1 Introduction
1.1 Background and motivation
When
$X$
is a scheme,
$G$
is an affine algebraic group acting on
$X$
,
$\unicode[STIX]{x1D712}:G\rightarrow \mathbb{G}_{m}$
is a character, and
$W:X\rightarrow \mathbb{A}^{1}$
is a
$\unicode[STIX]{x1D712}$
-semi-invariant regular function, we call data
$(X,\unicode[STIX]{x1D712},W)^{G}$
a gauged Landau–Ginzburg (LG) model. Following Positselski [Reference PositselskiPos11, Reference Efimov and PositselskiEP15], we consider the derived factorization category of
$(X,\unicode[STIX]{x1D712},W)^{G}$
, denoted by
$$\begin{eqnarray}\text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},W).\end{eqnarray}$$
Derived factorization categories are simultaneous generalizations of bounded derived categories of coherent sheaves on schemes, and of categories of (graded) matrix factorizations of (homogeneous) polynomials. Orlov proved the following semi-orthogonal decompositions between bounded derived categories of hypersurfaces in projective spaces and categories of graded matrix factorizations [Reference OrlovOrl09].
Theorem 1.1 [Reference OrlovOrl09, Theorem 40].
Let
$X\subset \mathbb{P}_{k}^{N-1}$
be the hypersurface defined by a section
$f\in \unicode[STIX]{x1D6E4}(\mathbb{P}_{k}^{N-1},{\mathcal{O}}(d))$
. Denote by
$F$
the corresponding homogeneous polynomial.
-
(1) If
$d<N$
, there is a semi-orthogonal decomposition
$$\begin{eqnarray}\text{D}^{\text{b}}(\text{coh}X)=\langle {\mathcal{O}}_{X}(d-N+1),\ldots ,{\mathcal{O}}_{X},\text{Dcoh}_{\mathbb{G}_{m}}(\mathbb{A}_{k}^{N},\unicode[STIX]{x1D712}_{d},F)\rangle .\end{eqnarray}$$
-
(2) If
$d=N$
, there is an equivalence
$$\begin{eqnarray}\text{D}^{\text{b}}(\text{coh}X)\cong \text{Dcoh}_{\mathbb{G}_{m}}(\mathbb{A}_{k}^{N},\unicode[STIX]{x1D712}_{d},F).\end{eqnarray}$$
-
(3) If
$d>N$
, there is a semi-orthogonal decomposition
$$\begin{eqnarray}\text{Dcoh}_{\mathbb{G}_{m}}(\mathbb{A}_{k}^{N},\unicode[STIX]{x1D712}_{d},F)=\langle k,\ldots ,k(N-d+1),\text{D}^{\text{b}}(\text{coh}X)\rangle .\end{eqnarray}$$
While Orlov’s approach was algebraic, there are geometric approaches to the above theorem [Reference ShipmanShi12, Reference Ballard, Favero and KatzarkovBFK12, Reference Ballard, Deliu, Favero, Isik and KatzarkovBDFIK14], where a version of Knörrer periodicity [Reference ShipmanShi12, Reference IsikIsi13] and homological variations of geometric invariant theory (GIT) quotients [Reference SegalSeg11, Reference Halpern-LeistnerHal15, Reference Ballard, Favero and KatzarkovBFK12] are the main tools. Combinations of Knörrer periodicity and the theory of variations of GIT quotients also imply homological projective dualities [Reference Ballard, Deliu, Favero, Isik and KatzarkovBDFIK13, Reference Addington, Donovan and SegalADS15, Reference Segal and ThomasST14, Reference RennemoRen15].
In this article, we prove another version of Knörrer periodicity [Reference KnörrerKnö87], which is a derived (or global) version, and we combine it with the theory of variations of GIT quotients by [Reference Ballard, Favero and KatzarkovBFK12] to obtain a gauged LG version of Orlov’s theorem.
1.2 Main results
Let
$k$
be an algebraically closed field of characteristic zero. Let
$X$
be a smooth quasi-projective variety over
$k$
, and let
$G$
be a reductive affine algebraic group acting on
$X$
. Let
${\mathcal{E}}$
be a
$G$
-equivariant locally free sheaf of finite rank, and choose a
$G$
-invariant regular section
$s\in \unicode[STIX]{x1D6E4}(X,{\mathcal{E}}^{\vee })^{G}$
. Denote by
$Z\subset X$
the zero scheme of
$s$
. Let
$\unicode[STIX]{x1D712}:G\rightarrow \mathbb{G}_{m}$
be a character of
$G$
, and set
${\mathcal{E}}(\unicode[STIX]{x1D712}):={\mathcal{E}}\otimes {\mathcal{O}}(\unicode[STIX]{x1D712}),$
where
${\mathcal{O}}(\unicode[STIX]{x1D712})$
is the
$G$
-equivariant invertible sheaf corresponding to
$\unicode[STIX]{x1D712}$
. Then
${\mathcal{E}}(\unicode[STIX]{x1D712})$
induces a vector bundle
$\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712}))$
over
$X$
with a
$G$
-action induced by the equivariant structure of
${\mathcal{E}}(\unicode[STIX]{x1D712})$
. Let
$q:\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712}))\rightarrow X$
and
$p:\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712}))|_{Z}\rightarrow Z$
be natural projections, and let
$i:\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712}))|_{Z}\rightarrow \text{V}({\mathcal{E}}(\unicode[STIX]{x1D712}))$
be a natural inclusion. The regular section
$s$
induces a
$\unicode[STIX]{x1D712}$
-semi-invariant regular function
$Q_{s}:\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712}))\rightarrow \mathbb{A}^{1}$
. The first main result in this paper is the following.
Theorem 1.2. Let
$W:X\rightarrow \mathbb{A}^{1}$
be a
$\unicode[STIX]{x1D712}$
-semi-invariant regular function, such that the restricted function
$W|_{Z}:Z\rightarrow \mathbb{A}^{1}$
is flat. Then there is an equivalence
$$\begin{eqnarray}i_{\ast }p^{\ast }:\text{Dcoh}_{G}(Z,\unicode[STIX]{x1D712},W|_{Z})\xrightarrow[{}]{{\sim}}\text{Dcoh}_{G}(\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712})),\unicode[STIX]{x1D712},q^{\ast }W+Q_{s}).\end{eqnarray}$$
The above result is an analogy of Shipman’s and Isik’s result, where they consider the case when
$G=\mathbb{G}_{m}$
, the
$G$
-action on
$X$
is trivial,
$\unicode[STIX]{x1D712}=\text{id}_{\mathbb{G}_{m}}$
, and
$W=0$
(see [Reference ShipmanShi12, Reference IsikIsi13]). Furthermore, the above theorem can be considered as a generalization of Knörrer periodicity to a derived and
$G$
-equivariant version. The proof of the above theorem is quite different from Shipman’s and Isik’s proofs, and we consider relative singularity categories introduced in [Reference Efimov and PositselskiEP15], which are equivalent to derived factorization categories, and use results from [Reference OrlovOrl06].
To state the next result, let
$S$
be a smooth quasi-projective variety over
$k$
with a
$\mathbb{G}_{m}$
-action, and let
$W:S\rightarrow \mathbb{A}^{1}$
be a
$\unicode[STIX]{x1D712}_{1}:=\text{id}_{\mathbb{G}_{m}}$
-semi-invariant regular function which is flat. Let
$d>1$
and
$N>0$
be positive integers, and consider
$\mathbb{G}_{m}$
-actions on
$\mathbb{A}_{S}^{N}:=S\times \mathbb{A}_{k}^{N}$
and on
$\mathbb{P}_{S}^{N-1}:=S\times \mathbb{P}_{k}^{N-1}$
given by
$$\begin{eqnarray}\displaystyle \mathbb{G}_{m}\times \mathbb{A}_{S}^{N}\ni t\times (s,v_{1},\ldots ,v_{N}) & \mapsto & \displaystyle (t^{d}\cdot s,tv_{1},\ldots ,tv_{N})\in \mathbb{A}_{S}^{N},\nonumber\\ \displaystyle \mathbb{G}_{m}\times \mathbb{P}_{S}^{N-1}\ni t\times (s,v_{1}:\cdots :v_{N}) & \mapsto & \displaystyle (t\cdot s,v_{1}:\cdots :v_{N})\in \mathbb{P}_{S}^{N-1}.\nonumber\end{eqnarray}$$
Denote by the same notation
$W:\mathbb{A}_{S}^{N}\rightarrow \mathbb{A}^{1}$
and
$W:\mathbb{P}_{S}^{N-1}\rightarrow \mathbb{A}^{1}$
the pull-backs of
$W:S\rightarrow \mathbb{A}^{1}$
by the natural projections, respectively. Combining the above-derived Knörrer periodicity with the theory of variations of GIT quotients, we obtain the following gauged LG version of the Orlov’s theorem.
Theorem 1.3. Let
$X\subset \mathbb{P}_{S}^{N-1}$
be the hypersurface defined by a
$\mathbb{G}_{m}$
-invariant section
$f\in \unicode[STIX]{x1D6E4}(\mathbb{P}_{S}^{N-1},{\mathcal{O}}(d))^{\mathbb{G}_{m}}$
, and assume that the morphism
$W:\mathbb{P}_{S}^{N-1}\rightarrow \mathbb{A}^{1}$
is flat on
$X$
. Denote by
$F:\mathbb{A}_{S}^{N}\rightarrow \mathbb{A}^{1}$
the regular function induced by
$f$
.
-
(1) If
$d<N$
, there are fully faithful functors and there is a semi-orthogonal decomposition
$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F7}:\text{Dcoh}_{\mathbb{G}_{m}}(\mathbb{A}_{S}^{N},\unicode[STIX]{x1D712}_{d},W+F)\rightarrow \text{Dcoh}_{\mathbb{G}_{m}}(X,\unicode[STIX]{x1D712}_{1},W), & \displaystyle \nonumber\\ \displaystyle & \displaystyle \unicode[STIX]{x1D6F6}:\text{Dcoh}_{\mathbb{G}_{m}}(S,\unicode[STIX]{x1D712}_{1},W)\rightarrow \text{Dcoh}_{\mathbb{G}_{m}}(X,\unicode[STIX]{x1D712}_{1},W), & \displaystyle \nonumber\end{eqnarray}$$
where
$$\begin{eqnarray}\text{Dcoh}_{\mathbb{G}_{m}}(X,\unicode[STIX]{x1D712}_{1},W)=\langle \unicode[STIX]{x1D6F6}_{d-N+1},\ldots ,\unicode[STIX]{x1D6F6}_{0},\unicode[STIX]{x1D6F7}(\text{Dcoh}_{\mathbb{G}_{m}}(\mathbb{A}_{S}^{N},\unicode[STIX]{x1D712}_{d},W+F))\rangle ,\end{eqnarray}$$
$\unicode[STIX]{x1D6F6}_{i}$
denotes the essential image of the composition
$(-)\otimes {\mathcal{O}}(i)\circ \unicode[STIX]{x1D6F6}$
.
-
(2) If
$d=N$
, there is an equivalence
$$\begin{eqnarray}\text{Dcoh}_{\mathbb{G}_{m}}(X,\unicode[STIX]{x1D712}_{1},W)\cong \text{Dcoh}_{\mathbb{G}_{m}}(\mathbb{A}_{S}^{N},\unicode[STIX]{x1D712}_{d},W+F).\end{eqnarray}$$
-
(3) If
$d>N$
, there are fully faithful functors and there is a semi-orthogonal decomposition
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F9}:\text{Dcoh}_{\mathbb{G}_{m}}(X,\unicode[STIX]{x1D712}_{1},W) & \rightarrow & \displaystyle \text{Dcoh}_{\mathbb{G}_{m}}(\mathbb{A}_{S}^{N},\unicode[STIX]{x1D712}_{d},W+F),\nonumber\\ \displaystyle \unicode[STIX]{x1D6F6}:\text{Dcoh}_{\mathbb{G}_{m}}(S,\unicode[STIX]{x1D712}_{1},W) & \rightarrow & \displaystyle \text{Dcoh}_{\mathbb{G}_{m}}(\mathbb{A}_{S}^{N},\unicode[STIX]{x1D712}_{d},W+F),\nonumber\end{eqnarray}$$
where
$$\begin{eqnarray}\text{Dcoh}_{\mathbb{G}_{m}}(\mathbb{A}_{S}^{N},\unicode[STIX]{x1D712}_{d},W+F)=\langle \unicode[STIX]{x1D6F6}_{0},\ldots ,\unicode[STIX]{x1D6F6}_{N-d+1},\unicode[STIX]{x1D6F9}(\text{Dcoh}_{\mathbb{G}_{m}}(X,\unicode[STIX]{x1D712}_{1},W))\rangle ,\end{eqnarray}$$
$\unicode[STIX]{x1D6F6}_{i}$
denotes the essential image of the composition
$(-)\otimes {\mathcal{O}}(\unicode[STIX]{x1D712}_{i})\circ \unicode[STIX]{x1D6F6}$
.
Since we have an equivalence
$$\begin{eqnarray}\text{D}^{\text{b}}(\text{coh}X)\cong \text{Dcoh}_{\mathbb{G}_{m}}(X,\unicode[STIX]{x1D712}_{1},0),\end{eqnarray}$$
where the
$\mathbb{G}_{m}$
-action on
$X$
is trivial, we can view Orlov’s theorem as the case when
$S=\operatorname{Spec}k$
and
$W=0$
in the above theorem.
1.3 Notation and conventions
-
∙ For an integer
$n\in \mathbb{Z}$
, we denote by
$\unicode[STIX]{x1D712}_{n}:\mathbb{G}_{m}\rightarrow \mathbb{G}_{m}$
the character of the one-dimensional algebraic torus
$\mathbb{G}_{m}$
defined by
$$\begin{eqnarray}\unicode[STIX]{x1D712}_{n}(t):=t^{n}.\end{eqnarray}$$
-
∙ For a character
$\unicode[STIX]{x1D712}:G\rightarrow \mathbb{G}_{m}$
of an algebraic group
$G$
, we denote by
${\mathcal{O}}(\unicode[STIX]{x1D712})$
the
$G$
-equivariant invertible sheaf
$({\mathcal{O}}_{X},\unicode[STIX]{x1D706}:\unicode[STIX]{x1D70B}^{\ast }{\mathcal{O}}_{X}\xrightarrow[{}]{{\sim}}\unicode[STIX]{x1D70E}^{\ast }{\mathcal{O}}_{X})$
associated to
$\unicode[STIX]{x1D712}$
, where
$\unicode[STIX]{x1D70B}:G\times X\rightarrow X$
and
$\unicode[STIX]{x1D70E}:G\times X\rightarrow X$
are the projection and the morphism defining the
$G$
-action, respectively. For any
$g\in G$
,
$\unicode[STIX]{x1D706}_{g}:=\unicode[STIX]{x1D706}|_{\{g\}\times X}:{\mathcal{O}}_{X}\xrightarrow[{}]{{\sim}}g^{\ast }{\mathcal{O}}_{X}$
is given as the composition of the multiplication by
$$\begin{eqnarray}{\mathcal{O}}_{X}\xrightarrow[{}]{\unicode[STIX]{x1D712}(g)}{\mathcal{O}}_{X}\xrightarrow[{}]{{\sim}}g^{\ast }{\mathcal{O}}_{X}\end{eqnarray}$$
$\unicode[STIX]{x1D712}(g)\in \mathbb{G}_{m}$
and the natural isomorphism
${\mathcal{O}}_{X}\xrightarrow[{}]{{\sim}}g^{\ast }{\mathcal{O}}_{X}$
.
For a
$G$
-equivariant quasi-coherent sheaf
$F$
on a
$G$
-scheme, we set
$$\begin{eqnarray}F(\unicode[STIX]{x1D712}):=F\otimes {\mathcal{O}}(\unicode[STIX]{x1D712}).\end{eqnarray}$$
-
∙ Throughout this article, unless stated otherwise, all schemes and categories are over an algebraically closed field
$k$
of characteristic zero.
2 Derived factorization categories
In this section, we provide the definition of the derived factorization category, which is introduced by Positselski (cf. [Reference PositselskiPos11, Reference Efimov and PositselskiEP15]), and its basic properties.
2.1 Derived factorization categories
Definition 2.1. A gauged LG model is data
$(X,\unicode[STIX]{x1D712},W)^{G}$
with
$X$
a scheme,
$\unicode[STIX]{x1D712}:G\rightarrow \mathbb{G}_{m}$
a character of
$G$
,
$G$
an affine algebraic group acting on
$X$
, and
$W:X\rightarrow \mathbb{A}^{1}$
a
$\unicode[STIX]{x1D712}$
-semi-invariant regular function, i.e.
