasi-compact, quasi-
separated, with an affine diagonal) if $f$ is flat (resp. locally of finite
type, quasi-compact, quasi-separated, with an affine diagonal). Moreover, if
$g$ is a smooth atlas, $f$ is flat (resp. locally of finite type, quasi-
compact, quasi-separated, with an affine diagonal) if $f^{\prime}$ is flat
(resp. locally of finite type, quasi-compact, quasi-separated, with an affine
diagonal).

  3. (3)

We consider a morphism of affine quotient stacks

$[f/\psi]:[X/G]\to[Y/H].$

The morphism $[f/\psi]$ is always quasi-compact. It is flat (resp. of locally
finite type) if $f$ is flat (resp. locally of finite type). It is quasi-
separated and with an affine diagonal if $\psi$ is faithfully flat.

  4. (4)

Given a quasi-compact morphism of algebraic stacks
$f:\mathcal{X}\to\mathcal{Y}$ such that $\mathcal{Y}$ is represented by an
affine scheme, then $\mathcal{X}$ has a smooth cover by an affine scheme.

######  Example 3.2.

The morphisms

$\theta^{l}:\Theta\to\Theta,\quad Bt^{l}:B\mathbb{G}_{m}\to B\mathbb{G}_{m}$

are both locally of finite type, flat, quasi-compact, quasi-separated, and
have affine diagonals.

###  3.3. Quasi-coherent sheaves on algebraic stacks

We refer to [17, Tag 06WU] for the abelian category of quasi-coherent sheaves
$\mathrm{QCoh}(\mathcal{Y})$ for an algebraic stack $\mathcal{Y}$. It has the
following properties:

  1. (1)

for an affine quotient stack $[X/G]$, we have a canonical equivalence:

$\mathrm{QCoh}([X/G])\cong\mathrm{QCoh}_{G}(X).$

  2. (2)

for a morphism of algebraic stacks $f:\mathcal{X}\to\mathcal{Y}$ the pull-back
$f^{*}$ preserves quasi-coherent sheaves and induces a functor

$f^{*}:\mathrm{QCoh}(\mathcal{Y})\to\mathrm{QCoh}(\mathcal{X}).$

Moreover, for a morphism $[f/\psi]:[X/G]\to[Y/H]$ of affine quotient stacks,
the pull-back functor $[f/\psi]^{*}$ is represented by $(f,\psi)^{*}$.

  3. (3)

when $f$ is quasi-compact and quasi-separated, it induces a functor

$f_{*}:\mathrm{QCoh}(\mathcal{X})\to\mathrm{QCoh}(\mathcal{Y})$

which is a right adjoint to $f^{*}$, following Proposition 103.11.1 of [17,
Tag 070A]. A similar construction will produce higher direct image functors
$R^{i}f_{*}:\mathrm{QCoh}(\mathcal{X})\to\mathrm{QCoh}(\mathcal{Y})$ for all
integers $i\geq 0$.

  4. (4)

if $\mathcal{X}$ is locally Noetherian, i.e. there is a smooth atlas of
$\mathcal{X}$ which is a locally Noetherian scheme, we can define the abelian
category of coherent sheaves $\mathrm{Coh}(\mathcal{X})$ which is a full
subcategory of $\mathrm{QCoh}(\mathcal{X})$, and $f^{*}$ induces a pull-back
functor for the category of coherent sheaves if $f$ is a morphism of locally
Noetherian algebraic stacks, which we also denote as $f^{*}$. Moreover, we
have a canonical equivalence:

$\mathrm{Coh}([X/G])\cong\mathrm{Coh}_{G}(X),$

if $R[X]$ is Noetherian.

  5. (5)

The abelian category of quasi-coherent sheaves satisfies the fppf descent in
the sense of flat-fppf sites (see [17, Tag 08MZ] for the introduction).

The quasi-coherent sheaves over algebraic stacks satisfy the following flat
base change theorem:

######  Theorem 3.3 (Flat base change theorem, Lemma 103.4.1 of [17, Tag
076W], Lemma 103.7.2 and Lemma 103.7.3 of [17, Tag 0760]).

Given a flat morphism $f:\mathcal{X}\to\mathcal{Y}$ of algebraic stacks, the
pullback functor

$f^{*}:\mathrm{QCoh}(\mathcal{Y})\to\mathrm{QCoh}(\mathcal{X})$

is an exact functor. Moreover, a Cartesian diagram of algebraic stacks:

${\mathcal{X}_{1}}$${\mathcal{X}_{2}}$${\mathcal{Y}_{1}}$${\mathcal{Y}_{2},}$$\scriptstyle{f^{\prime}}$$\scriptstyle{g^{\prime}}$$\scriptstyle{g}$$\scriptstyle{f}$

such that $g$ is quasi-compact and quasi-separated and $f$ is flat induces
canonical equivalence of natural transformations:

$f^{*}R^{i}g_{*}\cong
R^{i}g_{*}^{\prime}f^{\prime*}:\mathrm{QCoh}(\mathcal{X}_{2})\to\mathrm{QCoh}(\mathcal{Y}_{1})$

for all non-negative integers $i$.

###  3.4. Universally good morphisms

######  Definition 3.4 (Universally good morphisms).

Let $f:\mathcal{X}\to\mathcal{Y}$ be a morphism of algebraic stacks. We say
that $f$ is a universally good morphism if $f$ is locally of finite type,
quasi-compact, quasi-separated, with an affine diagonal, and for any Cartesian
diagram of algebraic stacks

(3.1)
${\mathcal{X}^{\prime}}$${\mathcal{X}}$${\mathcal{Y}^{\prime}}$${\mathcal{Y},}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f}$$\scriptstyle{g}$

  1. (1)

the functor

$f_{*}:QCoh(\mathcal{X})\to QCoh(\mathcal{Y}).$

is exact;

  2. (2)

if $\mathcal{Y}^{\prime}$ is locally Noetherian, then $f_{*}$ maps coherent
sheaves to coherent sheaves;

  3. (3)

the canonical morphism $\mathcal{O}_{\mathcal{Y}^{\prime}}\to
f_{*}\mathcal{O}_{\mathcal{X}^{\prime}}$ is an isomorphism,

######  Remark 3.5.

Obviously, the terminology “universally good” is from Alper’s “good moduli
space” in [2].

By the fppf descent of quasi-coherent sheaves (resp. coherent sheaves) over
algebraic stacks, we have

######  Lemma 3.6.

Universally good morphisms are stable under base change and compositions.
Moreover, given a Cartesian diagram of algebraic stacks in 3.1 such that $g$
is a fppf atlas of $\mathcal{Y}$, $f$ is universally good if and only if
$f^{\prime}$ is universally good.

Particularly, assuming a morphism of algebraic stacks
$f:\mathcal{X}\to\mathcal{Y}$ is locally of finite type, quasi-compact, quasi-
separated and with an affine diagonal, $f$ is universally good if and only if
for any Cartesian diagram of algebraic stacks in 3.1 such that
$\mathcal{Y}^{\prime}\cong\mathrm{Spec}(R)$, the functor

$f_{*}^{\prime}:\mathrm{QCoh}(\mathcal{X}^{\prime})\to R-\mathrm{Mod},\quad
F\to\Gamma(\mathcal{X}^{\prime},F)$

is exact, maps coherent sheaves to finite generated $R$-modules if $R$ is
Noetherian and the canonical morphism
$R\to\Gamma(\mathcal{X}^{\prime},\mathcal{O}_{\mathcal{X}^{\prime}})$ is an
isomorphism.

######  Example 3.7.

Given an $R$-affine group $G$, we say that $BG$ is linearly reductive if the
morphism $BG\to\mathrm{Spec}(R)$ is universally good. Particularly,
$\mathbb{G}_{m,R}$ and $\mu_{l,R}$ are all linearly reductive groups by
Example 2.4

######  Example 3.8.

Let $R[X]$ be an $\mathbb{Z}/l\mathbb{Z}$-graded $R$-algebra which is finite
generated as a $R[X]_{0}$-module. Then the quotient stack morphism

$\psi_{l,X}:[X/\mu_{l,R}]\to\mathrm{Spec}(R[X]_{0})$

is universally good by Example 2.8.

Particularly, we consider the Cartesian diagram of algebraic stacks:

${\left[\mathbb{A}^{1}/\mu_{l}\right]}$${\left[\mathbb{A}^{1}/\mathbb{G}_{m}\right]}$${\mathbb{A}^{1}}$${\left[\mathbb{A}^{1}/\mathbb{G}_{m}\right].}$$\scriptstyle{\left[id/\mu_{l}\right]}$$\scriptstyle{\left[x^{l}/t^{l}\right]}$$\scriptstyle{\left[x^{l}/t^{l}\right]}$

The morphism

$[x^{l}/t^{l}]:[\mathbb{A}^{1}/\mu_{l}]\to\mathbb{A}^{1}$

is universally good by the above argument, and thus the morphism

$\theta^{l}:\Theta\to\Theta$

is also universally good.

######  Lemma 3.9.

Given a universally good morphism $f:\mathcal{X}\to\mathcal{Y}$ of algebraic
stacks, the higher direct image vanishes for any $\mathcal{F}\in
QCoh(\mathcal{X})$, the higher direct image vanishes:

$R^{i}f_{*}(\mathcal{F})=0,\quad i>0.$

###### Proof.

As the higher direct image is fppf local, we can assume that $\mathcal{Y}$ is
an affine scheme. As $f$ is quasi-compact, we have a smooth cover
$u:U\to\mathcal{Y}$ where $U$ is an affine scheme. Then for any integer $n>0$,
the $n$-Cartesian product
$U_{n}:=\underbrace{U\times_{\mathcal{X}}\times\cdots\times_{\mathcal{X}}U}_{n\text{
times}}$ is also an affine scheme, as the diagonal
$\Delta_{\mathcal{X}}:\mathcal{X}\to\mathcal{X}\times\mathcal{X}$ is affine.
The projection morphisms $u_{n}:U_{n}\to\mathcal{X}$ are also affine morphisms
We denote $u_{n}:U_{n}\to\mathcal{X}$ as all $U_{n+1}$ are affine. Thus we
have the long exact sequence of Cech complexes in $\mathrm{QCoh}(\mathcal{Y})$

$0\to\mathcal{F}\to u_{1*}u_{1}^{*}\mathcal{F}\to
u_{2*}u_{2}^{*}\mathcal{F}\to\cdots.$

By the Grothendieck spectral sequence and the fact that all $U_{n}$ and
$u_{n}$ are affine, $R^{i}f_{*}\mathcal{F}$ is the cohomology of the following
complex

$f_{*}u_{1*}u_{1}^{*}\mathcal{F}\to f_{*}u_{2*}u_{2}^{*}\mathcal{F}\to\cdots$

and the higher cohomology vanishes because $f_{*}$ is an exact functor. ∎

###  3.5. Derived category of (quasi)-coherent sheaves on algebraic stacks

We refer to [17, Tag 07B5] for the derived category of (quasi)-coherent
sheaves on algebraic stacks. Given an algebraic stack $\mathcal{X}$, we denote
$D_{qcoh}(\mathcal{O}_{\mathcal{X}})$ (resp.
$D^{+}_{qcoh}(\mathcal{O}_{\mathcal{X}})$,
$D^{-}_{qcoh}(\mathcal{O}_{\mathcal{X}})$,
$D^{b}_{qcoh}(\mathcal{O}_{\mathcal{X}})$) as the derived category of quasi-
coherent sheaves (resp. with cohomology bounded from below, above and both
sides) on $\mathcal{X}$. We denote $D_{Coh}(\mathcal{O}_{\mathcal{X}})$ (resp.
$D^{+}_{coh}(\mathcal{O}_{\mathcal{X}})$,
$D^{-}_{coh}(\mathcal{O}_{\mathcal{X}})$,
$D^{b}_{coh}(\mathcal{O}_{\mathcal{X}})$) as the full subcategory of
$D_{qcoh}(\mathcal{O}_{\mathcal{X}})$ (resp.
$D^{+}_{qcoh}(\mathcal{O}_{\mathcal{X}})$,
$D^{-}_{qcoh}(\mathcal{O}_{\mathcal{X}})$,
$D^{b}_{qcoh}(\mathcal{O}_{\mathcal{X}})$) with coherent cohomologies.

Given a morphism $f:\mathcal{X}\to\mathcal{Y}$, the pull-back functor of
quasi-coherent sheaves induce the derived functors

$Lf^{*}:D_{qcoh}(\mathcal{O}_{\mathcal{Y}})\to
D_{qcoh}(\mathcal{O}_{\mathcal{X}}),$

which maps complexes with bounded from above (resp. coherent) cohomologies to
complexes with bounded from above (resp. coherent) cohomologies. Moreover, it
maps complexes with bounded from below (resp. bounded) cohomologies to
complexes with bounded from below (resp. bounded) cohomologies if $f$ is flat.

Moreover, when $f:\mathcal{X}\to\mathcal{Y}$ is quasi-compact and quasi-
separated, then the push-forward functor induces the derived functor

$Rf_{*}:D_{qcoh}^{+}(\mathcal{O}_{\mathcal{X}})\to
D_{qcoh}^{+}(\mathcal{O}_{\mathcal{Y}}).$

If $f$ is universally good, then $Rf_{*}$ maps complexes with bounded
cohomologies (resp. coherent cohomologies) to complexes with bounded
cohomologies (resp. coherent cohomologies).

Like the flat base change theorem for quasi-coherent sheaves, we also have the
following theorem for the derived category of quasi-coherent sheaves:

######  Theorem 3.10 (Proposition 2.3.2 of [8]).

A Cartesian diagram of algebraic stacks:

${\mathcal{X}_{1}}$${\mathcal{X}_{2}}$${\mathcal{Y}_{1}}$${\mathcal{Y}_{2},}$$\scriptstyle{f^{\prime}}$$\scriptstyle{g^{\prime}}$$\scriptstyle{g}$$\scriptstyle{f}$

such that $g$ is quasi-compact and quasi-separated and $f$ is flat induces
canonical equivalence of natural transformations:

$Lf^{*}Rg_{*}\cong
Rg_{*}^{\prime}Lf^{\prime*}:D^{+}_{qcoh}(\mathcal{X}_{2})\to
D^{+}_{qcoh}(\mathcal{Y}_{1})$

##  4\. Semir-orthogonal decomposition of the root stacks

In this section, we give a constructive proof of the semi-orthogonal
decomposition of the derived category of bounded-(from-below) (quasi)-coherent
sheaves on the root stack.