$W(g\cdot x)=\unicode[STIX]{x1D712}(g)W(x)$
for any
$g\in G$
and any
$x\in X$
. If
$G$
is trivial, we denote the gauged LG model by
$(X,W)$
, and call it an LG model.
For a gauged LG model, we consider its factorizations which are ‘twisted’ complexes.
Definition 2.2. Let
$(X,\unicode[STIX]{x1D712},W)^{G}$
be a gauged LG model. A factorization
$F$
of
$(X,\unicode[STIX]{x1D712},W)^{G}$
is a sequence
$$\begin{eqnarray}F=(F_{1}\xrightarrow[{}]{\unicode[STIX]{x1D711}_{1}^{F}}F_{0}\xrightarrow[{}]{\unicode[STIX]{x1D711}_{0}^{F}}F_{1}(\unicode[STIX]{x1D712})),\end{eqnarray}$$
where
$F_{i}$
is a
$G$
-equivariant quasi-coherent sheaf on
$X$
and
$\unicode[STIX]{x1D711}_{i}^{F}$
is a
$G$
-invariant homomorphism for
$i=0,1$
such that
$\unicode[STIX]{x1D711}_{0}^{F}\circ \unicode[STIX]{x1D711}_{1}^{F}=W\cdot \text{id}_{F_{1}}$
and
$\unicode[STIX]{x1D711}_{1}^{F}(\unicode[STIX]{x1D712})\circ \unicode[STIX]{x1D711}_{0}^{F}=W\cdot \text{id}_{F_{0}}$
. Equivariant quasi-coherent sheaves
$F_{0}$
and
$F_{1}$
in the above sequence are called components of the factorization
$F$
.
Definition 2.3. For a gauged LG model
$(X,\unicode[STIX]{x1D712},W)^{G}$
, we define a differential graded (dg) category
$$\begin{eqnarray}\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)\end{eqnarray}$$
whose objects are factorizations of
$(X,\unicode[STIX]{x1D712},W)^{G}$
, and whose complexes of morphisms are defined as follows: for two objects
$E,F\in \text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)$
, we define the complex
$\operatorname{Hom}(E,F)^{\bullet }$
of morphisms from
$E$
to
$F$
as the following graded vector space
$$\begin{eqnarray}\operatorname{Hom}(E,F)^{\bullet }:=\bigoplus _{n\in \mathbb{Z}}\operatorname{Hom}(E,F)^{n}\end{eqnarray}$$
with a differential
$d^{i}:\text{Hom}(E,F)^{i}\rightarrow \operatorname{Hom}(E,F)^{i+1}$
given by
$$\begin{eqnarray}d^{i}(f):=\unicode[STIX]{x1D711}^{F}\circ f-(-1)^{i}f\circ \unicode[STIX]{x1D711}^{E}\end{eqnarray}$$
where
$$\begin{eqnarray}\displaystyle \operatorname{Hom}(E,F)^{2m} & := & \displaystyle \operatorname{Hom}(E_{1},F_{1}(\unicode[STIX]{x1D712}^{m}))\oplus \operatorname{Hom}(E_{0},F_{0}(\unicode[STIX]{x1D712}^{m})),\nonumber\\ \displaystyle \operatorname{Hom}(E,F)^{2m+1} & := & \displaystyle \operatorname{Hom}(E_{1},F_{0}(\unicode[STIX]{x1D712}^{m}))\oplus \operatorname{Hom}(E_{0},F_{1}(\unicode[STIX]{x1D712}^{m+1})).\nonumber\end{eqnarray}$$
We define dg full subcategories
$\text{coh}_{G}(X,\unicode[STIX]{x1D712},W)$
,
$\text{Inj}_{G}(X,\unicode[STIX]{x1D712},W)$
,
$\text{LFr}_{G}(X,\unicode[STIX]{x1D712},W)$
, and
$\text{lfr}_{G}(X,\unicode[STIX]{x1D712},W)$
of
$\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)$
whose objects are factorizations whose components are coherent, injective quasi-coherent, locally free, and locally free of finite ranks, respectively. If
$G$
is trivial, dropping
$G$
and
$\unicode[STIX]{x1D712}$
from each notation, we denote the dg categories by
$\text{Qcoh}(X,W)$
,
$\text{coh}(X,W)$
, etc.
The dg category
$\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)$
induces two categories
$$\begin{eqnarray}\displaystyle Z^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))\quad \text{and}\quad H^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)). & & \displaystyle \nonumber\end{eqnarray}$$
Objects of these categories are same as objects of
$\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)$
, and sets of morphisms are defined as
$$\begin{eqnarray}\displaystyle \operatorname{Hom}_{Z^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))}(E,F) & := & \displaystyle Z^{0}(\operatorname{Hom}(E,F)^{\bullet })=\operatorname{Ker}(d^{0}),\nonumber\\ \displaystyle \operatorname{Hom}_{H^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))}(E,F) & := & \displaystyle H^{0}(\operatorname{Hom}(E,F)^{\bullet })=\operatorname{Ker}(d^{0})/\!\operatorname{Im}(d^{-1}).\nonumber\end{eqnarray}$$
Remark 2.4. We can write down the above sets of morphisms as follows:
$\operatorname{Hom}_{Z^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))}(E,F)$
is the set of pairs
$(f_{1},f_{0})$
such that
$f_{i}\in \operatorname{Hom}_{\text{Qcoh}_{G}X}(E_{i},F_{i})$
and that the following diagram is commutative.
The set of morphisms in the category
$H^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))$
can be described as the set of homotopy equivalence (denoted by
${\sim}$
) classes in
$\operatorname{Hom}_{Z^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))}(E,F)$
,
$$\begin{eqnarray}\operatorname{Hom}_{H^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))}(E,F)=\operatorname{Hom}_{Z^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))}(E,F)/\sim ,\end{eqnarray}$$
where two morphisms
$f=(f_{1},f_{0})$
and
$g=(g_{1},g_{0})$
in
$\operatorname{Hom}_{Z^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))}(E,F)$
are homotopy equivalent if there exist two morphisms
$$\begin{eqnarray}h_{0}:E_{0}\rightarrow F_{1}\quad \text{and}\quad h_{1}:E_{1}(\unicode[STIX]{x1D712})\rightarrow F_{0}\end{eqnarray}$$
such that
$f_{0}=\unicode[STIX]{x1D711}_{1}^{F}h_{0}+h_{1}\unicode[STIX]{x1D711}_{0}^{E}$
and
$f_{1}(\unicode[STIX]{x1D712})=\unicode[STIX]{x1D711}_{0}^{F}h_{1}+h_{0}(\unicode[STIX]{x1D712})\unicode[STIX]{x1D711}_{1}^{E}(\unicode[STIX]{x1D712})$
.
We easily see the following result (cf. [Reference HiranoHir17, Proposition 3.5]).
Proposition 2.5. The categories
$Z^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))$
and
$Z^{0}(\text{coh}_{G}(X,\unicode[STIX]{x1D712},W))$
are abelian, and the categories
$Z^{0}(\text{LFr}_{G}(X,\unicode[STIX]{x1D712},W))$
and
$Z^{0}(\text{lfr}_{G}(X,\unicode[STIX]{x1D712},W))$
are exact.
We next define the totalizations of bounded complexes of factorizations.
Definition 2.6. Let
$F^{\bullet }=(\cdots \rightarrow F^{i}\xrightarrow[{}]{\unicode[STIX]{x1D6FF}^{i}}F^{i+1}\rightarrow \cdots \,)$
be a bounded complex of
$Z^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))$
. For
$l=0,1$
, set
$$\begin{eqnarray}T_{l}:=\bigoplus _{i+j=-l}F_{\overline{j}}^{i}(\unicode[STIX]{x1D712}^{\lceil j/2\rceil }),\end{eqnarray}$$
and let
$$\begin{eqnarray}t_{l}:T_{l}\rightarrow T_{\overline{l+1}}\end{eqnarray}$$
be a
$G$
-invariant homomorphism given by
$$\begin{eqnarray}t_{l}|_{F_{\overline{j}}^{i}(\unicode[STIX]{x1D712}^{\lceil j/2\rceil })}:=\unicode[STIX]{x1D6FF}_{\overline{j}}^{i}(\unicode[STIX]{x1D712}^{\lceil j/2\rceil })+(-1)^{i}\unicode[STIX]{x1D711}_{\overline{j}}^{F^{i}}(\unicode[STIX]{x1D712}^{\lceil j/2\rceil }),\end{eqnarray}$$
where
$\overline{n}$
is
$n$
modulo
$2$
, and
$\lceil m\rceil$
is the minimum integer which is greater than or equal to a real number
$m$
. We define the totalization
$\operatorname{Tot}(F^{\bullet })\in Z^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))$
of
$F^{\bullet }$
as
$$\begin{eqnarray}\operatorname{Tot}(F^{\bullet }):=(T_{1}\xrightarrow[{}]{t_{1}}T_{0}\xrightarrow[{}]{t_{0}}T_{1}(\unicode[STIX]{x1D712})).\end{eqnarray}$$
In what follows, we will recall that the category
$H^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))$
has a structure of a triangulated category.
Definition 2.7. We define an automorphism
$T$
on
$H^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))$
, which is called the shift functor, as follows. For an object
$F\in H^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))$
, we define an object
$T(F)$
as
$$\begin{eqnarray}T(F):=(F_{0}\xrightarrow[{}]{-\unicode[STIX]{x1D711}_{0}^{F}}F_{1}(\unicode[STIX]{x1D712})\xrightarrow[{}]{-\unicode[STIX]{x1D711}_{1}^{F}(\unicode[STIX]{x1D712})}F_{0}(\unicode[STIX]{x1D712}))\end{eqnarray}$$
and for a morphism
$f=(f_{1},f_{0})\in \operatorname{Hom}(E,F)$
, we set
$T(f):=(f_{0},f_{1}(\unicode[STIX]{x1D712}))\in \operatorname{Hom}(T(E),T(F))$
. For any integer
$n\in \mathbb{Z}$
, denote by
$(-)[n]$
the functor
$T^{n}(-)$
.
Definition 2.8. Let
$f:E\rightarrow F$
be a morphism in
$Z^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))$
. We define its mapping cone Cone
$(f)$
to be the totalization of the complex
$$\begin{eqnarray}(\cdots \rightarrow 0\rightarrow E\xrightarrow[{}]{f}F\rightarrow 0\rightarrow \cdots \,)\end{eqnarray}$$
with
$F$
in degree zero.
A distinguished triangle is a sequence in
$H^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))$
which is isomorphic to a sequence of the form
$$\begin{eqnarray}E\xrightarrow[{}]{f}F\xrightarrow[{}]{i}\operatorname{Cone}(f)\xrightarrow[{}]{p}E[1],\end{eqnarray}$$
where
$i$
and
$p$
are natural injection and projection, respectively.
The following is well known.
Proposition 2.9. We denote by
$H^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))$
a triangulated category with respect to its shift functor and its distinguished triangles defined above. Full subcategories
$H^{0}(\text{coh}_{G}(X,\unicode[STIX]{x1D712},W))$
,
$H^{0}(\text{Inj}_{G}(X,\unicode[STIX]{x1D712},W))$
,
$H^{0}(\text{LFr}_{G}(X,\unicode[STIX]{x1D712},W))$
, and
$H^{0}(\text{lfr}_{G}(X,\unicode[STIX]{x1D712},W))$
are full triangulated subcategories.
Following Positselski [Reference PositselskiPos11, Reference Efimov and PositselskiEP15], we define derived factorization categories.
Definition 2.10. Denote by
$\text{Acycl}(\text{coh}_{G}(X,\unicode[STIX]{x1D712},W))$
the smallest thick subcategory of
$H^{0}(\text{coh}_{G}(X,\unicode[STIX]{x1D712},W))$
containing all totalizations of short exact sequences in
$Z^{0}(\text{coh}_{G}(X,\unicode[STIX]{x1D712},W))$
. We define the derived factorization category of
$(X,\unicode[STIX]{x1D712},W)^{G}$
as the Verdier quotient
$$\begin{eqnarray}\text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},W):=H^{0}(\text{coh}_{G}(X,\unicode[STIX]{x1D712},W))/\text{Acycl}(\text{coh}_{G}(X,\unicode[STIX]{x1D712},W)).\end{eqnarray}$$
Similarly, consider thick full subcategories
$\text{Acycl}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))$
,
$\text{Acycl}(\text{LFr}_{G}(X,\unicode[STIX]{x1D712},W))$
and
$\text{Acycl}(\text{lfr}_{G}(X,\unicode[STIX]{x1D712},W))$
of
$H^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))$
,
$H^{0}(\text{LFr}_{G}(X,\unicode[STIX]{x1D712},W))$
, and
$H^{0}(\text{lfr}_{G}(X,\unicode[STIX]{x1D712},W))$
respectively, and denote the corresponding Verdier quotients by
$$\begin{eqnarray}\displaystyle \text{DQcoh}_{G}(X,\unicode[STIX]{x1D712},W) & := & \displaystyle H^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))/\text{Acycl}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)),\nonumber\\ \displaystyle \text{DLFr}_{G}(X,\unicode[STIX]{x1D712},W) & := & \displaystyle H^{0}(\text{LFr}_{G}(X,\unicode[STIX]{x1D712},W))/\text{Acycl}(\text{LFr}_{G}(X,\unicode[STIX]{x1D712},W)),\nonumber\\ \displaystyle \text{Dlfr}_{G}(X,\unicode[STIX]{x1D712},W) & := & \displaystyle H^{0}(\text{lfr}_{G}(X,\unicode[STIX]{x1D712},W))/\text{Acycl}(\text{lfr}_{G}(X,\unicode[STIX]{x1D712},W)).\nonumber\end{eqnarray}$$
Objects in
$\text{Acycl}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))$
are called acyclic.
Denote by
$\text{Acycl}^{\text{co}}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))$
(respectively
$\text{Acycl}^{\text{co}}(\text{LFr}_{G}(X,\unicode[STIX]{x1D712},W))$
) the smallest thick subcategory of the triangulated category
$H^{0}(\text{Qcoh}_{G}(X,\!\unicode[STIX]{x1D712},W))$
(respectively
$H^{0}(\text{LFr}_{G}(X,\!\unicode[STIX]{x1D712},W))$
) which is closed under taking small direct sums and contain all totalizations of short exact sequences in
$Z^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))$
(respectively
$Z^{0}(\text{LFr}_{G}(X,\unicode[STIX]{x1D712},W))$
). Denote the Verdier quotients by
$$\begin{eqnarray}\displaystyle {\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W) & := & \displaystyle H^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))/\text{Acycl}^{\text{co}}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)),\nonumber\\ \displaystyle {\text{D}^{\text{co}}\text{LFr}}_{G}(X,\unicode[STIX]{x1D712},W) & := & \displaystyle H^{0}(\text{LFr}_{G}(X,\unicode[STIX]{x1D712},W))/\text{Acycl}^{\text{co}}(\text{LFr}_{G}(X,\unicode[STIX]{x1D712},W)).\nonumber\end{eqnarray}$$
Objects in
$\text{Acycl}^{\text{co}}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))$
are called coacyclic.
If
$G$
is trivial, we drop
$G$
and
$\unicode[STIX]{x1D712}$
from the above notation, and denote each triangulated category by
$\text{Dcoh}(X,W)$
,
$\text{DQcoh}^{\text{co}}(X,W)$
, etc.
Remark 2.11. If
$X$
is a regular Noetherian scheme of finite Krull dimension, then
$\text{Acycl}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))$
is cocomplete, i.e. admits arbitrary direct sums (cf. [Reference Lunts and SchnürerLS16, Corollary 2.23]). Hence, in that case, we have
$$\begin{eqnarray}{\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W)=\text{DQcoh}_{G}(X,\unicode[STIX]{x1D712},W).\end{eqnarray}$$
The following lemmas ensure the existence of derived functors between derived factorization categories.
Lemma 2.12. Assume that the scheme
$X$
is Noetherian. The natural functor
$$\begin{eqnarray}H^{0}(\text{Inj}_{G}(X,\unicode[STIX]{x1D712},W))\rightarrow {\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W)\end{eqnarray}$$
is an equivalence.
Proof. Since the abelian category
$\text{Qcoh}_{G}X$
of
$G$
-equivariant quasi-coherent sheaves is a locally Noetherian Grothendieck category, it has enough injective objects, and coproducts of injective objects are injective. Hence, the result follows from [Reference Ballard, Deliu, Favero, Isik and KatzarkovBDFIK16, Cororally 2.25].◻
Lemma 2.13 [Reference Ballard, Favero and KatzarkovBFK14, Proposition 3.14].
Assume that
$X$
is a smooth variety. Then the natural functor
$$\begin{eqnarray}\text{DLFr}_{G}(X,\unicode[STIX]{x1D712},W)\rightarrow \text{DQcoh}_{G}(X,\unicode[STIX]{x1D712},W)\end{eqnarray}$$
is an equivalence. This equivalence induces an equivalence
$$\begin{eqnarray}\text{Dlfr}_{G}(X,\unicode[STIX]{x1D712},W)\rightarrow \text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},W).\end{eqnarray}$$
2.2 Derived categories and derived factorization categories
In this section, we recall that derived factorization categories are generalizations of bounded derived categories of coherent sheaves on schemes.
Consider trivial
$\mathbb{G}_{m}$
-action on a scheme
$X$
and an exact functor between abelian categories
$$\begin{eqnarray}\unicode[STIX]{x1D6F6}:\text{Ch}(\text{Qcoh}X)\rightarrow Z^{0}(\text{Qcoh}_{\mathbb{ G}_{m}}(X,\unicode[STIX]{x1D712}_{1},0))\end{eqnarray}$$
given by
$$\begin{eqnarray}\unicode[STIX]{x1D6F6}(F^{\bullet },d_{F}^{\bullet }):=\biggl(\bigoplus _{i\in \mathbb{Z}}F^{2i-1}(\unicode[STIX]{x1D712}_{-i})\xrightarrow[{}]{\oplus \,d_{F}^{2i-1}(\unicode[STIX]{x1D712}_{-i})}\bigoplus _{i\in \mathbb{Z}}F^{2i}(\unicode[STIX]{x1D712}_{-i})\xrightarrow[{}]{\oplus \,d_{F}^{2i}(\unicode[STIX]{x1D712}_{-i})}\bigoplus _{i\in \mathbb{Z}}F^{2i-1}(\unicode[STIX]{x1D712}_{-i+1})\biggr).\end{eqnarray}$$
Then it is easy to see that the exact functor
$\unicode[STIX]{x1D6F6}$
is an equivalence, and it induces an equivalence between triangulated categories:
$$\begin{eqnarray}\unicode[STIX]{x1D6F6}:\text{K}(\text{Qcoh}X)\rightarrow H^{0}(\text{Qcoh}_{\mathbb{ G}_{m}}(X,\unicode[STIX]{x1D712}_{1},0)).\end{eqnarray}$$
Since the triangulated equivalence preserves coacyclic objects, we obtain the following.
Proposition 2.14. There is an equivalence
$$\begin{eqnarray}\unicode[STIX]{x1D6F6}:\text{D}^{\text{co}}(\text{Qcoh}X)\xrightarrow[{}]{{\sim}}\text{D}^{\text{co}}\text{Qcoh}_{\mathbb{G}_{m}}(X,\unicode[STIX]{x1D712}_{1},0),\end{eqnarray}$$
which induces an equivalence between full subcategories
$$\begin{eqnarray}\unicode[STIX]{x1D6F6}:\text{D}^{\text{b}}(\text{coh}X)\xrightarrow[{}]{{\sim}}\text{D}\text{coh}_{\mathbb{G}_{m}}(X,\unicode[STIX]{x1D712}_{1},0).\end{eqnarray}$$
Remark 2.15. See [Reference PositselskiPos12, Appendix A] for the definition and basic properties of coderived categories. By [Reference PositselskiPos12, Theorem 5.9.1(b)], if
$X$
is Noetherian, the thick subcategory
$\text{D}^{\text{b}}(\text{coh}X)$
of
$\text{D}^{\text{co}}(\text{Qcoh}X)$
is the full subcategory of compact objects. However, when
$X$
is a singular variety, there is an object in
$\text{D}^{\text{b}}(\text{coh}X)$
which is not compact in the usual derived category
$\text{D}(\text{Qcoh}X)$
. This is a remarkable difference between the usual derived category and the coderived category. On the other hand, if
$X$
is a smooth variety, these two kinds of derived categories are equivalent (see the argument in the proof of [Reference PositselskiPos12, Theorem 5.5.1(c)]).
2.3 Derived functors between derived factorization categories
We quickly review derived functors between derived factorization categories. See, for example, [Reference Lunts and SchnürerLS16, Reference Ballard, Favero and KatzarkovBFK14] or [Reference HiranoHir17] for more details.
2.3.1 Direct images and inverse images
Let
$X$
and
$Y$
be Noetherian schemes, and let
$G$
be an affine algebraic group acting on
$X$
and
$Y$
. Let
$f:X\rightarrow Y$
be a
$G$
-equivariant morphism, and choose a
$\unicode[STIX]{x1D712}$
-semi-invariant function
$W:Y\rightarrow \mathbb{A}^{1}$
.
The morphism
$f$
naturally induces a dg functor
$$\begin{eqnarray}f_{\ast }:\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},f^{\ast }W)\rightarrow \text{Qcoh}_{G}(Y,\unicode[STIX]{x1D712},W)\end{eqnarray}$$
defined by
$$\begin{eqnarray}f_{\ast }F:=(f_{\ast }(F_{1})\xrightarrow[{}]{f_{\ast }(\unicode[STIX]{x1D711}_{1}^{F})}f_{\ast }(F_{0})\xrightarrow[{}]{f_{\ast }(\unicode[STIX]{x1D711}_{0}^{F})}f_{\ast }(F_{1})(\unicode[STIX]{x1D712})).\end{eqnarray}$$
By Lemma 2.12, we can derive the dg functor
$f_{\ast }$
to obtain an exact functor
$$\begin{eqnarray}\mathbf{R}f_{\ast }:{\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},f^{\ast }W)\rightarrow {\text{D}^{\text{co}}\text{Qcoh}}_{G}(Y,\unicode[STIX]{x1D712},W).\end{eqnarray}$$
If
$f$
is a proper morphism, it preserves factorizations whose components are coherent sheaves:
$$\begin{eqnarray}\mathbf{R}f_{\ast }:\text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},f^{\ast }W)\rightarrow \text{Dcoh}_{G}(Y,\unicode[STIX]{x1D712},W).\end{eqnarray}$$
The morphism
$f$
also induces another dg functor
$$\begin{eqnarray}f^{\ast }:\text{Qcoh}_{G}(Y,\unicode[STIX]{x1D712},W)\rightarrow \text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},f^{\ast }W),\end{eqnarray}$$
defined by
$$\begin{eqnarray}f^{\ast }E:=(f^{\ast }(E_{1})\xrightarrow[{}]{f^{\ast }(\unicode[STIX]{x1D711}_{1}^{E})}f^{\ast }(E_{0})\xrightarrow[{}]{f^{\ast }(\unicode[STIX]{x1D711}_{0}^{E})}f^{\ast }(E_{1})(\unicode[STIX]{x1D712})).\end{eqnarray}$$
If
$Y$
is a smooth variety, by Lemma 2.13, we have the derived functor of
$f^{\ast }$
$$\begin{eqnarray}\mathbf{L}f^{\ast }:\text{DQcoh}_{G}(Y,\unicode[STIX]{x1D712},W)\rightarrow {\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},f^{\ast }W).\end{eqnarray}$$
This functor maps coherent factorizations to coherent factorizations:
$$\begin{eqnarray}\mathbf{L}f^{\ast }:\text{Dcoh}_{G}(Y,\unicode[STIX]{x1D712},W)\rightarrow \text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},f^{\ast }W).\end{eqnarray}$$
It is standard that the direct image
$\mathbf{R}f_{\ast }$
and the inverse image
$\mathbf{L}f^{\ast }$
are adjoint.
Remark 2.16. If
$f$
is an affine morphism, we do not need to take the derived functor, and the dg functor
$f_{\ast }$
naturally defines an exact functor
$$\begin{eqnarray}f_{\ast }:{\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},f^{\ast }W)\rightarrow {\text{D}^{\text{co}}\text{Qcoh}}_{G}(Y,\unicode[STIX]{x1D712},W).\end{eqnarray}$$
Similarly, if
$f$
is a flat morphism, we do not have to assume that
$Y$
is a smooth variety and take the derived functor, and the dg functor
$f^{\ast }$
naturally defines an exact functor
$$\begin{eqnarray}f^{\ast }:{\text{D}^{\text{co}}\text{Qcoh}}_{G}(Y,\unicode[STIX]{x1D712},W)\rightarrow {\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},f^{\ast }W).\end{eqnarray}$$
2.3.2 Tensor products
Let
$(X,\unicode[STIX]{x1D712},W)^{G}$
be a gauged LG model, and let
$V:X\rightarrow \mathbb{A}^{1}$
be another
$\unicode[STIX]{x1D712}$
-semi-invariant regular function. Fix an object
$F\in \text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},V)$
. We define a dg functor
$$\begin{eqnarray}(-)\otimes F:\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)\rightarrow \text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W+V)\end{eqnarray}$$
given by
$$\begin{eqnarray}E\otimes F:=\biggl(\bigoplus _{i=0,1}(F_{i}\otimes E_{\overline{i+1}})\xrightarrow[{}]{\unicode[STIX]{x1D711}_{1}^{E\,\otimes \,F}}\bigoplus _{i=0,1}(F_{i}\otimes E_{i})(\unicode[STIX]{x1D712}^{i})\xrightarrow[{}]{\unicode[STIX]{x1D711}_{0}^{E\,\otimes \,F}}\bigoplus _{i=0,1}(F_{i}\otimes E_{\overline{i+1}})(\unicode[STIX]{x1D712})\biggr),\end{eqnarray}$$
where
$\overline{n}$
is
$n$
modulo 2, and
$$\begin{eqnarray}\unicode[STIX]{x1D711}_{1}^{E\,\otimes \,F}=\left(\begin{array}{@{}cc@{}}\unicode[STIX]{x1D711}_{1}^{E}\otimes 1 & 1\otimes \unicode[STIX]{x1D711}_{1}^{F}\\ -1\otimes \unicode[STIX]{x1D711}_{0}^{F} & \unicode[STIX]{x1D711}_{0}^{E}\otimes 1\end{array}\right)\end{eqnarray}$$
and
$$\begin{eqnarray}\unicode[STIX]{x1D711}_{0}^{E\,\otimes \,F}=\left(\begin{array}{@{}cc@{}}\unicode[STIX]{x1D711}_{0}^{E}\otimes 1 & -(1\otimes \unicode[STIX]{x1D711}_{1}^{F})(\unicode[STIX]{x1D712})\\ 1\otimes \unicode[STIX]{x1D711}_{0}^{F} & (\unicode[STIX]{x1D711}_{1}^{E}\otimes 1)(\unicode[STIX]{x1D712})\end{array}\right).\end{eqnarray}$$
By Lemma 2.13, if
$X$
is a smooth variety, we have the derived functor
$$\begin{eqnarray}(-)\otimes ^{\mathbf{L}}F:\text{DQcoh}_{G}(X,\unicode[STIX]{x1D712},W)\rightarrow \text{DQcoh}_{G}(X,\unicode[STIX]{x1D712},W+V).\end{eqnarray}$$
If
$F$
is a coherent factorization, this functor preserves coherent factorizations:
$$\begin{eqnarray}(-)\otimes ^{\mathbf{L}}F:\text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},W)\rightarrow \text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},W+V).\end{eqnarray}$$
Consider a natural exact functor between abelian categories
$$\begin{eqnarray}\unicode[STIX]{x1D70F}:\text{Qcoh}_{G}(X)\rightarrow Z^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},0))\end{eqnarray}$$
defined by
$$\begin{eqnarray}\unicode[STIX]{x1D70F}(F):=(0\rightarrow F\rightarrow 0).\end{eqnarray}$$
Then we denote by
$\unicode[STIX]{x1D6F4}$
the following composition of functors
$$\begin{eqnarray}\unicode[STIX]{x1D6F4}:\text{D}^{\text{b}}(\text{Qcoh}_{G}X)\xrightarrow[{}]{\unicode[STIX]{x1D70F}}\text{D}^{\text{b}}(Z^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},0)))\xrightarrow[{}]{\operatorname{Tot}}\text{DQcoh}_{G}(X,\unicode[STIX]{x1D712},0).\end{eqnarray}$$
For a complex
$F^{\bullet }\in \text{D}^{\text{b}}(\text{Qcoh}_{G}X)$
, we define the tensor product
$$\begin{eqnarray}(-)\otimes ^{\mathbf{L}}F^{\bullet }:\text{DQcoh}_{G}(X,\unicode[STIX]{x1D712},W)\rightarrow \text{DQcoh}_{G}(X,\unicode[STIX]{x1D712},W)\end{eqnarray}$$
by the following
$$\begin{eqnarray}(-)\otimes ^{\mathbf{L}}F^{\bullet }:=(-)\otimes ^{\mathbf{L}}\unicode[STIX]{x1D6F4}(F^{\bullet }).\end{eqnarray}$$
Remark 2.17. If the components of
$F$
are flat sheaves, we do not have to assume that
$X$
is a smooth variety and take the derived functor, and the dg functor
$(-)\otimes F$
induces an exact functor
$$\begin{eqnarray}(-)\otimes F:{\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W)\rightarrow {\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W+V).\end{eqnarray}$$
Furthermore, if
$F$
is a coherent factorization, the tensor product preserves coherent factorizations:
$$\begin{eqnarray}(-)\otimes F:\text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},W)\rightarrow \text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},W+V).\end{eqnarray}$$
2.3.3 Integral functors
We define integral functors between derived factorization categories. For simplicity, we consider the case when
$G$
is trivial. Let
$X_{1}$
and
$X_{2}$
be Gorenstein quasi-projective schemes, and let
$W_{i}:X_{i}\rightarrow \mathbb{A}^{1}$
be a regular function. We denote the projection by
$\unicode[STIX]{x1D70B}_{i}:X_{1}\times X_{2}\rightarrow X_{i}$
for each
$i=1,2$
.
In order to define integral functors, we need the following lemma.
Lemma 2.18 [Reference Efimov and PositselskiEP15, Corollaries 2.3.e and 2.4.a].
Let
$(X,W)$
be a LG model. Assume that the scheme
$X$
is a Gorenstein separated scheme of finite Krull dimension with an ample line bundle. Then the functor
$$\begin{eqnarray}\text{D}^{\text{co}}\text{LFr}(X,W)\rightarrow \text{D}^{\text{co}}\text{Qcoh}(X,W)\end{eqnarray}$$
induced by the embedding of dg functor
$\text{LFr}(X,W)\rightarrow \text{Qcoh}(X,W)$
is an equivalence.
Now we define integral functors. Let
$P\in \text{D}^{\text{co}}\text{Qcoh}(X_{1}\times X_{2},\unicode[STIX]{x1D70B}_{2}^{\ast }W-\unicode[STIX]{x1D70B}_{1}^{\ast }W)$
be an object. Since
$X_{1}$
and
$X_{2}$
are Gorenstein, so is
$X_{1}\times X_{2}$
(cf. [Reference Tousi and YassemiTY03]). By the above lemma, we have the derived tensor product
$$\begin{eqnarray}(-)\otimes ^{\mathbf{L}}P:\text{D}^{\text{co}}\text{Qcoh}(X_{1}\times X_{2},\unicode[STIX]{x1D70B}_{1}^{\ast }W)\rightarrow \text{D}^{\text{co}}\text{Qcoh}(X_{1}\times X_{2},\unicode[STIX]{x1D70B}_{2}^{\ast }W).\end{eqnarray}$$
Definition 2.19. For an object
$P\in \text{D}^{\text{co}}\text{Qcoh}(X_{1}\times X_{2},\unicode[STIX]{x1D70B}_{2}^{\ast }W-\unicode[STIX]{x1D70B}_{1}^{\ast }W)$
, we define the integral functor with respect to
$P$
, denoted by
$\unicode[STIX]{x1D6F7}_{P}$
, as the following functor
$$\begin{eqnarray}\mathbf{R}\unicode[STIX]{x1D70B}_{2\ast }(\unicode[STIX]{x1D70B}_{1}^{\ast }(-)\otimes ^{\mathbf{L}}P):\text{D}^{\text{co}}\text{Qcoh}(X_{1},W_{1})\rightarrow \text{D}^{\text{co}}\text{Qcoh}(X_{2},W_{2}).\end{eqnarray}$$
We call the object
$P$
the kernel of the integral functor
$\unicode[STIX]{x1D6F7}_{P}$
.