###  4.1. Semi-orthogonal decomposition of the $\Theta_{R}$

We first give a semi-orthogonal decomposition for the derived category of
(quasi)-coherent sheaves of

$\Theta_{R}:=\Theta\times_{\mathrm{Spec}(\mathbb{Z})}\mathrm{Spec}(R).$

We recall a lemma about effective Cartier divisor:

######  Lemma 4.1.

[17, Tag 0B4A] Let $f:X\to Z$ be an effective Cartier divisor such that $Z$ is
an algebraic stack. Then the right adjoint functor of

$Rf_{*}:D^{+}_{qcoh}(X)\to D^{+}_{qcoh}(Z)$

which we denote as $Lf^{!}$ is $Lf^{*}(-\otimes^{L}\mathcal{O}(X))[-1]$.
Moreover, the ad junction formula induces a canonical triangle of derived
functors

$L_{f}^{*}R_{f*}\to id\to L_{f}^{!}R_{f*}.$

We consider the $\mathbb{G}_{m,R}$-equivariant morphism:

$i_{R}:B\mathbb{G}_{m,R}\to\Theta_{R},\quad R[X]\to R,\quad x\to 0.$

and the following commutative diagram

(4.1)
${B\mathbb{G}_{m,R}}$${\Theta_{R}}$${B\mathbb{G}_{m,R}}$${\Theta_{R}}$$\scriptstyle{Bt^{l}_{R}}$$\scriptstyle{i_{R}}$$\scriptstyle{\theta_{R}^{l}}$$\scriptstyle{i_{R}}$

and denote the following derived functors:

$\triangleright_{m,R}:=RBi_{R*}(-\otimes_{R}\mathcal{L}_{R}^{m})LBt_{R}^{l*},\quad\triangleleft_{m,R}:=RBt_{R*}^{l}(-\otimes_{R}\mathcal{L}_{R}^{m})Li_{R}^{*}.$

If $l>1$, by Example 2.4 and Example 2.8, we have

(4.2)
$R\theta_{R*}^{l}\mathcal{O}_{\Theta_{l,R}}\cong\mathcal{O}_{\Theta_{l,R}},\quad
RBt^{l}_{R*}\mathcal{L}_{R}^{i}\cong\mathcal{L}_{R}^{i/l}.$

######  Lemma 4.2.

The derived functor $\triangleleft_{i+1,R}[-1]$ is the right adjoint functor
of $\triangleright_{i,R}$ and $R\theta_{R*}^{l}$ is the right-adjoint functor
of $L\theta_{R}^{l*}$. Moreover, we have

(4.3) $R\theta_{R*}^{l}L\theta_{R}^{l*}\cong id.$ (4.4)
$\triangleleft_{-j,R}\circ\triangleright_{i,R}\cong-\otimes(\mathcal{L}_{R}^{(i-j)/l}\oplus\mathcal{L}_{R}^{(i-j+1)/l}[-1])$
(4.5) $R\theta_{R*}^{l}\circ\triangleright_{i,R}\cong
Ri_{R*}\circ(-\otimes\mathcal{(}L)_{R}^{i/l}),\quad\triangleleft_{i,R}\circ
L\theta_{R}^{l*}\cong(-\otimes\mathcal{L}_{R}^{i/l})Li_{R}^{*}.$ (4.6)
$\triangleright_{i+l,R}\cong\triangleright_{i,R}\circ(-\otimes_{R}\mathcal{L}_{R}),\quad\triangleright_{i+l,R}\cong(-\otimes_{R}\mathcal{L}_{R})\circ\triangleright_{i,R}.$

###### Proof.

The equation 4.2 follows from Example 2.4 and Example 2.8. The equations 4.3,
4.4,4.5,Section 4.1 follows from the projection formula, Lemma 4.1 and 4.2. ∎

######  Theorem 4.3.

There are functors:

$\overline{\tau_{m,n,R}}:D^{+}_{qcoh}(\Theta)\to D^{+}_{qcoh}(\Theta),\quad
0\leq n\leq m\leq l-1$

which maps complexes with bounded cohomologies (resp. coherent cohomologies)
to complexes with bounded cohomologies (resp. coherent cohomologies) and
satisfies the following properties:

  1. (1)

we have

$\overline{\tau_{n,n,R}}\cong\otimes\mathcal{L}_{\mathbb{A}_{R}^{1}}^{n},\quad\overline{\tau_{0,l-1,R}}\cong
L\theta_{R}^{l*}R\theta_{R*}^{l}$

  2. (2)

for any $0\leq n<m\leq l-1$, we have canonical triangles:

$\displaystyle\overline{\tau_{n,m-1,R}}\to\overline{\tau_{n,m,R}}\to\bigoplus_{i=0}^{m}\triangleright_{m-i,R}\triangleleft_{i,R}$
$\displaystyle\overline{\tau_{n,m,R}}\to\overline{\tau_{n+1,m,R}}\to\bigoplus_{i=n+1}^{l-1}\triangleright_{i-1-n,R}\triangleleft_{i,R}.$

.

  3. (3)

All those functors map complexes with bounded cohomologies (resp. coherent
cohomologies) to complexes with bounded cohomologies (resp. coherent
cohomologies).

###### Proof.

We consider the equivariant morphisms in Section 2.5

$\displaystyle\alpha_{1}:(\mathbb{G}^{m}\times\mu_{l},\alpha_{l})\to(\mathbb{G}_{m},\mathbb{A}^{1}),$
$\displaystyle\quad((t,\mu),(x,y))\to(t,x),$
$\displaystyle\alpha_{2}:(\mathbb{G}^{m}\times\mu_{l},\alpha_{l})\to(\mathbb{G}_{m},\mathbb{A}^{1}),$
$\displaystyle\quad((t,\mu),(x,y))\to(t\mu,y).$

which induces morphisms of quotient stacks from
$[\alpha_{l}/\mathbb{G}_{m}\times\mu_{l}]$ to $\Theta$ and we abuse the
notation to still denote them as $\alpha_{1},\alpha_{2}$ respectively. Then we
have the Cartesian diagram of affine quotient stacks:

${\left[\alpha_{l}/\mathbb{G}^{m}\times\mu_{l}\right]}$${\Theta}$${\Theta}$${\Theta.}$$\scriptstyle{\alpha_{1}}$$\scriptstyle{\alpha_{2}}$$\scriptstyle{\theta^{l}}$$\scriptstyle{\theta^{l}}$

Such that the morphisms $\alpha_{1}$ and $\alpha_{2}$ are both flat,
universally cohomologically affine and universally cohomologically proper. The
diagonal $\Delta_{\theta_{l}}:\Theta\to\alpha_{l}$ is represented by
$\Delta_{t^{l},x^{l}}$ in Section 2.5 and Example 3.1. We denote
$\alpha_{1,R}:=\alpha_{1}\otimes R$ and $\alpha_{2,R}:=\alpha_{2}\otimes R$.

For

$x\in D^{b}_{coh}([\alpha_{l,R}/\mathbb{G}^{m}_{R}\times\mu_{l,R}]),$

we consider the Fourier-Mukai transform:

$\bar{x}:R_{\alpha_{2,R}*}(x\otimes^{L}L\alpha_{1,R}^{*}(-)):D^{+}_{qcoh}(\Theta_{R})\to
D^{+}_{qcoh}(\Theta_{R}).$

which maps complexes with bounded or coherent cohomologies to complexes with
bounded or coherent cohomologies.

Now we define $\tau_{m,n,R}:=\tau_{m,n}\otimes R$, where $\tau_{m,n}$ is
defined in Example 3.1. By the flat base change theorem, we have
$\overline{\tau_{0,l-1,R}}\cong L\theta^{l*}R\theta^{l}_{*}$ and

$R_{\alpha_{2,R}*}(R<m,n>\otimes^{L}L\alpha_{1,R}^{*}(-))\cong\triangleright_{n-m,R}\triangleleft_{m,R}.$

Thus Theorem 4.3 follows from Example 3.1. ∎

######  Corollary 4.4.

If $l>1$, the following functors are fully faithful:

$\displaystyle L\theta^{*}_{l}:D_{qcoh}^{+}(\Theta_{R})\to
D_{qcoh}^{+}(\Theta_{R}),\quad\triangleright_{i,R}:D_{qcoh}^{+}(B\mathbb{G}_{m,R})\to
D_{qcoh}^{+}(\Theta_{R}).$

Moreover, we denote
$D_{l,R}^{i}:=\triangleright_{i,R}D_{qcoh}^{+}(B\mathbb{G}_{m,R})$. Then

$D_{l,R}^{i}\cong D_{l,R}^{i+l}$

and for any $0\leq i\leq l-1$, we have the semi-orthogonal decomposition

(4.7)
$<D_{l,R}^{i-l+1},\cdots,D_{l,R}^{-1},L\theta^{*}_{l}D^{+}_{qcoh}(\Theta_{R}),D_{l,R}^{0},\cdots,D_{l,R}^{i-1}>$

which is a fully subcategory of $D^{+}_{qcoh}(\Theta)$. Moreover, similar
arguments also hold for for derived category of complexes with bounded or
coherent cohomologies.

###  4.2. The semi-orthogonal decomposition for root stacks

Now we give a proof for the semi-orthogonal decomposition of the derived
category of (below)-bounded (quasi)-coherent sheaves of root stacks.

Let $\mathcal{X}$ be an algebraic stack. Given a line bundle $\mathcal{L}$
over $\mathcal{X}$, it induces a canonical morphism from $\mathcal{X}\to
B\mathbb{G}_{m}$ such that $\mathcal{L}$ is the pull-back of
$\mathcal{L}_{\mathbb{Z}}$. Given an integer $l>1$, we denote

$\mathcal{X}_{\mathcal{L},l}:=\mathcal{X}\times_{B{\mathbb{G}_{m}},Bt^{l}}B\mathbb{G}_{m}.$

and denote $\mathcal{L}_{l}\in Pic(\mathcal{X}_{\mathcal{L},l})$ as the pull-
back of $\mathcal{L}_{\mathbb{Z}}$ along the second projection
$\mathcal{X}_{\mathcal{L},l}\to B\mathbb{G}_{m}$.

Let

$r_{\mathbb{Z}}:\mathcal{L}_{\mathbb{A}_{\mathbb{Z}}^{1}}\to\mathcal{O}_{\Theta}$

be the canonical on $\Theta$ which maps $1$ to $x$. Given a line bundle
$\mathcal{L}$ with a co-section

$r:\mathcal{L}\to\mathcal{O}_{\mathcal{X}}$

over $\mathcal{X}$, it induces a canonical morphism from $\mathcal{X}$ to
$\Theta$ such that $r$ is the pull-back of $r_{\mathbb{Z}}$. We define the
$l$-th root stack of $r$ as

$\mathcal{X}_{r,l}:=\mathcal{X}\times_{\Theta,\theta^{l}}\Theta.$

Now we assume that $r$ is generated by an effective divisor $\mathcal{D}$,
i.e.

$r=(\mathcal{O}_{\mathcal{X}}(-\mathcal{D})\subset\mathcal{O}_{\mathcal{X}}).$

It induces a Cartesian diagram

${\mathcal{D}}$${B\mathbb{G}_{m}}$${\mathcal{X}}$${\Theta.}$$\scriptstyle{i_{\mathcal{X}}}$$\scriptstyle{i_{\mathbb{Z}}}$$\scriptstyle{r}$

such that the morphism from $\mathcal{D}$ to $B\mathbb{G}_{m}$ is induced by
the line bundle
$\mathcal{L}_{\mathcal{D}}:=\mathcal{O}_{\mathcal{X}}(\mathcal{-D})|_{\mathcal{D}}$.
We denote

$\mathcal{X}_{\mathcal{D},l}:=\mathcal{X}_{r,l},\quad\mathcal{D}_{l}:=\mathcal{D}_{\mathcal{L}_{\mathcal{D}},l}$

and consider the pull-back of 4.1 for $R=\mathbb{Z}$ along $r$, and have the
following Cartesian diagram

${\mathcal{D}_{l}}$${\mathcal{X}_{\mathcal{D},l}}$${\mathcal{D}}$${\mathcal{X}.}$$\scriptstyle{i_{\mathcal{X}_{\mathcal{D},l}}}$$\scriptstyle{Bt_{\mathcal{D}}^{l}}$$\scriptstyle{\theta_{\mathcal{X}}^{l}}$$\scriptstyle{i_{\mathcal{X}}}$

We notice that $Bt^{l}_{\mathcal{D}}$ and $\theta_{\mathcal{X}}^{l}$ are all
universally good morphisms by the base change. For any integer $m$, we define
the following functors

$\displaystyle\triangleleft_{m,\mathcal{X}}:=RBt_{\mathcal{D}*}^{l}\circ(-\otimes\mathcal{L}_{\mathcal{D},l}^{m})\circ
Li_{\mathcal{X}_{\mathcal{D},l}}^{*}:D^{+}_{qcoh}(\mathcal{X}_{\mathcal{D},l})\to
D_{qcoh}^{+}(\mathcal{D}),$
$\displaystyle\triangleright_{m,\mathcal{X}}:=Ri_{\mathcal{X}_{\mathcal{D},l}*}\circ(-\otimes\mathcal{L}_{D,l}^{m})\circ
LBt_{\mathcal{D}}^{l*}:D^{+}_{qcoh}(\mathcal{D})\to
D_{qcoh}^{+}(\mathcal{X}_{\mathcal{D},l}).$