In the following, we recall that the composition of integral functors is also an integral functor. Let
$X_{3}$
be another Gorenstein quasi-projective scheme, and let
$W_{3}:X_{3}\rightarrow \mathbb{A}^{1}$
be a regular function. Consider the following diagram
where all morphisms are projections. For two objects
$$\begin{eqnarray}\displaystyle P & \in & \displaystyle \text{D}^{\text{co}}\text{Qcoh}(X_{1}\times X_{2},\unicode[STIX]{x1D70B}_{2}^{\ast }W_{2}-\unicode[STIX]{x1D70B}_{1}^{\ast }W_{1}),\nonumber\\ \displaystyle Q & \in & \displaystyle \text{D}^{\text{co}}\text{Qcoh}(X_{2}\times X_{3},p_{3}^{\ast }W_{3}-p_{2}^{\ast }W_{2}),\nonumber\end{eqnarray}$$
we define another object
$$\begin{eqnarray}P\star Q:=\mathbf{R}\unicode[STIX]{x1D70B}_{13\ast }(\unicode[STIX]{x1D70B}_{12}^{\ast }P\otimes ^{\mathbf{L}}\unicode[STIX]{x1D70B}_{23}^{\ast }Q)\in \text{D}^{\text{co}}\text{Qcoh}(X_{1}\times X_{3},q_{3}^{\ast }W_{3}-q_{1}^{\ast }W_{1}).\end{eqnarray}$$
Proposition 2.20. The notation is the same as above. The composition of integral functors
$$\begin{eqnarray}\text{D}^{\text{co}}\text{Qcoh}(X_{1},W_{1})\xrightarrow[{}]{\unicode[STIX]{x1D6F7}_{P}}\text{D}^{\text{co}}\text{Qcoh}(X_{2},W_{2})\xrightarrow[{}]{\unicode[STIX]{x1D6F7}_{Q}}\text{D}^{\text{co}}\text{Qcoh}(X_{3},W_{3})\end{eqnarray}$$
is isomorphic to the following integral functor
$$\begin{eqnarray}\text{D}^{\text{co}}\text{Qcoh}(X_{1},W_{1})\xrightarrow[{}]{\unicode[STIX]{x1D6F7}_{P\star Q}}\text{D}^{\text{co}}\text{Qcoh}(X_{3},W_{3}).\end{eqnarray}$$
Proof. The result follows from the projection formula and base change formula for derived factorization categories. ◻
2.4 Support properties of factorizations
Following [Reference Efimov and PositselskiEP15, §1.10], we consider set-theoretic supports of factorizations. In this section,
$X$
is a Noetherian scheme.
Definition 2.21. Let
$(X,\unicode[STIX]{x1D712},W)^{G}$
be a gauged LG model, and let
$Z\subset X$
be a
$G$
-invariant closed subset of
$X$
. We say that a factorization
$F\in \text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)$
is set-theoretically supported on
$Z$
if the supports
$\operatorname{Supp}(F_{i})$
of components of
$F$
are contained in
$Z$
.
Denote by
$$\begin{eqnarray}\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}\end{eqnarray}$$
the dg subcategory of
$\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)$
consisting of factorizations set-theoretically supported on
$Z$
.
$H^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z})$
is a full triangulated subcategory of
$H^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))$
. Denote by
$\text{Acycl}^{\text{co}}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z})$
the smallest thick subcategory of
$H^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z})$
which is closed under small direct sums and contains all totalizations of short exact sequences in
$Z^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z})$
. Set
$$\begin{eqnarray}{\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}:=H^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z})/\text{Acycl}^{\text{co}}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}).\end{eqnarray}$$
Similarly, denote by
$$\begin{eqnarray}\displaystyle \text{coh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}\quad \text{and}\quad \text{Inj}_{G}(X,\unicode[STIX]{x1D712},W)_{Z} & & \displaystyle \nonumber\end{eqnarray}$$
the dg subcategory of
$\text{coh}_{G}(X,\unicode[STIX]{x1D712},W)$
and
$\text{Inj}_{G}(X,\unicode[STIX]{x1D712},W)$
, respectively, consisting of factorizations set-theoretically supported on
$Z$
. Let
$\text{Acycl}(\text{coh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z})$
be the smallest thick subcategory of
$H^{0}(\text{coh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z})$
containing all totalizations of short exact sequences in
$Z^{0}(\text{coh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z})$
, and consider the Verdier quotient
$$\begin{eqnarray}\text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}:=H^{0}(\text{coh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z})/\text{Acycl}(\text{coh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}).\end{eqnarray}$$
Lemma 2.22. Let
$A\in \text{Acycl}^{\text{co}}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z})$
and
$I\in H^{0}(\text{Inj}_{G}(X,\unicode[STIX]{x1D712},W)_{Z})$
. Then we have
$$\begin{eqnarray}\operatorname{Hom}_{H^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z})}(A,I)=0.\end{eqnarray}$$
Proof. Since arbitrary direct sums of short exact sequences are exact and the totalization functor commutes with arbitrary direct sums, it is enough to show that for a short exact sequence
$A^{\bullet }:0\rightarrow A^{1}\rightarrow A^{2}\rightarrow A^{3}\rightarrow 0$
in
$Z^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z})$
, we have
$\operatorname{Hom}_{H^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z})}(\operatorname{Tot}(A^{\bullet }),I)=0$
. This follows from a similar argument as in the proof of [Reference Lunts and SchnürerLS16, Lemma 2.13].◻
By the above lemma, we see that every morphism from
$\text{Acycl}^{\text{co}}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z})$
to
$\text{Inj}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}$
factors through the zero object in
$H^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z})$
. Hence, by [Reference Lunts and SchnürerLS16, Proposition B.2], we have the following lemma.
Lemma 2.23. Let
$F\in H^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z})$
and
$I\in H^{0}(\text{Inj}_{G}(X,\unicode[STIX]{x1D712},W)_{Z})$
. Then the natural map
$$\begin{eqnarray}\operatorname{Hom}_{H^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z})}(F,I)\xrightarrow[{}]{{\sim}}\operatorname{Hom}_{{\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}}(F,I)\end{eqnarray}$$
is an isomorphism.
Furthermore, we have the following result.
Lemma 2.24. The natural functor
$$\begin{eqnarray}H^{0}(\text{Inj}_{G}(X,\unicode[STIX]{x1D712},W)_{Z})\rightarrow {\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}\end{eqnarray}$$
is an equivalence.
Proof. This follows from [Reference Ballard, Deliu, Favero, Isik and KatzarkovBDFIK16, Cororally 2.25]. ◻
The following two propositions are
$G$
-equivariant versions of results in [Reference Efimov and PositselskiEP15, §1.10].
Proposition 2.25 (Cf. [Reference Efimov and PositselskiEP15, Proposition 1.10]).
$(1)$
The natural functor
$$\begin{eqnarray}\text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}\rightarrow {\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}\end{eqnarray}$$
is fully faithful, and its image is a set of compact generators.
$(2)$
The natural functor
$$\begin{eqnarray}\unicode[STIX]{x1D704}_{Z}:{\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}\rightarrow {\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W)\end{eqnarray}$$
induced by the embedding of dg categories
$\unicode[STIX]{x1D704}_{Z}:\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}\rightarrow \text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)$
is fully faithful.
$(3)$
The functor
$$\begin{eqnarray}\unicode[STIX]{x1D704}_{Z}:\text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}\rightarrow \text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},W)\end{eqnarray}$$
induced by the embedding of dg categories
$\unicode[STIX]{x1D704}_{Z}:\text{coh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}\rightarrow \text{coh}_{G}(X,\unicode[STIX]{x1D712},W)$
is fully faithful.
Proof. (1) It is enough to prove that any morphism
$F\rightarrow A$
in
$H^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z})$
from
$F\in H^{0}(\text{coh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z})$
to
$A\in \text{Acycl}^{\text{co}}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z})$
factors through some object in
$\text{Acycl}(\text{coh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z})$
. This follows from a similar argument as in the proof of [Reference Lunts and SchnürerLS16, Lemma 2.15].
We show that
$\text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}$
generates
${\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}$
by using a similar discussion in the proof of [Reference PositselskiPos11, Theorem 3.11.2]. By Lemmas 2.23 and 2.24, it is enough to show that for an object
$I\in H^{0}(\text{Inj}_{G}(X,\unicode[STIX]{x1D712},W)_{Z})$
if
$$\begin{eqnarray}\operatorname{Hom}_{H^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z})}(F,I)=0\end{eqnarray}$$
for any
$F\in \text{coh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}$
, then
$\text{id}_{I}:I\rightarrow I$
is homotopic to zero. Consider the partially ordered set of pairs
$(M,h)$
, where
$M\subset I$
is a subfactorization of
$I$
and
$h:M\rightarrow I$
is a contracting homotopy of the embedding
$i:M{\hookrightarrow}I$
, i.e.
$d(h)=i$
. By Zorn’s lemma, the partially ordered set contains a maximal element. Hence, it suffices to show that given
$(M,h)$
with
$M\neq I$
, there exists
$(M^{\prime },h^{\prime })$
with
$M\subsetneq M^{\prime }$
and
$h^{\prime }|_{M}=h$
. Take a subfactorization
$M^{\prime }\subset I$
such that
$M\subsetneq M^{\prime }$
and
$M^{\prime }/M\in \text{coh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}$
. Since the components of
$I$
are injective sheaves, the morphism
$h:M\rightarrow I$
of degree
$-1$
can be extended to a morphism
$h^{\prime \prime }:M^{\prime }\rightarrow I$
. Denote by
$i:M{\hookrightarrow}I$
and
$i^{\prime }:M^{\prime }{\hookrightarrow}I$
the embeddings. Since the map
$i^{\prime }-d(h^{\prime \prime })$
is a closed degree-zero morphism and vanishes on
$M$
, it induces a closed degree-zero morphism
$g:M^{\prime }/M\rightarrow I$
. By the assumption,
$g$
is homotopic to zero, i.e. there exists a homotopy
$c:M^{\prime }/M\rightarrow I$
such that
$d(c)=g$
. Then
$h^{\prime }=h^{\prime \prime }+c\circ p:M^{\prime }\rightarrow I$
is a contracting homotopy for
$i^{\prime }$
extending
$h$
, where
$p:M^{\prime }\rightarrow M^{\prime }/M$
is the natural projection.
The compactness of objects in
$\text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}$
follows from Lemmas 2.23 and 2.24. Parts (2) and (3) follows from Lemma 2.24 and part (1).◻
Proposition 2.26 (Cf. [Reference Efimov and PositselskiEP15, Theorem 1.10]).
Let
$U:=X\setminus Z$
be the complement of
$Z\subset X$
, and let
$j:U\rightarrow X$
be the open immersion.
-
(1) The restriction
is the Verdier localization by the thick subcategory
$$\begin{eqnarray}j^{\ast }:{\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W)\rightarrow {\text{D}^{\text{co}}\text{Qcoh}}_{G}(U,\unicode[STIX]{x1D712},W|_{U})\end{eqnarray}$$
${\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}$
.
-
(2) The restriction
is the Verdier localization by the triangulated subcategory
$$\begin{eqnarray}j^{\ast }:\text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},W)\rightarrow \text{Dcoh}_{G}(U,\unicode[STIX]{x1D712},W|_{U})\end{eqnarray}$$
$\text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}$
. The kernel of
$j^{\ast }$
is the thick envelope of
$\text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}$
in
$\text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},W)$
.
Proof. We can prove this by a standard discussion as in the proof of [Reference Efimov and PositselskiEP15, Theorem 1.10].
(1) Since
$j^{\ast }$
has a right adjoint
$\mathbf{R}j_{\ast }:{\text{D}^{\text{co}}\text{Qcoh}}_{G}(U,\unicode[STIX]{x1D712},W|_{U})\rightarrow {\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W)$
which is fully faithful, we see that
$j^{\ast }$
is the Verdier (Bousfield) localization by its kernel which is generated by cones of adjunctions
$F\rightarrow \mathbf{R}j_{\ast }j^{\ast }F$
for any
$F\in {\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W)$
.
We show that
$\operatorname{Ker}(j^{\ast })={\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}$
. Since the inclusion
${\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}\subset \operatorname{Ker}(j^{\ast })$
is trivial, it is enough to show that the cone of the adjunction
$F\rightarrow \mathbf{R}j_{\ast }j^{\ast }F$
, for any
$F\in {\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W)$
, is contained in
${\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}$
. By Lemma 2.12, we may take
$F$
as an factorization whose components are injective quasi-coherent sheaves. Then the adjunction comes from a closed morphism
$F\rightarrow j_{\ast }j^{\ast }F$
in
$Z^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))$
. Since its kernel and cokernel are objects in
$\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}$
, so is the cone of the adjunction by an equivariant version of [Reference Lunts and SchnürerLS16, Lemma 2.7.c].
(2) By Proposition 2.25(1) and [Reference NeemanNee92], we have a fully faithful functor
$$\begin{eqnarray}\overline{\text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},W)}/\overline{\text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}}\longrightarrow \overline{\text{Dcoh}_{G}(U,\unicode[STIX]{x1D712},W|_{U})},\end{eqnarray}$$
where
$\overline{(-)}$
denotes the idempotent completion of
$(-)$
. Since every morphism
$D\rightarrow E$
from
$D\in \text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},W)$
to
$E\in \overline{\text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}}$
factors through an object in
$\text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}$
, the natural functor
$$\begin{eqnarray}\text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},W)/\text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}\rightarrow \overline{\text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},W)}/\overline{\text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}}\end{eqnarray}$$
is fully faithful. Hence, we see that the natural functor
$$\begin{eqnarray}\text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},W)/\text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},W)_{Z}\rightarrow \text{Dcoh}_{G}(U,\unicode[STIX]{x1D712},W|_{U})\end{eqnarray}$$
is also fully faithful. This functor is essentially surjective since for every
$G$
-equivariant coherent
$F\in \text{coh}_{G}U$
there exists a
$G$
-equivariant coherent sheaf
$\overline{F}\in \text{coh}_{G}X$
such that
$j^{\ast }\overline{F}\cong F$
and the coherent sheaves generate
$\text{Dcoh}_{G}(U,\unicode[STIX]{x1D712},W|_{U})$
by [Reference Ballard, Deliu, Favero, Isik and KatzarkovBDFIK16, Corollary 2.29].◻
2.5 Koszul factorizations
Let
$(X,\unicode[STIX]{x1D712},W)^{G}$
be a gauged LG model such that
$X$
is a smooth variety. Let
${\mathcal{E}}$
be a
$G$
-equivariant locally free sheaf on
$X$
of rank
$r$
, and let
$$\begin{eqnarray}s:{\mathcal{E}}\rightarrow {\mathcal{O}}_{X}\quad \text{and}\quad t:{\mathcal{O}}_{X}\rightarrow {\mathcal{E}}(\unicode[STIX]{x1D712})\end{eqnarray}$$
be morphisms in
$\text{coh}_{G}X$
such that
$t\circ s=W\cdot \text{id}_{{\mathcal{E}}}$
and
$s(\unicode[STIX]{x1D712})\circ t=W$
. Let
$Z_{s}\subset X$
be the zero scheme of the section
$s\in \unicode[STIX]{x1D6E4}(X,{\mathcal{E}}^{\vee })^{G}$
. We say that
$s$
is regular if the codimension of
$Z_{s}$
in
$X$
equals the rank
$r$
.
Definition 2.27. We define an object
$K(s,t)\in \text{lfr}_{G}(X,\unicode[STIX]{x1D712},W)$
as
$$\begin{eqnarray}K(s,t):=(K_{1}\xrightarrow[{}]{k_{1}}K_{0}\xrightarrow[{}]{k_{0}}K_{1}(\unicode[STIX]{x1D712}))\end{eqnarray}$$
where
$$\begin{eqnarray}K_{1}:=\bigoplus _{n=0}^{\lceil r/2\rceil -1}\biggl(\bigwedge ^{2n+1}{\mathcal{E}}\biggr)(\unicode[STIX]{x1D712}^{n}),\quad K_{0}:=\bigoplus _{n=0}^{\lfloor r/2\rfloor }\biggl(\bigwedge ^{2n}{\mathcal{E}}\biggr)(\unicode[STIX]{x1D712}^{n})\end{eqnarray}$$
and
$$\begin{eqnarray}k_{i}:=t\wedge (-)\oplus s\vee (-).\end{eqnarray}$$
The following property will be necessary in § 4.1.