By the fppf descent of quasi-coherent sheaves, Lemma 4.1 and Lemma 4.2, we
have the following lemmas

######  Lemma 4.5.

If $l>1$, we have

(4.8)
$R\theta_{\mathcal{X}*}^{l}\mathcal{O}_{\mathcal{X}_{\mathcal{D},l}}\cong\mathcal{O}_{\mathcal{X}},\quad
RBt^{l}_{\mathcal{D}*}(\mathcal{L}_{\mathcal{D},l}^{i})\cong\mathcal{L}_{\mathcal{D}}^{i/l}$

The derived functor $\triangleleft_{i+1,\mathcal{X}}[-1]$ is the right adjoint
functor of $\triangleright_{i,\mathcal{X}}$ and $R\theta_{\mathcal{X}*}^{l}$
is the right-adjoint functor of $L\theta_{\mathcal{X}}^{l*}$. Moreover, we
have

(4.9) $R\theta_{\mathcal{X}*}^{l}L\theta_{\mathcal{X}}^{l*}\cong id.$ (4.10)
$\triangleleft_{-j,\mathcal{X}}\circ\triangleright_{i,\mathcal{X}}\cong-\otimes(\mathcal{L}_{\mathcal{D}}^{(i-j)/l}\oplus\mathcal{L}_{\mathcal{D}}^{(i-j+1)/l}[-1])$
(4.11) $R\theta_{\mathcal{X}*}^{l}\circ\triangleright_{i,\mathcal{X}}\cong
Ri_{\mathcal{X}*}\circ(-\otimes\mathcal{L}_{\mathcal{D}}^{i/l}),\quad\triangleleft_{i,\mathcal{X}}\circ
L\theta_{\mathcal{X}}^{l*}\cong(-\otimes\mathcal{L}_{\mathcal{D}}^{i/l})Li_{\mathcal{X}}^{*}.$
(4.12)
$\triangleright_{i+l,\mathcal{X}}\cong\triangleright_{i,\mathcal{X}}\circ(-\otimes\mathcal{L}_{\mathcal{D}}),\quad\triangleright_{i+l,\mathcal{X}}\cong(-\otimes\mathcal{L}_{\mathcal{D}})\circ\triangleright_{i,\mathcal{X}}.$

######  Theorem 4.6.

Given $0\leq n\leq m\leq n-1$, we consider

$\tau_{n,m,\mathcal{X}}:=r^{*}\tau_{n,m}\in
D^{b}_{coh}(\mathcal{X}_{\mathcal{D},l}\times_{\theta_{\mathcal{X}}^{l},\mathcal{X},\theta_{\mathcal{X}}^{l}}\mathcal{X}_{\mathcal{D},l}).$

and define

$\overline{\tau}_{n,m,\mathcal{X}}:D^{+}_{qcoh}(\mathcal{X}_{\mathcal{D},l})\to
D^{+}_{qcoh}(\mathcal{X}_{\mathcal{D},l})$

as the Fourier-Mukai transform generated by the Fourier-Mukai kernels
$\tau_{n,m,\mathcal{X}}$. Then which maps complexes with bounded cohomologies
(resp. coherent cohomologies) to complexes with bounded cohomologies (resp.
coherent cohomologies) and satisfy the following properties:

  1. (1)

we have

$\overline{\tau_{n,n,\mathcal{X}}}\cong\otimes\mathcal{O}_{\mathcal{X}_{\mathcal{D},l}}(-\mathcal{D}_{l})^{n},\quad\overline{\tau_{0,l-1,\mathcal{X}}}\cong
L\theta_{\mathcal{X}}^{l*}R\theta_{\mathcal{X}*}^{l}$

  2. (2)

for any $0\leq n<m\leq l-1$, we have canonical triangles:

$\displaystyle\overline{\tau_{n,m-1,\mathcal{X}}}\to\overline{\tau_{n,m,\mathcal{X}}}\to\bigoplus_{i=0}^{m}\triangleright_{m-i,\mathcal{X}}\triangleleft_{i,\mathcal{X}}$
$\displaystyle\overline{\tau_{n,m,\mathcal{X}}}\to\overline{\tau_{n+1,m,\mathcal{X}}}\to\bigoplus_{i=n+1}^{l-1}\triangleright_{i-1-n,\mathcal{X}}\triangleleft_{i,\mathcal{X}}.$

.

  3. (3)

All those functors map complexes with bounded cohomologies (resp. coherent
cohomologies) to complexes with bounded cohomologies (resp. coherent
cohomologies).

By Lemma 4.5 and Theorem 4.6, we have the following theorem

######  Theorem 4.7.

If $l>1$, the following functors are fully faithful:

$\displaystyle L\theta^{l*}_{\mathcal{X}}:D_{qcoh}^{+}(\mathcal{X})\to
D_{qcoh}^{+}(\mathcal{X}_{\mathcal{D},l}),$
$\displaystyle\triangleright_{i,\mathcal{X}}:D_{qcoh}^{+}(\mathcal{D})\to
D_{qcoh}^{+}(\mathcal{X}_{\mathcal{D},l}).$

Moreover, we denote
$D_{l,\mathcal{D}}^{i}:=\triangleright_{i,\mathcal{X}}D_{qcoh}^{+}(\mathcal{D})$.
Then

$D_{l,\mathcal{D}}^{i}\cong D_{l,\mathcal{D}}^{i+l}$

and for any $0\leq i\leq l-1$, we have the semi-orthogonal decomposition

(4.13)
$D^{+}_{qcoh}(\mathcal{X}_{\mathcal{D},l}):=<D_{l,\mathcal{D}}^{i-l+1},\cdots,D_{l,\mathcal{D}}^{-1},L\theta^{*}_{l}D^{+}_{qcoh}(\mathcal{X}),D_{l,\mathcal{D}}^{0},\cdots,D_{l,\mathcal{D}}^{i-1}>.$

Similar arguments also hold for complexes with bounded or coherent
cohomologies.

## References

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Kavli Institute for the Physics and Mathematics of the Universe (WPI),
University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan.

E-mail address<EMAIL_ADDRESS>


# On the origin of metal-insulator transitions in the parent compounds of
ruthenium-pnictide superconductors

Niraj Aryal<EMAIL_ADDRESS>Condensed Matter Physics and Materials Science
Division, Brookhaven National Laboratory, Upton, New York 11973, USA  Emil S.
Bozin  Condensed Matter Physics and Materials Science Division, Brookhaven
National Laboratory, Upton, New York 11973, USA  Weiguo Yin<EMAIL_ADDRESS>Condensed Matter Physics and Materials Science Division, Brookhaven National
Laboratory, Upton, New York 11973, USA

###### Abstract

We study the interplay of the structural phase transition, flat electronic
band dispersion, and metal-to-insulator transition (MIT) in the parent
compounds of the Ru-pnictide superconductors by using first-principles
calculations. Our electron and phonon calculations reveal that Ru(P,As)
undergo MIT accompanied by orthorhombic to monoclinic distortion at low
temperature, but RuSb stays orthorhombic and metallic in agreement with the
experimental findings. We find that although small monoclinic distortion can
remove the van Hove singularity at the Fermi level, it does not immediately
gap out the Fermi surface and a large value of monoclinic distortion is
necessary for a clear MIT suggesting the possibility of an intermediate
pseudogapped monoclinic metallic phase. Furthermore, we predict a light-
induced two-step insulator-to-metal and structural transitions in the
monoclinic phases of RuP and RuAs, which can be tested in future ultrafast
pump-probe experiments as an alternative ideal play ground to VO2.