Lemma 2.28 [Reference Ballard, Favero and KatzarkovBFK14, Lemma 3.21 and Proposition 3.20].
(1) have a natural isomorphism
$$\begin{eqnarray}K(s,t)^{\vee }\cong K(t^{\vee },s^{\vee }).\end{eqnarray}$$
(2) If
$s$
is regular, we have natural isomorphisms in
$\text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},W)$
,
$$\begin{eqnarray}{\mathcal{O}}_{Z_{s}}\cong K(s,t)\quad \text{and}\quad {\mathcal{O}}_{Z_{s}}\otimes \bigwedge ^{r}{\mathcal{E}}^{\vee }(\unicode[STIX]{x1D712}^{-1})[-r]\cong K(s,t)^{\vee },\end{eqnarray}$$
where
${\mathcal{O}}_{Z_{s}}:=(0\rightarrow {\mathcal{O}}_{Z_{s}}\rightarrow 0)$
and
$\bigwedge ^{r}{\mathcal{E}}^{\vee }(\unicode[STIX]{x1D712}^{-1})[-r]$
is a complex in
$\text{coh}_{G}X$
.
2.6 Restriction and induction functors
We construct restriction and induction functors. Let
$G$
be an affine algebraic group acting on a scheme
$X$
. Let
$G\times ^{l}X$
and
$G\times ^{d}X$
be schemes
$G\times X$
with different
$G$
-actions which are defined as follows:
and
Let
$\unicode[STIX]{x1D704}:X\rightarrow G\times X$
be a morphism defined by
$$\begin{eqnarray}X\ni x\longmapsto (1_{G},x)\in G\times X.\end{eqnarray}$$
We define an exact functor
$\unicode[STIX]{x1D704}^{\ast }:\text{Qcoh}_{G}(G\times ^{l}X)\rightarrow \text{Qcoh}X$
as
Since
$G\times ^{l}X$
is a principal
$G$
-bundle over
$X$
, the above functor
$\unicode[STIX]{x1D704}^{\ast }$
is an equivalence.
The following morphisms
and
are
$G$
-equivariant. The
$G$
-action
$\unicode[STIX]{x1D70E}:G\times X\rightarrow X$
on
$X$
is the composition
$\unicode[STIX]{x1D70B}\circ \unicode[STIX]{x1D711}$
. Since the morphism
$\unicode[STIX]{x1D711}$
is an isomorphism, the following functors are equivalences:
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}^{\ast }:\text{Qcoh}_{G}(G\times ^{d}X) & \xrightarrow[{}]{{\sim}} & \displaystyle \text{Qcoh}_{G}(G\times ^{l}X),\nonumber\\ \displaystyle \unicode[STIX]{x1D711}_{\ast }:\text{Qcoh}_{G}(G\times ^{l}X) & \xrightarrow[{}]{{\sim}} & \displaystyle \text{Qcoh}_{G}(G\times ^{d}X).\nonumber\end{eqnarray}$$
Since the morphism
$\unicode[STIX]{x1D70B}$
is flat and affine, the following functors are exact:
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70B}^{\ast }:\text{Qcoh}_{G}(X) & \rightarrow & \displaystyle \text{Qcoh}_{G}(G\times ^{d}X),\nonumber\\ \displaystyle \unicode[STIX]{x1D70B}_{\ast }:\text{Qcoh}_{G}(G\times ^{d}X) & \rightarrow & \displaystyle \text{Qcoh}_{G}(X).\nonumber\end{eqnarray}$$
Definition 2.29. We define the restriction functor
$\text{Res}_{G}:\text{Qcoh}_{G}X\rightarrow \text{Qcoh}X$
and the induction functor
$\text{Ind}_{G}:\text{Qcoh}X\rightarrow \text{Qcoh}_{G}X$
as
$$\begin{eqnarray}\text{Res}_{G}:=\unicode[STIX]{x1D704}^{\ast }\circ \unicode[STIX]{x1D70E}^{\ast }\quad \text{and}\quad \text{Ind}_{G}:=\unicode[STIX]{x1D70E}_{\ast }\circ (\unicode[STIX]{x1D704}^{\ast })^{-1}.\end{eqnarray}$$
Remark 2.30. (1) The restriction functor
$\text{Res}_{G}:\text{Qcoh}_{G}X\rightarrow \text{Qcoh}X$
is isomorphic to the forgetful functor, i.e.
$\text{Res}_{G}({\mathcal{F}},\unicode[STIX]{x1D703})\cong {\mathcal{F}}$
.
(2) Although the functor
$\text{Res}_{G}$
sends equivariant coherent sheaves to coherent sheaves for arbitrary group
$G$
, the functor
$\text{Ind}_{G}$
does not preserve coherent sheaves if
$G$
is not a finite group.
Since
$\unicode[STIX]{x1D704}^{\ast }$
is an equivalence, the adjoint pair
$\unicode[STIX]{x1D70E}^{\ast }\dashv \unicode[STIX]{x1D70E}_{\ast }$
induces the adjoint pair
$$\begin{eqnarray}\text{Res}_{G}\dashv \text{Ind}_{G}.\end{eqnarray}$$
For a
$\unicode[STIX]{x1D712}$
-semi-invariant regular function
$W:X\rightarrow \mathbb{A}^{1}$
, the restriction functor
$\text{Res}_{G}:\text{Qcoh}_{G}X\rightarrow \text{Qcoh}X$
and the induction functor
$\text{Ind}_{G}:\text{Qcoh}X\rightarrow \text{Qcoh}_{G}X$
induce dg functors between factorization categories, which we denote by the same notation:
$$\begin{eqnarray}\displaystyle & \displaystyle \text{Res}_{G}:\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)\rightarrow \text{Qcoh}(X,W), & \displaystyle \nonumber\\ \displaystyle & \displaystyle \text{Ind}_{G}:\text{Qcoh}(X,W)\rightarrow \text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W). & \displaystyle \nonumber\end{eqnarray}$$
These functors are also adjoint to each other:
$$\begin{eqnarray}\text{Res}_{G}\dashv \text{Ind}_{G}.\end{eqnarray}$$
Since the restriction and the induction functors are exact, we obtain the following induced functors between bounded derived categories
$$\begin{eqnarray}\displaystyle & \displaystyle \text{Res}_{G}:\text{D}^{\text{b}}(\text{Qcoh}_{G}X)\rightarrow \text{D}^{\text{b}}(\text{Qcoh}X), & \displaystyle \nonumber\\ \displaystyle & \displaystyle \text{Ind}_{G}:\text{D}^{\text{b}}(\text{Qcoh}X)\rightarrow \text{D}^{\text{b}}(\text{Qcoh}_{G}X) & \displaystyle \nonumber\end{eqnarray}$$
and the functors between coderived factorization categories
$$\begin{eqnarray}\displaystyle & \displaystyle \text{Res}_{G}:{\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W)\rightarrow \text{D}^{\text{co}}\text{Qcoh}(X,W), & \displaystyle \nonumber\\ \displaystyle & \displaystyle \text{Ind}_{G}:\text{D}^{\text{co}}\text{Qcoh}(X,W)\rightarrow {\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W). & \displaystyle \nonumber\end{eqnarray}$$
The adjunction of the restriction and the induction functors between
$\text{Qcoh}_{G}X$
and
$\text{Qcoh}X$
induces the adjunction of the above induced functors between triangulated categories:
$$\begin{eqnarray}\text{Res}_{G}\dashv \text{Ind}_{G}.\end{eqnarray}$$
We recall the definition of (linearly) reductiveness of algebraic groups.
Definition 2.31. Let
$H$
be an affine algebraic group over a field
$K$
:
-
(1)
$H$
is called reductive if the radical of
$H$
is a torus; -
(2)
$H$
is called linearly reductive if every rational representations of
$H$
over
$K$
is completely reducible.
The following says that the above two properties are equivalent if the characteristic of the base field is zero.
Proposition 2.32 [Reference Mumford, Forgarty and KirwanMFK94, Appendix A].
Let
$H$
be an affine algebraic group over a field
$K$
of characteristic zero. Then
$H$
is reductive if and only if it is linearly reductive.
Lemma 2.33. Assume that
$G$
is linearly reductive.
-
(1) The adjunction morphism
is a split mono, i.e. there exists a functor morphism
$$\begin{eqnarray}\unicode[STIX]{x1D700}:\text{id}_{\text{Qcoh}_{G}X}\rightarrow \text{Ind}_{G}\circ \text{Res}_{G}\end{eqnarray}$$
$\unicode[STIX]{x1D702}:\text{Ind}_{G}\circ \text{Res}_{G}\rightarrow \text{id}_{\text{Qcoh}_{G}X}$
such that
$\unicode[STIX]{x1D702}\circ \unicode[STIX]{x1D700}=\text{id}$
. The adjunction morphism is also a split mono.
$$\begin{eqnarray}\text{id}_{\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)}\rightarrow \text{Ind}_{G}\circ \text{Res}_{G}\end{eqnarray}$$
-
(2) The restriction functors
are faithful.
$$\begin{eqnarray}\displaystyle & \displaystyle \text{Res}_{G}:\text{D}^{\text{b}}(\text{Qcoh}_{G}X)\rightarrow \text{D}^{\text{b}}(\text{Qcoh}X), & \displaystyle \nonumber\\ \displaystyle & \displaystyle \text{Res}_{G}:{\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W)\rightarrow \text{D}^{\text{co}}\text{Qcoh}(X,W) & \displaystyle \nonumber\end{eqnarray}$$
Proof. (1) This follows from the argument in the proof of [Reference HiranoHir17, Lemma 4.56].
(2) We will prove that the upper functor
$\text{Res}_{G}:\text{D}^{\text{b}}(\text{Qcoh}_{G}X)\rightarrow \text{D}^{\text{b}}(\text{Qcoh}X)$
is faithful; the proof of the faithfulness of the lower functor is similar. The functor morphism
$\unicode[STIX]{x1D702}:\text{Ind}_{G}\circ \text{Res}_{G}\rightarrow \text{id}_{\text{Qcoh}_{G}X}$
constructed in part (1) naturally induces the functor morphism
$\overline{\unicode[STIX]{x1D702}}:\text{Ind}_{G}\circ \text{Res}_{G}\rightarrow \text{id}_{\text{D}^{\text{b}}(\text{Qcoh}_{G}X)}$
such that the composition with the adjunction morphism
$$\begin{eqnarray}\text{id}_{\text{D}^{\text{b}}(\text{Qcoh}_{G}X)}\rightarrow \text{Ind}_{G}\circ \text{Res}_{G}\xrightarrow[{}]{\overline{\unicode[STIX]{x1D702}}}\text{id}_{\text{D}^{\text{b}}(\text{Qcoh}_{G}X)}\end{eqnarray}$$
is the identity. Hence, any morphism
$f$
in
$\text{D}^{\text{b}}(\text{Qcoh}_{G}X)$
factors through
$\text{Ind}_{G}\circ \text{Res}_{G}(f)$
, and so
$f=0$
if
$\text{Res}(f)=0$
.◻
3 Relative singularity categories
Relative singularity categories are introduced in [Reference Efimov and PositselskiEP15], and it is shown that derived factorization categories (with some conditions on regular functions) are equivalent to relative singularity categories. In this section, we recall the definition and properties of relative singularity categories.
3.1 Triangulated categories of relative singularities
Let
$X$
be a quasi-projective scheme, and let
$G$
be an affine algebraic group acting on
$X$
. Throughout this section, we assume that
$X$
has a
$G$
-equivariant ample line bundle. If
$X$
is normal, this condition is satisfied by [Reference ThomasonTho87, Lemma 2.10]. The equivariant triangulated category of singularities
$\text{D}_{G}^{\text{sg}}(X)$
of
$X$
is defined as the Verdier quotient of
$\text{D}^{\text{b}}(\text{coh}_{G}X)$
by the thick subcategory
$\text{Perf}_{G}(X)$
of equivariant perfect complexes. Following [Reference OrlovOrl04], we consider a larger category
$\text{D}_{G}^{\text{cosg}}(X)$
defined as the Verdier quotient of
$\text{D}^{\text{b}}(\text{Qcoh}_{G}X)$
by the thick subcategory
$\text{Lfr}_{G}(X)$
of complexes which is quasi-isomorphic to bounded complexes of equivariant locally free sheaves (not necessarily of finite ranks). If
$G$
is trivial, we denote the singularity categories by
$\text{D}^{\text{cosg}}(X)$
or
$\text{D}^{\text{sg}}(X)$
.
We recall relative singularity categories following [Reference Efimov and PositselskiEP15]. Let
$Z\subset X$
be a
$G$
-invariant closed subscheme of
$X$
such that
${\mathcal{O}}_{Z}$
has finite
$G$
-flat dimension as an
${\mathcal{O}}_{X}$
-module, i.e. the
$G$
-equivariant sheaf
${\mathcal{O}}_{Z}\in \text{coh}_{G}(X)$
has a finite resolution
$F^{\bullet }\rightarrow {\mathcal{O}}_{Z}$
of
$G$
-equivariant flat sheaves on
$X$
. Under the assumption, we have the derived inverse image
$\mathbf{L}i^{\ast }:\text{D}^{\text{b}}(\text{Qcoh}_{G}X)\rightarrow \text{D}^{\text{b}}(\text{Qcoh}_{G}Z)$
between bounded derived categories for the closed immersion
$i:Z\rightarrow X$
. This functor preserves complexes of coherent sheaves:
$\mathbf{L}i^{\ast }:\text{D}^{\text{b}}(\text{coh}X)\rightarrow \text{D}^{\text{b}}(\text{coh}Z)$
.
Definition 3.1 [Reference Efimov and PositselskiEP15, §2.1].
We define the following Verdier quotients
$$\begin{eqnarray}\displaystyle & \displaystyle \text{D}_{G}^{\text{cosg}}(Z/X):=\text{D}^{\text{b}}(\text{Qcoh}_{G}Z)/\langle \operatorname{Im}(\mathbf{L}i^{\ast }:\text{D}^{\text{b}}(\text{Qcoh}_{G}X)\rightarrow \text{D}^{\text{b}}(\text{Qcoh}_{G}Z))\rangle ^{\oplus }, & \displaystyle \nonumber\\ \displaystyle & \displaystyle \text{D}_{G}^{\text{sg}}(Z/X):=\text{D}^{\text{b}}(\text{coh}_{G}Z)/\langle \operatorname{Im}(\mathbf{L}i^{\ast }:\text{D}^{\text{b}}(\text{coh}_{G}X)\rightarrow \text{D}^{\text{b}}(\text{coh}_{G}Z))\rangle , & \displaystyle \nonumber\end{eqnarray}$$
where
$\langle -\rangle$
(respectively
$\langle -\rangle ^{\oplus }$
) denotes the smallest thick subcategory containing objects in
$(-)$
(respectively and closed under infinite direct sums which exist in
$\text{D}^{\text{b}}(\text{Qcoh}_{G}Z)$
). The quotient category
$\text{D}_{G}^{\text{sg}}(Z/X)$
is called the equivariant triangulated category of singularities of
$Z$
relative to
$X$
. If
$G$
is trivial, we denote the categories defined above by
$\text{D}^{\text{cosg}}(Z/X)$
or
$\text{D}^{\text{sg}}(Z/X)$
.
Proposition 3.2. Assume that
$G$
is reductive. We have natural Verdier localizations by thick subcategories
$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70B}^{\text{co}}:\text{D}_{G}^{\text{cosg}}(Z)\rightarrow \text{D}_{G}^{\text{cosg}}(Z/X), & \displaystyle \nonumber\\ \displaystyle & \displaystyle \unicode[STIX]{x1D70B}:\text{D}_{G}^{\text{sg}}(Z)\rightarrow \text{D}_{G}^{\text{sg}}(Z/X). & \displaystyle \nonumber\end{eqnarray}$$
Proof. It is enough to show that
$\text{Lfr}_{G}(Z)\subset \langle \operatorname{Im}(\mathbf{L}i^{\ast }:\text{D}^{\text{b}}(\text{Qcoh}_{G}X)\rightarrow \text{D}^{\text{b}}(\text{Qcoh}_{G}Z))\rangle ^{\oplus }$
and
$\text{Perf}_{G}(Z)\subset \langle \operatorname{Im}(\mathbf{L}i^{\ast }:\text{D}^{\text{b}}(\text{coh}_{G}X)\rightarrow \text{D}^{\text{b}}(\text{coh}_{G}Z))\rangle$
. These inclusions follow from the assumption that
$X$
has a
$G$
-equivariant ample line bundle
$L$
. The proofs of the inclusions are similar, and we prove the only former inclusion. It is enough to show that any
$G$
-equivariant locally free sheaf
$E$
on
$Z$
is a direct summand of a bounded complex whose terms are direct sums of invertible sheaves of the form
$i^{\ast }L^{\otimes n}$
. By [Reference ThomasonTho87, Lemma 1.4], there is a bounded above locally free resolution
$E^{\bullet }\xrightarrow[{}]{{\sim}}E$
whose terms are as above. For any
$n>0$
, we have the following triangle in
$\text{D}^{\text{b}}(\text{Qcoh}_{G}Z)$
$$\begin{eqnarray}\unicode[STIX]{x1D70E}^{{\geqslant}-n}E^{\bullet }\rightarrow E\rightarrow H^{-n}(\unicode[STIX]{x1D70E}^{{\geqslant}-n}E^{\bullet })[n+1]\rightarrow \unicode[STIX]{x1D70E}^{{\geqslant}-n}E^{\bullet }[1],\end{eqnarray}$$
where
$\unicode[STIX]{x1D70E}^{{\geqslant}-n}$
denotes the brutal truncation. If we choose a sufficiently large
$n\gg 0$
, we have
$$\begin{eqnarray}\operatorname{Hom}_{\text{D}^{\text{b}}(\text{Qcoh}_{G}Z)}(E,H^{-n}(\unicode[STIX]{x1D70E}^{{\geqslant}-n}E^{\bullet })[n+1])=0\end{eqnarray}$$
by [Reference OrlovOrl04, Lemma 1.12], since the restriction functor
$\text{Res}_{G}:\text{D}^{\text{b}}(\text{Qcoh}_{G}Z)\rightarrow \text{D}^{\text{b}}(\text{Qcoh}Z)$
is faithful by Lemma 2.33(2). Hence, the above triangle splits, and
$E$
is a direct summand of the complex
$\unicode[STIX]{x1D70E}^{{\geqslant}-n}E^{\bullet }$
.◻
Remark 3.3. Note that, if
$X$
is regular, then the thick subcategory
$\langle \operatorname{Im}(\mathbf{L}i^{\ast })\rangle \subset \text{D}^{\text{b}}(\text{coh}_{G}Z)$
coincides with its thick subcategory
$\text{Perf}_{G}(Z)$
of equivariant perfect complexes of
$Z$
. Hence, the quotient category
$\text{D}_{G}^{\text{sg}}(Z/X)$
is the same as
$\text{D}_{G}^{\text{sg}}(Z)$
. Similarly,
$\text{D}_{G}^{\text{cosg}}(Z/X)$
is also same as
$\text{D}_{G}^{\text{cosg}}(Z)$
when
$X$
is regular.
The exact functors
$\text{Res}_{G}:\text{Qcoh}_{G}Z\rightarrow \text{Qcoh}Z$
and
$\text{Ind}_{G}:\text{Qcoh}Z\rightarrow \text{Qcoh}_{G}Z$
, defined in Definition 2.29, induce functors between relative singularity categories
$$\begin{eqnarray}\displaystyle \text{Res}_{G}:\text{D}_{G}^{\text{cosg}}(Z/X) & \rightarrow & \displaystyle \text{D}^{\text{cosg}}(Z/X),\nonumber\\ \displaystyle \text{Ind}_{G}:\text{D}^{\text{cosg}}(Z/X) & \rightarrow & \displaystyle \text{D}_{G}^{\text{cosg}}(Z/X).\nonumber\end{eqnarray}$$
We need the following lemma in the proof of the main result.
Lemma 3.4. Assume that
$G$
is reductive. Then the restriction functor
$$\begin{eqnarray}\text{Res}_{G}:\text{D}_{G}^{\text{cosg}}(Z/X)\rightarrow \text{D}^{\text{cosg}}(Z/X)\end{eqnarray}$$
is faithful.
Proof. This follows from a similar argument as in the proof of Lemma 2.33(2). ◻
3.2 Direct images and inverse images in relative singularity categories
Let
$X_{1}$
and
$X_{2}$
be quasi-projective schemes with actions of an affine algebraic group
$G$
. Assume that
$X_{1}$
and
$X_{2}$
have
$G$
-equivariant ample line bundles. Let
$\tilde{f}:X_{2}\rightarrow X_{1}$
be a
$G$
-equivariant morphism. Let
$Z_{1}$
be a
$G$
-invariant closed subscheme of
$X_{1}$
such that
${\mathcal{O}}_{Z_{1}}$
has finite
$G$
-flat dimension as a
${\mathcal{O}}_{X_{1}}$
-module, and let
$Z_{2}$
be the fiber product
$Z_{1}\times _{X_{1}}X_{2}$
. Denote by
$f$
the restriction
$\tilde{f}|_{Z_{2}}:Z_{2}\rightarrow Z_{1}$
of
$\tilde{f}$
to
$Z_{2}$
. We assume that the cartesian square
is exact in the sense of [Reference KuznetsovKuz06]. Then,
${\mathcal{O}}_{Z_{2}}$
also has finite
$G$
-flat dimension as a
${\mathcal{O}}_{X_{2}}$
-module. Furthermore, we assume that
$\tilde{f}$
has finite G-flat dimension, i.e. the derived inverse image
$\mathbf{L}\tilde{f}^{\ast }:\text{D}^{-}(\text{Qcoh}_{G}X_{1})\rightarrow \text{D}^{-}(\text{Qcoh}_{G}X_{2})$
maps
$\text{D}^{\text{b}}(\text{Qcoh}_{G}X_{1})$
to
$\text{D}^{\text{b}}(\text{Qcoh}_{G}X_{2})$
. Then
$f$
also has finite
$G$
-flat dimension.
In the above setting, the derived inverse image
$\mathbf{L}f^{\ast }:\text{D}^{\text{b}}(\text{Qcoh}_{G}Z_{1})\rightarrow \text{D}^{\text{b}}(\text{Qcoh}_{G}Z_{2})$
induces exact functors
$$\begin{eqnarray}\displaystyle f^{\circ }:\text{D}_{G}^{\text{cosg}}(Z_{1}/X_{1}) & \rightarrow & \displaystyle \text{D}_{G}^{\text{cosg}}(Z_{2}/X_{2}),\nonumber\\ \displaystyle f^{\circ }:\text{D}_{G}^{\text{sg}}(Z_{1}/X_{1}) & \rightarrow & \displaystyle \text{D}_{G}^{\text{sg}}(Z_{2}/X_{2}),\nonumber\end{eqnarray}$$
and the derived direct image
$\mathbf{R}f_{\ast }:\text{D}^{\text{b}}(\text{Qcoh}_{G}Z_{2})\rightarrow \text{D}^{\text{b}}(\text{Qcoh}_{G}Z_{1})$
induces a right adjoint functor of
$f^{\circ }:\text{D}_{G}^{\text{cosg}}(Z_{1}/X_{1})\rightarrow \text{D}_{G}^{\text{cosg}}(Z_{2}/X_{2})$
$$\begin{eqnarray}f_{\circ }:\text{D}_{G}^{\text{cosg}}(Z_{2}/X_{2})\rightarrow \text{D}_{G}^{\text{cosg}}(Z_{1}/X_{1}).\end{eqnarray}$$
If
$f$
is a proper morphism, the direct image
$\mathbf{R}f_{\ast }:\text{D}^{\text{b}}(\text{coh}_{G}Z_{2})\rightarrow \text{D}^{\text{b}}(\text{coh}_{G}Z_{1})$
between bounded complexes of coherent sheaves induces a right adjoint functor
$$\begin{eqnarray}f_{\circ }:\text{D}_{G}^{\text{sg}}(Z_{2}/X_{2})\rightarrow \text{D}_{G}^{\text{sg}}(Z_{1}/X_{1})\end{eqnarray}$$
of
$f^{\circ }:\text{D}_{G}^{\text{sg}}(Z_{1}/X_{1})\rightarrow \text{D}_{G}^{\text{sg}}(Z_{2}/X_{2})$
.
Let
$X$
be a quasi-projective scheme with an action of an affine algebraic group
$G$
, and let
$U\subset X$
be a
$G$
-invariant open subscheme. Let
$Z\subset X$
be a
$G$
-invariant closed subscheme such that
${\mathcal{O}}_{Z}$
has finite
$G$
-flat dimension, and consider the fiber product
$U_{Z}:=Z\times _{X}U$
. Denote by
$\tilde{l}:U\rightarrow X$
and
$l:U_{Z}\rightarrow Z$
the open immersions. Then we have the following exact cartesian square.
Lemma 3.5. The inverse image
$$\begin{eqnarray}l^{\circ }:\text{D}_{G}^{\text{cosg}}(Z/X)\rightarrow \text{D}_{G}^{\text{cosg}}(U_{Z}/U)\end{eqnarray}$$
is a Verdier localization by the kernel of
$l^{\circ }$
.
Proof. The direct image
$\mathbf{R}l_{\ast }:\text{D}^{\text{b}}(\text{Qcoh}U_{Z})\rightarrow \text{D}^{\text{b}}(\text{Qcoh}Z)$
is fully faithful and right adjoint to the inverse image
$l^{\ast }:\text{D}^{\text{b}}(\text{Qcoh}Z)\rightarrow \text{D}^{\text{b}}(\text{Qcoh}U_{Z})$
. By [Reference OrlovOrl06, Lemma 1.1], the direct image functor
$l_{\circ }:\text{D}_{G}^{\text{cosg}}(Z/X)\rightarrow \text{D}_{G}^{\text{cosg}}(U_{Z}/U)$
is fully faithful. Hence,
$l^{\circ }$
admits a right adjoint functor which is fully faithful, and this implies the result.◻
3.3 Relative singularity categories and derived factorization categories
In this section,
$X$
and
$G$
are the same as in § 3.1, and we assume that
$G$
is reductive. Let
$\unicode[STIX]{x1D712}:G\rightarrow \mathbb{G}_{m}$
be a character of
$G$
, and let
$W:X\rightarrow \mathbb{A}^{1}$
be a
$\unicode[STIX]{x1D712}$
-semi-invariant regular function. In this section, we assume that the corresponding
$G$
-invariant section
$W:{\mathcal{O}}_{X}\rightarrow {\mathcal{O}}(\unicode[STIX]{x1D712})$
is injective. For example, if
$W$
is flat, this condition is satisfied. Denote by
$X_{0}$
the fiber of
$W$
over
$0\in \mathbb{A}^{1}$
, and let
$i:X_{0}\rightarrow X$
be the closed immersion. We have an exact functor
$\unicode[STIX]{x1D70F}:\text{Qcoh}_{G}X_{0}\rightarrow Z^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))$
defined by
$$\begin{eqnarray}\unicode[STIX]{x1D70F}(F):=(0\rightarrow i_{\ast }(F)\rightarrow 0).\end{eqnarray}$$
We define a natural functor
$$\begin{eqnarray}\unicode[STIX]{x1D6F6}:\text{D}^{\text{b}}(\text{Qcoh}_{G}X_{0})\rightarrow {\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W)\end{eqnarray}$$
as the composition of functors
$$\begin{eqnarray}\text{D}^{\text{b}}(\text{Qcoh}_{G}X_{0})\xrightarrow[{}]{\unicode[STIX]{x1D70F}}\text{D}^{\text{b}}(Z^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)))\xrightarrow[{}]{\operatorname{Tot}}{\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W).\end{eqnarray}$$
The functor
$\unicode[STIX]{x1D6F6}$
annihilates the thick category
$\langle \operatorname{Im}(\mathbf{L}i^{\ast })\rangle ^{\oplus }\subset \text{D}^{\text{b}}(\text{Qcoh}_{G}X_{0})$
, since its non-equivariant functor
$\unicode[STIX]{x1D6F6}:\text{D}^{\text{b}}(\text{Qcoh}X_{0})\rightarrow \text{D}^{\text{co}}\text{Qcoh}(X,W)$
annihilates
$\text{Res}_{G}(\langle \operatorname{Im}(\mathbf{L}i^{\ast })\rangle ^{\oplus })$
(see [Reference Efimov and PositselskiEP15, proofs of Theorems 2.7 and 2.8]) and the restriction functor
$\text{Res}_{G}:{\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W)\rightarrow \text{D}^{\text{co}}\text{Qcoh}(X,W)$
is faithful. Hence, it induces an exact functor
$$\begin{eqnarray}\unicode[STIX]{x1D6F6}:\text{D}_{G}^{\text{cosg}}(X_{0}/X)\rightarrow {\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W).\end{eqnarray}$$
Similarly, we have the following exact functor
$$\begin{eqnarray}\unicode[STIX]{x1D6F6}:\text{D}_{G}^{\text{sg}}(X_{0}/X)\rightarrow \text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},W),\end{eqnarray}$$
and the following diagram is commutative,
where the vertical arrows are natural inclusion functors (which are fully faithful).
Theorem 3.6 (Cf. [Reference Efimov and PositselskiEP15, Theorems 2.7 and 2.8]).
The functors
$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F6}:\text{D}_{G}^{\text{cosg}}(X_{0}/X)\rightarrow {\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W), & \displaystyle \nonumber\\ \displaystyle & \displaystyle \unicode[STIX]{x1D6F6}:\text{D}_{G}^{\text{sg}}(X_{0}/X)\rightarrow \text{Dcoh}_{G}(X,\unicode[STIX]{x1D712},W) & \displaystyle \nonumber\end{eqnarray}$$
are equivalences.
In order to prove the above theorem, we need to construct the quasi-inverse of
$\unicode[STIX]{x1D6F6}$
. We say that a
$G$
-equivariant quasi-coherent sheaf
$F\in \text{Qcoh}_{G}X$
is
$\mathbf{W}$
-flat, if the morphism of sheaves
$W\,:\,F\,\rightarrow \,F\,\otimes \,L$
is injective. Denote by
$\text{Flat}_{G}^{W}(X,\unicode[STIX]{x1D712},W)$
the dg full subcategory of
$\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)$
consisting of factorizations whose components are
$W$
-flat. Then
$H^{0}(\text{Flat}_{G}^{W}(X,\unicode[STIX]{x1D712},W))$
is a full triangulated subcategory of
$H^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))$
. Denote by
$\text{Acycl}^{\text{co}}(\text{Flat}_{G}^{W}(X,\unicode[STIX]{x1D712},W))$
the smallest thick subcategory of
$H^{0}(\text{Flat}_{G}^{W}(X,\unicode[STIX]{x1D712},W))$
containing all totalizations of short exact sequences in the exact category
$Z^{0}(\text{Flat}_{G}^{W}(X,\unicode[STIX]{x1D712},W))$
. Consider the corresponding Verdier quotients
$$\begin{eqnarray}{\text{D}^{\text{co}}\text{Flat}}_{G}^{W}(X,\unicode[STIX]{x1D712},W):=H^{0}(\text{Flat}_{G}^{W}(X,\unicode[STIX]{x1D712},W))/\text{Acycl}^{\text{co}}(\text{Flat}_{G}^{W}(X,\unicode[STIX]{x1D712},W)).\end{eqnarray}$$
The restriction functor
$\text{Res}_{G}:\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)\rightarrow \text{Qcoh}(X,W)$
and the induction functor
$\text{Ind}_{G}:\text{Qcoh}(X,W)\rightarrow \text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)$
preserve factorizations whose components are
$W$
-flat sheaves since
$\text{Res}_{G}:\text{Qcoh}_{G}X\rightarrow \text{Qcoh}X$
and
$\text{Ind}_{G}:\text{Qcoh}X\rightarrow \text{Qcoh}_{G}X$
are exact functors. Hence, the restriction and the induction functors induce the following functors
$$\begin{eqnarray}\displaystyle & \displaystyle \text{Res}_{G}:{\text{D}^{\text{co}}\text{Flat}}_{G}^{W}(X,\unicode[STIX]{x1D712},W)\rightarrow {\text{D}^{\text{co}}\text{Flat}}^{W}(X,W), & \displaystyle \nonumber\\ \displaystyle & \displaystyle \text{Ind}_{G}:{\text{D}^{\text{co}}\text{Flat}}^{W}(X,W)\rightarrow {\text{D}^{\text{co}}\text{Flat}}_{G}^{W}(X,\unicode[STIX]{x1D712},W), & \displaystyle \nonumber\end{eqnarray}$$
and these functors are adjoint to each other:
$$\begin{eqnarray}\text{Res}_{G}\dashv \text{Ind}_{G}.\end{eqnarray}$$
Lemma 3.7. The natural functor
$$\begin{eqnarray}{\text{D}^{\text{co}}\text{Flat}}_{G}^{W}(X,\unicode[STIX]{x1D712},W)\rightarrow {\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W)\end{eqnarray}$$
is an equivalence.