_Introduction._ —Understanding the origin of superconductivity and its
interplay with charge-density-wave (CDW) instability, band topology, and
metal-to-insulator transition (MIT) continues to be an important problem in
condensed matter physics Fradkin  _et al._ (2015); Kusmartseva  _et al._
(2009); Yu  _et al._ (2021). The discoveries of high-temperature
superconductivity and many competing phases in layered copper oxides Bednorz
and Müller (1986); Wu  _et al._ (1987) and iron pnictides (FePn) Kamihara  _et
al._ (2008); Lee  _et al._ (2008); Chen  _et al._ (2008a, b) have triggered
numerous investigations to find superconductivity in similar materials such as
layered nickelates Li  _et al._ (2019); Pan  _et al._ (2022); Sun  _et al._
(2023) and pnictides Wu  _et al._ (2014); Kotegawa  _et al._ (2014); Cheng
_et al._ (2015) for possible hints on the mechanism of high-temperature
superconductivity and its relationship with the interplay of the charge, spin,
orbital, and lattice degrees of freedom. To discern this complex interplay via
light-induced transient phase transition is of particular interest due to
recent advancements in ultrafast measurements Fausti  _et al._ (2011);
Mankowsky  _et al._ (2014); Nevola  _et al._ (2023); however, it turns out to
be highly challenging to even understand the light-induced MIT and melting of
CDW in VO2, especially concerning the emergence of the instantaneous
monoclinic metallic phase Xu  _et al._ (2023); Li  _et al._ (2022); Wall  _et
al._ (2018); Morrison  _et al._ (2014); Wegkamp  _et al._ (2014); Baum  _et
al._ (2007). Here, we show that the recently discovered ruthenium-pnictide
(RuPn) family of superconductors Ru1-xRhxAs, Ru1-xRhxP, and RuSb near a CDW-
driven MIT Hirai  _et al._ (2012); Chen  _et al._ (2015); Fan  _et al._
(2015); Li  _et al._ (2017); Nakajima  _et al._ (2019); Hirai  _et al._ (2022)
provides an alternative ideal play ground.

RuPn (Pn=P, As, Sb) materials crystallize in the MnP-type orthorhombic
structure with space group of Pnma (no. 62) at high temperature. Upon cooling,
two successive MITs have been reported for Ru(P,As). At the first transition
temperature $T_{1}$, these two materials undergo metal to pseudo-gap (PG)
phase transition and at $T_{2}$, there is a transition to the non-magnetic
monoclinic insulating phase. The presence of the PG feature and dome-shaped
superconductivity upon doping is reminiscent of the behavior of FePn Ishida
_et al._ (2008); Sato  _et al._ (2008); Shimojima  _et al._ (2014) and cuprate
superconductors Warren  _et al._ (1989); Alloul  _et al._ (1989); Norman  _et
al._ (2005). In addition, recently Koch et al. reported the observation of
fluctuating local Ru-trimers well above $T_{1}$ in synchrotron x-ray total
scattering measurements Koch  _et al._ (2022). This finding, in contrast to
VO2 where no such evidence of high-temperature fluctuating local V-dimers are
present Corr  _et al._ (2010), might facilitate the emergence of the much
debated instantaneous monoclinic metallic phase in pump-probe experiments as
well as could shed light on the PG behaviour. RuSb, on the other hand, stays
metallic with no structural phase transition and is superconducting without
doping. Unexpectedly, experiments have indicated higher density of states at
the Fermi level in RuSb; hence the finding of monoclincity in RuP and RuAs
appears to contradict the common notion that higher DOS drives instability
Hirai  _et al._ (2012). Therefore, it is important to thoroughly investigate
the electronic structure, the driving mechanism for the CDW transition and its
relation to the MIT in these family of materials. As Ru lies one row below Fe
in the periodic table, RuPn and FePn could share similar properties; one the
other hand, the $4d$ valence electronic states in RuPn are expected to be less
localized and less magnetic than the $3d$ states in FePn, making RuPn more
tractable theoretically.

In this letter, we perform a systematic study of the electronic bands, phonon
bands, and electron-lattice coupling in Ru(P,As,Sb) materials using first-
principles calculations in order to understand the origin of metal-to-
insulator transition. We find that the high temperature orthorhombic phase of
these systems hosts symmetry-enforced multiple Dirac band crossings and multi-
fold band degeneracies in the vicinity of the Fermi level. For RuP and RuAs,
the symmetry-enforced degenerate flat bands are present right at the Fermi
level giving rise to the van Hove singularity whereas the bands are less flat
and shifted away from the Fermi level for RuSb. The monoclinic distorted
phases in Ru(P,As) are energetically favorable compared to the orthorhombic
phase whereas RuSb prefers orthorhombicity which is consistent with
experiments. Furthermore, we predict a light-induced two-step insulator-to-
metal and structural transitions in the monoclinic phases of RuP and RuAs—with
an intermediate monoclinic metallic phase—which can be attested in future
ultrafast pump-probe experiments as an alternative ideal play ground to VO2.

Figure 1: (a) and (b) Crystal structure of the Ru-pnictides for orthorhombic
and monoclinic phases, respectively, shown from different perspectives (see
text for details). (c) and (d) Density of states plot for RuP and RuAs showing
comparison between the orthorhombic and monoclinic phases. The inset plots
show DOS in $\pm$0.5 eV window around the Fermi level.  Figure 2: Comparison
of the electronic band structure, Fermi surface and phonon band dispersion
between the orthorhombic phases of RuP and RuSb. Red (blue) lines in panels
(a) and (e) are obtained with (without) the inclusion of SOC. Panels (b) and
(f) show Ru-d band character plots in a small window around the flat bands
with the following color convention: black ($d_{xy}$), red ($d_{z^{2}}$),
green ($d_{x^{2}-y^{2}}$), blue ($d_{xz}$), and yellow ($d_{yz}$). (c) and (f)
show Fermi surface plots. The red-green (blue-magenta) colors show the hole
(electron) Fermi surface pieces. (d) and (h) show the phonon band dispersion.
The colors denote the weight of the normal mode atomic displacements along the
crystallographic aortho(red), bortho(green) and cortho(blue) direction for Ru-
atoms. Black color shows the weight of pnictogen atoms.