Proof. First, we prove that the functor is essentially surjective. Let
$F\in {\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W)$
be an object. Since
$X$
has a
$G$
-equivariant ample line bundle, there are
$G$
-equivariant locally free sheaf
$E_{i}$
and a surjective morphism
$p_{i}:E_{i}\rightarrow F_{i}$
in
$\text{Qcoh}_{G}X$
for each
$i=0,1$
. Let
$E\in \text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W)$
be the factorization of the following form
$$\begin{eqnarray}E:=(E_{1}\oplus E_{0}\xrightarrow[{}]{W\,\oplus \,\text{id}_{E_{0}}}E_{1}(\unicode[STIX]{x1D712})\oplus E_{0}\xrightarrow[{}]{\text{id}_{E_{1}(\unicode[STIX]{x1D712})}\,\oplus \,W}E_{1}(\unicode[STIX]{x1D712})\oplus E_{0}(\unicode[STIX]{x1D712})).\end{eqnarray}$$
Then
$p_{1}$
and
$p_{0}$
define a natural surjective morphism
$p:E\rightarrow F$
in
$Z^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))$
. The kernel
$K:=\operatorname{Ker}(p)$
of
$p$
is in
$Z^{0}(\text{Flat}_{G}^{W}(X,\unicode[STIX]{x1D712},W))$
since the components of
$K$
are subsheaves of
$W$
-flat sheaves. Hence, the totalization
$\operatorname{Tot}(C^{\bullet })$
of the complex
$$\begin{eqnarray}C^{\bullet }:\cdots \rightarrow 0\rightarrow K{\hookrightarrow}E\rightarrow 0\rightarrow \cdots\end{eqnarray}$$
with the cohomological degree of
$E$
zero is in
${\text{D}^{\text{co}}\text{Flat}}_{G}^{W}(X,\unicode[STIX]{x1D712},W)$
, and we see that the natural morphism
$\operatorname{Tot}(C^{\bullet })\rightarrow F$
induced by
$p$
is an isomorphism in
${\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W)$
.
To show the functor
${\text{D}^{\text{co}}\text{Flat}}_{G}^{W}(X,\unicode[STIX]{x1D712},W)\rightarrow {\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W)$
is fully faithful, it suffices to prove that for any morphism
$f:E\rightarrow F$
in
$H^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))$
with
$F\in H^{0}(\text{Flat}_{G}^{W}(X,\unicode[STIX]{x1D712},W))$
and the cone of
$f$
in
$\text{Acycl}^{\text{co}}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))$
, there exists a morphism
$g:F^{\prime }\rightarrow E$
with
$F^{\prime }\in H^{0}(\text{Flat}_{G}^{W}(X,\unicode[STIX]{x1D712},W))$
such that the cone of
$f\circ g$
is in
$\text{Acycl}^{\text{co}}(\text{Flat}_{G}^{W}(X,\unicode[STIX]{x1D712},W))$
(see [Reference Lunts and SchnürerLS16, Proposition B.2.
$(ff1)^{op}$
]). By the above argument in the previous paragraph, we can find a morphism
$g:F^{\prime }\rightarrow E$
with
$F^{\prime }\in H^{0}(\text{Flat}_{G}^{W}(X,\unicode[STIX]{x1D712},W))$
such that the cone of
$g$
is in
$\text{Acycl}^{\text{co}}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))$
, and then the cone of
$f\circ g$
is in
$H^{0}(\text{Flat}_{G}^{W}(X,\unicode[STIX]{x1D712},W))\cap \text{Acycl}^{\text{co}}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))$
. Hence, it is enough to show that
$$\begin{eqnarray}H^{0}(\text{Flat}_{G}^{W}(X,\unicode[STIX]{x1D712},W))\cap \text{Acycl}^{\text{co}}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))\subseteq \text{Acycl}^{\text{co}}(\text{Flat}_{G}^{W}(X,\unicode[STIX]{x1D712},W)).\end{eqnarray}$$
For this, let
$A\in H^{0}(\text{Flat}_{G}^{W}(X,\unicode[STIX]{x1D712},W))\cap \text{Acycl}^{\text{co}}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))$
be an object. We already know that
$\text{Res}_{G}(A)\in \text{Acycl}^{\text{co}}(\text{Flat}^{W}(X,W))$
by [Reference Efimov and PositselskiEP15, Corollary 2.6(a)]. Note that the restriction functor
$\text{Res}_{G}:{\text{D}^{\text{co}}\text{Flat}}_{G}^{W}(X,\unicode[STIX]{x1D712},W)\rightarrow {\text{D}^{\text{co}}\text{Flat}}^{W}(X,W)$
is faithful by a similar argument as in the proof of Lemma 2.33(2). Hence, the fact that
$\text{Res}_{G}(A)\in \text{Acycl}^{\text{co}}(\text{Flat}^{W}(X,W))$
implies that
$A\in \text{Acycl}^{\text{co}}(\text{Flat}_{G}^{W}(X,\unicode[STIX]{x1D712},W))$
.◻
For an object
$F=(F_{1}\xrightarrow[{}]{\unicode[STIX]{x1D711}_{1}^{F}}F_{0}\xrightarrow[{}]{\unicode[STIX]{x1D711}_{0}^{F}}F_{1}(\unicode[STIX]{x1D712}))\in Z^{0}(\text{Flat}_{G}^{W}(X,\unicode[STIX]{x1D712},W))$
, define an object
$\unicode[STIX]{x1D6EF}(F)\in \text{D}_{G}^{\text{cosg}}(X_{0}/X)$
by
$\unicode[STIX]{x1D6EF}(F):=\text{Cok}(\unicode[STIX]{x1D711}_{1}^{F})$
. It is easy to see that this defines the following exact functor
$$\begin{eqnarray}\unicode[STIX]{x1D6EF}:H^{0}(\text{Flat}_{G}^{W}(X,\unicode[STIX]{x1D712},W))\rightarrow \text{D}_{G}^{\text{cosg}}(X_{0}/X).\end{eqnarray}$$
If
$G$
is trivial, this exact functor annihilates
$\text{Acycl}^{\text{co}}(\text{Flat}^{W}(X,W))$
by [Reference Efimov and PositselskiEP15, Theorems 2.7 and 2.8]. Hence, since
$\text{Res}_{G}:\text{D}_{G}^{\text{cosg}}(X_{0}/X)\rightarrow \text{D}^{\text{cosg}}(X_{0}/X)$
is faithful, we obtain the exact functor
$\unicode[STIX]{x1D6EF}:{\text{D}^{\text{co}}\text{Flat}}_{G}^{W}(X,\unicode[STIX]{x1D712},W)\rightarrow \text{D}_{G}^{\text{cosg}}(X_{0}/X)$
. By Lemma 3.7, we have the left derived functor of
$\unicode[STIX]{x1D6EF}$
;
$$\begin{eqnarray}\mathbf{L}\unicode[STIX]{x1D6EF}:{\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W)\rightarrow \text{D}_{G}^{\text{cosg}}(X_{0}/X).\end{eqnarray}$$
Proof of Theorem 3.6.
We will show that the functors
$\unicode[STIX]{x1D6F6}$
and
$\mathbf{L}\unicode[STIX]{x1D6EF}$
are mutually inverse. Let
$E\in {\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W)$
be an object. By Lemma 3.7 we may assume that
$E\in {\text{D}^{\text{co}}\text{Flat}}_{G}^{W}(X,\unicode[STIX]{x1D712},W)$
. Then
$$\begin{eqnarray}\unicode[STIX]{x1D6F6}\mathbf{L}\unicode[STIX]{x1D6EF}(E)\cong \unicode[STIX]{x1D6F6}\unicode[STIX]{x1D6EF}(E)=(0\rightarrow \text{Cok}(\unicode[STIX]{x1D711}_{1}^{E})\rightarrow 0),\end{eqnarray}$$
and the surjective morphism
$E_{0}{\twoheadrightarrow}\text{Cok}(\unicode[STIX]{x1D711}_{1}^{E})$
induces the natural surjective morphism
$\unicode[STIX]{x1D719}_{E}:E\rightarrow \unicode[STIX]{x1D6F6}\unicode[STIX]{x1D6EF}(E)$
in
$Z^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))$
. Since the kernel of
$\unicode[STIX]{x1D719}_{E}$
is the factorization
$(E_{1}=E_{1}\xrightarrow[{}]{W}E_{1}(\unicode[STIX]{x1D712}))$
and it is isomorphic to the zero object in
$H^{0}(\text{Flat}_{G}^{W}(X,\unicode[STIX]{x1D712},W))$
, the morphism
$\unicode[STIX]{x1D719}_{E}:E\rightarrow \unicode[STIX]{x1D6F6}\unicode[STIX]{x1D6EF}(E)$
is an isomorphism in
${\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W)$
. It is easy to see that the isomorphisms
$\unicode[STIX]{x1D719}_{(-)}$
define an isomorphism of functors
$$\begin{eqnarray}\unicode[STIX]{x1D719}:\text{id}_{{\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W)}\xrightarrow[{}]{{\sim}}\unicode[STIX]{x1D6F6}\mathbf{L}\unicode[STIX]{x1D6EF}.\end{eqnarray}$$
Let
$F\in \text{D}_{G}^{\text{cosg}}(X_{0}/X)$
be an object. Then we may assume that
$F\in \text{Qcoh}_{G}X_{0}$
. Take a surjective morphism
$p:P{\twoheadrightarrow}i_{\ast }F$
with
$P$
locally free. Set
$K:=\operatorname{Ker}(p)\in \text{Qcoh}_{G}X$
and
$Q:=(K\xrightarrow[{}]{i}P\xrightarrow[{}]{W}K(\unicode[STIX]{x1D712}))\in \text{Qcoh}_{G}(X\unicode[STIX]{x1D712},W)$
, where
$i:K\rightarrow P$
is the natural inclusion. Consider the natural surjective morphism
$\unicode[STIX]{x1D70B}:Q\rightarrow (0\rightarrow i_{\ast }F\rightarrow 0)$
in
$Z^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))$
. Then the kernel of
$\unicode[STIX]{x1D70B}$
is the factorization
$(K=K\xrightarrow[{}]{W}K(\unicode[STIX]{x1D712}))$
, and it is isomorphic to the zero object in
$H^{0}(\text{Qcoh}_{G}(X,\unicode[STIX]{x1D712},W))$
. Hence,
$\unicode[STIX]{x1D70B}$
is an isomorphism in
${\text{D}^{\text{co}}\text{Qcoh}}_{G}(X,\unicode[STIX]{x1D712},W)$
, and so we have a natural isomorphism
$\unicode[STIX]{x1D713}_{F}:\mathbf{L}\unicode[STIX]{x1D6EF}\unicode[STIX]{x1D6F6}(F)\xrightarrow[{}]{{\sim}}F$
in
$\text{D}_{G}^{\text{cosg}}(X_{0}/X)$
defined as the composition
$\mathbf{L}\unicode[STIX]{x1D6EF}\unicode[STIX]{x1D6F6}(F)\xrightarrow[{}]{{\sim}}\unicode[STIX]{x1D6EF}\unicode[STIX]{x1D6F6}(Q)=\text{Cok}(i)=F$
. We need to show that the isomorphisms
$\unicode[STIX]{x1D713}_{(-)}$
are functorial in
$(-)$
. Since the restriction functor
$\text{Res}_{G}$
is isomorphic to the forgetful functor
$\text{Forg}_{G}$
, we have a natural isomorphism of functors
$\unicode[STIX]{x1D70E}:\text{Res}_{G}\mathbf{L}\unicode[STIX]{x1D6EF}\unicode[STIX]{x1D6F6}\xrightarrow[{}]{{\sim}}\mathbf{L}\unicode[STIX]{x1D6EF}\unicode[STIX]{x1D6F6}\text{Res}_{G}$
defined by the composition
$$\begin{eqnarray}\text{Res}_{G}\mathbf{L}\unicode[STIX]{x1D6EF}\unicode[STIX]{x1D6F6}\xrightarrow[{}]{{\sim}}\text{Forg}_{G}\mathbf{L}\unicode[STIX]{x1D6EF}\unicode[STIX]{x1D6F6}=\mathbf{L}\unicode[STIX]{x1D6EF}\unicode[STIX]{x1D6F6}\text{Forg}_{G}\xrightarrow[{}]{{\sim}}\mathbf{L}\unicode[STIX]{x1D6EF}\unicode[STIX]{x1D6F6}\text{Res}_{G},\end{eqnarray}$$
and the following diagram is commutative.
Hence, we see that the isomorphisms
$\unicode[STIX]{x1D713}_{(-)}$
are functorial by the fact that the isomorphisms
$\unicode[STIX]{x1D713}_{(-)}$
are functorial if
$G$
is trivial and that the functor
$\text{Res}_{G}$
is faithful. This completes the proof of the former equivalence.
The latter equivalence follows from [Reference Efimov and PositselskiEP15, Remark 2.7], which is a generalized result of [Reference Efimov and PositselskiEP15, Theorem 2.7]. ◻
4 Derived Knörrer periodicity
Let
$X$
be a smooth quasi-projective variety, and let
$G$
be a reductive affine algebraic group acting on
$X$
. Let
${\mathcal{E}}$
be a
$G$
-equivariant locally free sheaf of rank
$r$
, and let
$s\in \unicode[STIX]{x1D6E4}(X,{\mathcal{E}}^{\vee })^{G}$
be a
$G$
-invariant section of
${\mathcal{E}}^{\vee }$
. Denote by
$Z\subset X$
the zero scheme of
$s$
. We assume that
$s$
is regular, i.e. the codimension of
$Z$
in
$X$
is
$r$
. Let
$$\begin{eqnarray}\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712})):=\text{}\underline{\operatorname{Spec}}(\operatorname{Sym}({\mathcal{E}}(\unicode[STIX]{x1D712})^{\vee }))\end{eqnarray}$$
be a vector bundle over
$X$
with the
$G$
-action induced by the equivariant structure of the locally free sheaf
${\mathcal{E}}(\unicode[STIX]{x1D712})$
. Denote by
$\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712}))|_{Z}$
the restriction of the vector bundle
$\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712}))$
to
$Z$
. Let
$j:Z{\hookrightarrow}X$
and
$i:\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712}))|_{Z}{\hookrightarrow}\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712}))$
be the closed immersions, and let
$q:\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712}))\rightarrow X$
and
$p:\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712}))|_{Z}\rightarrow Z$
be the projections. Now we have the following commutative diagram.
The invariant section
$s$
induces a
$\unicode[STIX]{x1D712}$
-semi-invariant regular function
$$\begin{eqnarray}Q_{s}:\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712}))\rightarrow \mathbb{A}^{1}.\end{eqnarray}$$
Let
$W:X\rightarrow \mathbb{A}^{1}$
be a
$\unicode[STIX]{x1D712}$
-semi-invariant regular function on
$X$
. The function
$W$
induces
$\unicode[STIX]{x1D712}$
-semi-invariant functions on
$Z$
,
$\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712}))$
and
$\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712}))|_{Z}$
, which we denote by the same notation
$W$
(by abuse of notation). Since the inverse image
$p^{\ast }$
and the direct image
$i_{\ast }$
are exact and commutative with arbitrary direct sums as functors between categories of quasi-coherent sheaves, these induce (underived) functors
$$\begin{eqnarray}\displaystyle & \displaystyle p^{\ast }:{\text{D}^{\text{co}}\text{Qcoh}}_{G}(Z,\unicode[STIX]{x1D712},W)\rightarrow {\text{D}^{\text{co}}\text{Qcoh}}_{G}(\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712}))|_{Z},\unicode[STIX]{x1D712},W), & \displaystyle \nonumber\\ \displaystyle & \displaystyle i_{\ast }:{\text{D}^{\text{co}}\text{Qcoh}}_{G}(\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712}))|_{Z},\unicode[STIX]{x1D712},W)\rightarrow {\text{D}^{\text{co}}\text{Qcoh}}_{G}(\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712})),\unicode[STIX]{x1D712},W+Q_{s}). & \displaystyle \nonumber\end{eqnarray}$$
Restricting the composition
$i_{\ast }p^{\ast }:{\text{D}^{\text{co}}\text{Qcoh}}_{G}(Z,\unicode[STIX]{x1D712},W)\rightarrow {\text{D}^{\text{co}}\text{Qcoh}}_{G}(\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712})),\unicode[STIX]{x1D712},W+Q_{s})$
to
$\text{Dcoh}_{G}(Z,\unicode[STIX]{x1D712},W)$
, we obtain an exact functor
$$\begin{eqnarray}i_{\ast }p^{\ast }:\text{Dcoh}_{G}(Z,\unicode[STIX]{x1D712},W)\rightarrow \text{Dcoh}_{G}(\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712})),\unicode[STIX]{x1D712},W+Q_{s}).\end{eqnarray}$$
Shipman proved that the above functor
$i_{\ast }p^{\ast }$
is an equivalence when
$G=\mathbb{G}_{m}$
trivially acts on
$X$
and
$W=0$
(see also [Reference IsikIsi13]).