_Crystal structure and lattice relaxation._ —Fig. 1(a) and (b) shows the
crystal structure of Ru-pnictides in both orthorhombic and monoclinic
distorted phase. Above $T_{1}$, the system crystallizes in an orthorhombic
Pnma space group with 4 formula units in an unit cell. The Ru-atoms are
surrounded by 6 Pn-atoms in a distorted octahedral environment with face-
sharing octahedra along the aortho-direction and edge-sharing ones along the
bortho-direction and bc-plane. A zig-zag chain of nearest neighbour (nn) Ru-
atoms runs along the a-direction. In addition, the next nn and third nn Ru-
atoms form a triangular lattice in the bc-plane. Below $T_{2}$, P and As
members of the family undergo orthorhombic to monoclinic distortion. The unit
cell triples both along the orthorhombic c-direction and in the ab-plane
thereby increasing the number of atoms in the monoclinic unit cell by 9 times.
The transformation matrix that maps from orthorhombic to monoclinic unit cell
is given by $\mathrm{V_{mono}}=\mathcal{M}\mathrm{V_{ortho}}$ where,
$\mathrm{V}$’s are the lattice vectors for the monoclinic and orthorhombic
unit cells nciteCommentCrystalConvention and
$\mathcal{M}=\begin{bmatrix}1&2&0\\\ -1&1&0\\\ 0&0&3\\\ \end{bmatrix}.$ The
monoclinic distorted phase is characterized by trimerization along the bortho-
direction consisting of two short (S1, S2) and one long (L) bonds in the order
—LS1S2LS1S2L—. Such trimerization results in tightly bound Ru6 clusters or
hexamers. In addition to the trimerization and hexamerization, the zig-zag
chain along aorthodirection has a weak modulation of the bond distances with a
period of 6.

Fig. 1 (c) and (d) shows the density of states comparison between the
orthorhombic and monoclinic distorted phase for both RuP and RuAs obtained
from Density Functional Theory (DFT) calculation. The orthorhombic phases show
peak DOS at the Fermi level thereby signalling the presence of lattice
instability. As we will see below, such peak DOS arises from the presence of
Ru-$d_{xy}$ derived flat bands at the Fermi level. The fully relaxed
monoclinic distorted phases correctly reproduces the insulating phase for both
systems with gap-size comparable to the experimental findings which is $\sim
100$ meV for RuP Li  _et al._ (2017); Ootsuki  _et al._ (2020); Koch  _et al._
(2022), in agreement with a recent DFT study of RuP Hirai  _et al._ (2022),
and $\sim 30$ meV for RuAs Kotegawa  _et al._ (2018). In contrast, the
experimentally refined lattice parameters fail to show the gap opening in both
systems within DFT calculations despite using different flavors of exchange
correlation functional Koch  _et al._ (2022); Kotegawa  _et al._ (2018). It is
indeed incredible that the relaxed structures can reproduce the gap-sizes
correctly despite the well-known issue of underestimation of gap-size in plain
LDA/GGA calculations, in sharp contrast to the DFT studies of VO2, which
remains metallic in the LDA/GGA and requires additional cares of electronic
correlation to become an insulator Zhu and Schwingenschlögl (2012); Yuan  _et
al._ (2012); Grau-Crespo  _et al._ (2012); Eyert (2011). We attribute the
difference to the general expectation that the $4d$ valence electronic states
in RuPn are less localized and less magnetic than the $3d$ states in VO2,
making RuPn more tractable in DFT.

The lattice parameters and energy differences for various orthorhombic and
monoclinic phases obtained from the DFT calculations are shown in Table S1 of
Supplementary Material Sup . For RuP and RuAs, the DFT relaxed monoclinic
phases are lower in energy compared to the orthorhombic phases which is
consistent with the fact that the systems prefer monoclinic distortion at low
temperature. For RuSb, our calculation finds that the system prefers the
orthorhombic phase even when the relaxation with large monoclinic distortion
was attempted. Our results from phonon calculation which is discussed below
are consistent with the above findings thereby unifying our discussion. The
DFT relaxed lattice parameters and unit-cell volume obtained using GGA
exchange-correlation functional are consistently larger than the experimental
values due to the well-known delocalization problem resulting in the
overestimation of the lattice paramaters in GGA. The LDA calculations, on the
other hand, show decrease in the unit-cell volume due to the well-known
overestimation of the binding energy problem Cohen  _et al._ (2012). Despite
the opposite trends in the structural optimization, both functionals correctly
reproduces the MIT.

_Electronic bands, Fermi surface and phonon dispersion._ —Fig. 2 presents the
electronic bands and Fermi surface plots for RuP and RuSb in the orthorhombic
phase. Similar plots for RuAs are presented in the SM Sup . RuP and RuAs share
similar electronic structure with degenerate flat dispersion close to the
Fermi level, however the bands are less flat and slightly downshifted for
RuSb. The presence of such flat dispersion gives rise to the van Hove
singularity right at the Fermi level. The flat bands are derived mainly from
Ru-$d_{xy}$ and Ru-$d_{z^{2}}$ orbitals Loc . The effect of spin-orbit
coupling (SOC) is small; its main role being lifting of the degeneracy of the
band crossings throughout the spectrum. However, the flat band and peak DOS
remains intact [see DOS comparison with and without SOC in SM]. Hence, in the
following, we will present the results in the absence of the SOC. Note that
the presence of non-symmorphic symmetry gives rise to band degeneracies and
band crossing along different high symmetry directions. Some of the band
crossings are topological in the sense that they are unavoidable and protected
by symmetries. Such phenomena of “band sticking” has been studied in detail in
other systems belonging to the same space group Mozaffari  _et al._ (2020);
Cuono  _et al._ (2019).

Figs. 2(c) and (f) show the Fermi surface plots for RuP and RuSb,
respectively. The FS of RuP consists of a large corrugated cylinder-like
electron pocket at the BZ center and small dispersionless hole FS sheets at
the BZ boundary. RuAs shows similar FS topology. However, the FS of RuSb has
more structure; the electron FS at the BZ center is smaller and more
elliptical, whereas the hole FS covers a much larger volume with a complicated
topology. The nesting property of the FS will be investigated in the
subsequent sections.

Figure 3: Evolution of the electronic structure as a function of the
monoclinic distortion ($x$). (a-d) Unfolded band dispersion of the monoclinic
phases of RuP. (e-h) & (i-l) Unfolded RuP Fermi surface on the $k_{x}$-$k_{z}$
plane for $k_{y}=0.0$ and $0.5$ planes (in units of $2\pi/b)$, respectively.
The intensity shows the spectral weights. (m) Total energy cost per formula
unit for RuP and RuAs. The dashed lines are quadratic fits to the energy. (n)
Band gap and DOS at the Fermi level. All phases, except $x$=0, have monoclinic
distortion. See text for details.

Figs. 2(d) and (h) present the phonon dispersion for RuP and RuSb along high
symmetry direction. For RuP, there is a clear separation of the phonon modes;
the low lying modes correspond to the Ru-atom displacement and high-energy
modes are from the lighter P atoms. However, such distinction is less clear in
RuAs and becomes even more fuzzy for RuSb owing to the larger mass of
pnictogens. Consistent with our conclusions from the lattice relaxation
calculations and experimental findings, we find that orthorhombic phases of
RuP and RuAs give imaginary phonon frequencies (soft modes) signalling
dynamical lattice instability. We find four soft phonon modes with Ru-
displacements mainly along the zig-zag direction (bortho). The soft mode at T
high symmetry point introduces dimerization between Ru-atoms in the zig-zag
direction and is more likely to be the driver of the lattice instability.
RuSb, on the contrary, does not show such imaginary phonon modes confirming
that it’s orthorhombic phase is stable at low temperature.

In order to understand more about the potential driving factors for the CDW
instability, we first assess the role of the Fermi surface nesting (FSN) by
calculating the real and imaginary part of the bare charge susceptibility
within constant matrix approximation Pickett and Allen (1977); Johannes and
Mazin (2008) [see SM for definition]. The CDW vectors in the orthorhombic
basis are ${\bf q}=(\frac{1}{3},\frac{1}{3},0)$ and $(0,0,\frac{1}{3})$.
Hence, if FSN is the driving mechanism for the CDW transition, peaks are
expected in both the real and imaginary parts of the susceptibility at these
q-values Johannes  _et al._ (2006); Johannes and Mazin (2008) which correspond
to the $\Gamma$-S, $\Gamma$-Z and $\Gamma$-R directions. However, we find that
the peak position of the real and imaginary parts of charge susceptibility has
no correspondence to the CDW vector which excludes FSN as the driving force
for the CDW transition (see Fig. S2 in SM Sup ). The other possible mechanism
for CDW transition is the electron-phonon coupling (EPC) effect. For this, we
compared the relative strength of the electron-phonon coupling constant for
RuP and RuSb. As orthorhombic phase of RuP gives imaginary phonon modes, this
creates problem while calculating coupling constant. However, we were able to
stabilize the orthorhombic phase by means of electronic smearing [see Fig.S4
in SM Sup ]; such electronically smeared orthorhombic phases can give an
estimate of the strength of the EPC parameter $\lambda$ Wang  _et al._ (2020);
Si  _et al._ (2020). For RuP, we estimate $\lambda$ to be $\sim$ 1.2, whereas
for RuSb, we find $\lambda$ to be less than 1. This hints that EPC could be
responsible for the CDW instability in RuP and RuAs.

Figure 4: Variation of the Ru-Ru nearest neighbour bond distance (upper
panels) and bond angle (lower panels) along the trimerization direction for
(a, b) RuP and (c, d) RuAs as a pngction of the electronic temperature
($T_{e}$). The gray dashed lines are guide to the eye. The black dashed line
shows the MIT.

_Monoclinic distortion._ — To explore the relationship between the monoclinic
distortion and MIT, we study the evolution of the electronic structure and
total energy cost as a function of the strength of the monoclinic distortion
by linearly combination of the orthorhombic and monoclinic phases in the ratio
of $(1-x)$ and $x$, respectively. Figs. 3(a-d) and (e-l) show the unfolded
band structure and Fermi surface plots, respectively, in the BZ of the
orthorhombic unit cell, for various values of $x$. The folded band structure
in the BZ of the monoclinic unit cell along with the orbital character is
presented in the SM Sup . We find that the flat bands do not immediately gap-
out; instead there is rather a large critical value of monoclinic distortion
after which the flat bands open gap. The initial reaction of the system to the
monoclinic distortion is to gap-out the $\Gamma$-centered electron Fermi
surface [Figs. 3(e-h)] rather than gapping out the flat bands located on the
$k_{y}=0.5$ plane and the flat bands undergo massive reconstruction. At
$x\approx 0.5$, the FS gaps out completely at $k_{y}=0$ plane but does not gap
out at $k_{y}=0.5$ plane showing pseudo-gap like feature. The DOS at the Fermi
surface [Fig. 3(f)] keeps decreasing when monoclinic distortion increases with
a noticeable sharp drop at the value of distortion when the flat bands gap-
out. The total energy gain as a function of $x$ [Fig. 3(f)] confirms that the
system favors monoclinic distortion. In addition, we find that closer to the
insulating phase ($x>0.4$), quadratic equation of form
$E(x)\sim-\frac{1}{2}gx+\frac{1}{2}kx^{2}$, where $g$ and $k$ denote the
electron-lattice (e-l) and lattice-lattice (l-l) coupling, respectively,
yields excellent fit to the data points, suggesting that e-l coupling
stabilizes the monoclinic phase Yin  _et al._ (2006). From the fits, we find
the value of $g$ and $k$ to be of similar order for RuP and RuAs.

_Ultrafast laser induced MIT._ —Owing to the small band-gap in the monoclinic
phases of RuP and RuAs, it is plausible to induce insulator-to-metal
transition (IMT) in these systems under ultra-fast laser irradiation similar
to VO2 Xu  _et al._ (2023); Li  _et al._ (2022); Wall  _et al._ (2018);
Morrison  _et al._ (2014); Wegkamp  _et al._ (2014); Baum  _et al._ (2007). In
order to mimic the lattice response of the system to the ultrafast high energy
high fluence pump, we performed DFT lattice relaxation calculations at several
values of electronic temperatures ($T_{e}$) and studied the variation of the
atomic coordinates as a function of $T_{e}$ Li  _et al._ (2022); Wall  _et
al._ (2018); Wang  _et al._ (2021). We found that a nominal $T_{e}$ of $\sim
0.1$ eV is sufficient to induce insulator to metal transition in RuAs whereas
for RuP, the critical $T_{e}$ is $\sim 0.15$ eV owing to its slightly larger
band-gap. In Fig. 4, we show the variation of the bond distances and bond
angles along the trimerization direction as a function of $T_{e}$ for RuP and
RuAs. The trimer bond distances and bond angles decrease monotonically with
$T_{e}$ and finally reach the non-trimerized phase for larger values of
$T_{e}$. However, the IMT happens well before the system reaches the non-
trimerized phase, i.e., the monoclinic insulating phase can enter the
_monoclinic metallic_ phase—similar to what has been reported in Ref. Morrison
_et al._ (2014) on VO2—with less than 10% decrease in the trimerization bond
distance and bond angle. This is consistent with the results from linear
interpolation which shows similar behavior [Fig. 3(n)]. It will be interesting
to verify this prediction of two-step light-induced IMT in future ultrafast
pump-probe experiments.

In summary, we perform a systematic study of the electron, phonon, and
electron-lattice interplay in the parent compounds of RuPn superconductors
using first-principles calculations. Our calculations reveal a direct
correlation between the coexistence of the flat-bands, CDW instability and MIT
in this family. The electron and phonon calculations find that RuP and RuAs
members of the family host flat bands at the Fermi surface and undergo
monoclinic distortion at lower temperature accompanied by MIT, whereas RuSb
remains orthorhombic and metallic at low temperature which is in agreement
with the experimental observation. We predict a two-step insulator to metal
and structural phase transition in the monoclinic phases of RuP and RuAs which
can be attested in future ultrafast experiments.In addition, it will be
interesting in the future to study the stability of the monoclinic phases of
RuP and RuAs and the possible emergence of monoclinicility in RuSb by means of
small external perturbations like pressure, chemical substitution, etc.

This work was supported by the U.S. Department of Energy (DOE) the Office of
Basic Energy Sciences, Materials Sciences and Engineering Division under
Contract No. DE-SC0012704.

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