Theorem 4.1 [Reference ShipmanShi12, Theorem 3.4].
The composition
$$\begin{eqnarray}i_{\ast }p^{\ast }:\text{Dcoh}_{\mathbb{ G}_{m}}(Z,\unicode[STIX]{x1D712}_{1},0)\xrightarrow[{}]{{\sim}}\text{Dcoh}_{\mathbb{G}_{m}}(\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712}_{1})),\unicode[STIX]{x1D712}_{1},Q_{s})\end{eqnarray}$$
is an equivalence.
The goal of this section is to show the following main result which is an analogy of the above theorem.
Theorem 4.2. Assume that
$W|_{Z}:Z\rightarrow \mathbb{A}^{1}$
is flat. The functor
$$\begin{eqnarray}i_{\ast }p^{\ast }:\text{Dcoh}_{G}(Z,\unicode[STIX]{x1D712},W)\rightarrow \text{Dcoh}_{G}(\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712})),\unicode[STIX]{x1D712},W+Q_{s})\end{eqnarray}$$
is an equivalence.
Remark 4.3. Let
$S$
be a smooth quasi-projective variety, and let
$G$
be an affine reductive group acting on
$S$
. Let
$W:S\rightarrow \mathbb{A}^{1}$
be a
$\unicode[STIX]{x1D712}:=\unicode[STIX]{x1D712}_{1}+\unicode[STIX]{x1D712}_{2}$
-semi-invariant non-constant regular function for some characters
$\unicode[STIX]{x1D712}_{i}:G\rightarrow \mathbb{G}_{m}$
. Let
$X:=\text{V}({\mathcal{O}}(\unicode[STIX]{x1D712}_{1}))\cong S\times \mathbb{A}_{x_{1}}^{1}$
be the
$G$
-vector bundle over
$S$
, and let
$s\in \unicode[STIX]{x1D6E4}(X,{\mathcal{O}}(\unicode[STIX]{x1D712}_{1}))^{G}$
be the section corresponding to the
$\unicode[STIX]{x1D712}_{1}$
-semi-invariant function
$S\times \mathbb{A}_{x_{1}}^{1}\rightarrow \mathbb{A}^{1}$
which is defined as the projection
$(s,x_{1})\mapsto x_{1}$
. Then,
$S$
is isomorphic to the zero scheme of
$s$
, and the
$G$
-vector bundle
$\text{V}({\mathcal{O}}(-\unicode[STIX]{x1D712}_{1})(\unicode[STIX]{x1D712}))$
over
$X$
is isomorphic to the
$G$
-variety
$S\times \mathbb{A}_{x_{1},x_{2}}^{2}$
, where the
$G$
-weights of
$x_{i}$
are given by
$\unicode[STIX]{x1D712}_{i}$
. By Theorem 4.2, we have the following equivalence
$$\begin{eqnarray}\text{Dcoh}_{G}(S,\unicode[STIX]{x1D712},W)\simeq \text{Dcoh}_{G}(S\times \mathbb{A}_{x_{1},x_{2}}^{2},\unicode[STIX]{x1D712},W+x_{1}x_{2}).\end{eqnarray}$$
This kind of equivalence is known as Knörrer periodicity, so the above theorem is considered as a generalization of the original Knörrer periodicity [Reference KnörrerKnö87, Theorem 3.1].
4.1 Lemmas for the main theorem
In this section, we provide some lemmas for the main result. Throughout this section, we consider the case when
$G$
is trivial.
Set
$$\begin{eqnarray}\unicode[STIX]{x1D714}_{j}:=\bigwedge ^{r}({\mathcal{I}}_{Z}/{\mathcal{I}}_{Z}^{2})^{\vee }\quad \text{and}\quad \unicode[STIX]{x1D714}_{i}:=p^{\ast }\unicode[STIX]{x1D714}_{j},\end{eqnarray}$$
where
${\mathcal{I}}_{Z}$
is the ideal sheaf of
$Z$
in
$X$
. These are invertible sheaves on
$Z$
and
$\text{V}({\mathcal{E}})|_{Z}$
, respectively. We define an exact functor
$$\begin{eqnarray}i^{!}:\text{D}^{\text{co}}\text{Qcoh}(\text{V}({\mathcal{E}}),W+Q_{s})\rightarrow \text{D}^{\text{co}}\text{Qcoh}(\text{V}({\mathcal{E}})|_{Z},W)\end{eqnarray}$$
as
$i^{!}(-):=\mathbf{L}i^{\ast }(-)\otimes \unicode[STIX]{x1D714}_{i}[-r]$
. By [Reference Efimov and PositselskiEP15, Theorem 3.8], the above functor
$i^{!}$
is right adjoint to
$i_{\ast }:\text{D}^{\text{co}}\text{Qcoh}(\text{V}({\mathcal{E}})|_{Z},W)\rightarrow \text{D}^{\text{co}}\text{Qcoh}(\text{V}({\mathcal{E}}),W+Q_{s})$
. Let
$$\begin{eqnarray}K:=K(q^{\ast }s,t)\in \text{lfr}(\text{V}({\mathcal{E}}),Q_{s})\end{eqnarray}$$
be the Koszul factorization of
$q^{\ast }s\,\in \,\unicode[STIX]{x1D6E4}(\text{V}({\mathcal{E}}),q^{\ast }{\mathcal{E}}^{\vee })$
and
$t\,\in \,\unicode[STIX]{x1D6E4}(\text{V}({\mathcal{E}}),q^{\ast }{\mathcal{E}})$
, where
$t$
is the tautological section. By abuse of notation, we denote by
${\mathcal{O}}_{Z}$
the object in
$\text{coh}(Z,0)$
of the following form
$$\begin{eqnarray}(0\rightarrow {\mathcal{O}}_{Z}\rightarrow 0).\end{eqnarray}$$
Lemma 4.4. Consider the case when
$W=0$
. We have isomorphisms
$$\begin{eqnarray}i_{\ast }p^{\ast }({\mathcal{O}}_{Z})\cong K\quad \text{and}\quad p_{\ast }i^{!}(K)\cong {\mathcal{O}}_{Z}\end{eqnarray}$$
in
$\text{Dcoh}(\text{V}({\mathcal{E}}),Q_{s})$
and in
$\text{D}^{\text{co}}\text{Qcoh}(Z,0)$
, respectively.
Proof. These isomorphisms follow from Lemma 2.28. In particular, the former isomorphism is an immediate consequence. Note that
$\unicode[STIX]{x1D714}_{i}\cong i^{\ast }\bigwedge ^{r}q^{\ast }{\mathcal{E}}^{\vee }$
. We obtain the latter isomorphism as follows:
$$\begin{eqnarray}p_{\ast }i^{!}(K)\cong p_{\ast }\mathbf{L}i^{\ast }\biggl({\mathcal{O}}_{Z}\otimes \bigwedge ^{r}q^{\ast }{\mathcal{E}}^{\vee }[-r]\biggr)\cong p_{\ast }\mathbf{L}i^{\ast }(K^{\vee })\cong p_{\ast }\mathbf{L}i^{\ast }({\mathcal{O}}_{Z_{t^{\vee }}})\cong {\mathcal{O}}_{Z},\end{eqnarray}$$
where the last isomorphism follows from the fact that the zero section
$Z\subset \text{V}({\mathcal{E}})$
is isomorphic to the fiber product of closed subschemes
$\text{V}({\mathcal{E}})|_{Z}{\hookrightarrow}\text{V}({\mathcal{E}})$
and
$Z_{t^{\vee }}{\hookrightarrow}\text{V}({\mathcal{E}})$
.◻
Lemma 4.5. The functor
$$\begin{eqnarray}i_{\ast }p^{\ast }:\text{D}^{\text{co}}\text{Qcoh}(Z,W)\rightarrow \text{D}^{\text{co}}\text{Qcoh}(\text{V}({\mathcal{E}}),W+Q_{s})\end{eqnarray}$$
is fully faithful.
Proof. The functors
$i_{\ast }p^{\ast }$
and
$p_{\ast }i^{!}$
can be represented as integral functors
$$\begin{eqnarray}i_{\ast }p^{\ast }\cong \unicode[STIX]{x1D6F7}_{k_{\ast }{\mathcal{O}}_{\text{V}({\mathcal{E}})|_{Z}}}\quad \text{and}\quad p_{\ast }i^{!}\cong \unicode[STIX]{x1D6F7}_{k_{\ast }\unicode[STIX]{x1D714}_{i}[-r]},\end{eqnarray}$$
where
$k:=p\times i:\text{V}({\mathcal{E}})|_{Z}\rightarrow Z\times \text{V}({\mathcal{E}})$
and kernels
${\mathcal{O}}_{\text{V}({\mathcal{E}})|_{Z}}$
and
$\unicode[STIX]{x1D714}_{i}[-r]$
are objects in
$\text{Dcoh}(\text{V}({\mathcal{E}})|_{Z},0)$
. By easy computation, we see that there exists an object
$P\in \text{D}^{\text{co}}\text{Qcoh}(Z,0)$
such that
$p_{\ast }i^{!}\circ i_{\ast }p^{\ast }\cong \unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6E5}_{\ast }P}\cong (-)\otimes P$
, where
$\unicode[STIX]{x1D6E5}:Z\rightarrow Z\times Z$
is the diagonal embedding. Substituting
$W=0$
, by Lemma 4.4, we have an isomorphism
$P\cong {\mathcal{O}}_{Z}$
. But
$P$
does not depend on the function
$W$
. Hence, for any
$W$
, we have an isomorphism of functors
$p_{\ast }i^{!}\circ i_{\ast }p^{\ast }\cong \unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6E5}_{\ast }P}\cong \text{id}_{\text{D}^{\text{co}}\text{Qcoh}(Z,W)}$
. By the following lemma, this implies that the functor
$i_{\ast }p^{\ast }:\text{D}^{\text{co}}\text{Qcoh}(Z,W)\rightarrow \text{D}^{\text{co}}\text{Qcoh}(\text{V}({\mathcal{E}}),W+Q_{s})$
is fully faithful.◻
The following lemma is an opposite version of [Reference JohnstoneJoh02, Lemma 1.1.1].Footnote 1 We give a proof for the reader’s convenience.
Lemma 4.6 (Cf. [Reference JohnstoneJoh02, Lemma 1.1.1]).
Let
$F:{\mathcal{A}}\rightarrow {\mathcal{B}}$
be a functor between (arbitrary) categories, and suppose that
$F$
admits a right adjoint functor
$G:{\mathcal{B}}\rightarrow {\mathcal{A}}$
. Then, if there exists an isomorphism of functors
$\unicode[STIX]{x1D6FC}:\text{id}_{{\mathcal{A}}}\xrightarrow[{}]{{\sim}}GF$
(
$\unicode[STIX]{x1D6FC}$
is not necessarily the adjunction morphism), then
$F$
is fully faithful.
Proof. The isomorphism
$\unicode[STIX]{x1D6FC}$
implies that the following composition of maps is bijective:
$$\begin{eqnarray}\operatorname{Hom}(A,A^{\prime })\xrightarrow[{}]{F}\operatorname{Hom}(F(A),F(A^{\prime }))\xrightarrow[{}]{G}\operatorname{Hom}(GF(A),GF(A^{\prime })).\end{eqnarray}$$
Hence, it is enough to show that
$G$
is fully faithful on the image of
$F$
. Since the above composition is bijective,
$G$
is full on the image of
$F$
. Let
$\unicode[STIX]{x1D700}:\text{id}_{{\mathcal{A}}}\rightarrow GF$
and
$\unicode[STIX]{x1D6FF}:FG\rightarrow \text{id}_{{\mathcal{B}}}$
be the adjunction morphisms. For any
$f\in \operatorname{Hom}(F(A),F(A^{\prime }))$
we have
$$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{F(A^{\prime })}\circ FG(f)\circ F(\unicode[STIX]{x1D700}_{A})=f\circ \unicode[STIX]{x1D6FF}_{F(A)}\circ F(\unicode[STIX]{x1D700}_{A})=f,\end{eqnarray}$$
where the first equation follows from the functoriality of
$\unicode[STIX]{x1D6FF}$
and the second equation follows from the property of the adjunction morphisms. Hence, the following diagram is commutative
and hence
$G$
is faithful on the image of
$F$
.◻
4.2 Proof of the main theorem
In this section, we prove the main theorem. Recall that
$G$
is a reductive affine algebraic group acting on a smooth quasi-projective variety
$X$
. Since
$X$
is smooth, there is a
$G$
-equivariant ample line bundle on
$X$
. In what follows, we assume that
$W|_{Z}:Z\rightarrow \mathbb{A}^{1}$
is flat.
First, we consider relative singularity categories. Let
$Z_{0}$
,
$V|_{Z_{0}}$
and
$V_{0}$
be the fibers of
$W:Z\rightarrow \mathbb{A}^{1}$
,
$W:\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712}))|_{Z}\rightarrow \mathbb{A}^{1}$
and
$W+Q_{s}:\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712}))\rightarrow \mathbb{A}^{1}$
over
$0\in \mathbb{A}^{1}$
, respectively. Denote by
$p_{0}:V|_{Z_{0}}\rightarrow Z_{0}$
and
$i_{0}:V|_{Z_{0}}\rightarrow V_{0}$
the restrictions of
$p$
and
$i$
, respectively. By [Reference KuznetsovKuz06, Corollary 2.27], the following cartesian squares are exact.
Since
$p$
and
$i$
have finite flat dimensions, we have exact functors of relative singularity categories
$$\begin{eqnarray}\displaystyle {p_{0}}^{\circ }:\text{D}_{G}^{\text{cosg}}(Z_{0}/Z) & \rightarrow & \displaystyle \text{D}_{G}^{\text{cosg}}(V|_{Z_{0}}/\text{V}({\mathcal{E}})|_{Z}),\nonumber\\ \displaystyle {i_{0}}_{\circ }:\text{D}_{G}^{\text{cosg}}(V|_{Z_{0}}/\text{V}({\mathcal{E}})|_{Z}) & \rightarrow & \displaystyle \text{D}^{\text{cosg}}(V_{0}/\text{V}({\mathcal{E}}))=\text{D}_{G}^{\text{cosg}}(V_{0}).\nonumber\end{eqnarray}$$
Then the following diagram is commutative.
Furthermore, we compactify
$V_{0}$
and
$V|_{Z_{0}}$
. The compactifying technique appeared in [Reference OrlovOrl06]. Let
$$\begin{eqnarray}P:=\mathbb{P}({\mathcal{E}}(\unicode[STIX]{x1D712})\oplus {\mathcal{O}}_{X})=\text{}\underline{\operatorname{Proj}}(\operatorname{Sym}({\mathcal{E}}(\unicode[STIX]{x1D712})\oplus {\mathcal{O}}_{X})^{\vee })\end{eqnarray}$$
be the projective space bundle over
$X$
with a
$G$
-action induced by the equivariant structure of
${\mathcal{E}}(\unicode[STIX]{x1D712})\oplus {\mathcal{O}}_{X}$
. Then we have a natural equivariant open immersion
$$\begin{eqnarray}l:\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712}))\rightarrow P.\end{eqnarray}$$
Denote by
$l|_{Z}:\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712}))|_{Z}\rightarrow P|_{Z}$
the pull-back of
$l$
by the closed immersion
$\overline{i}:P|_{Z}\rightarrow P$
. Now we have the following cartesian square.
Denote by
$\overline{q}:P\rightarrow X$
the natural projection, and let
$\overline{p}:=\overline{q}|_{Z}:P|_{Z}\rightarrow Z$
be the pull-back of
$\overline{q}$
by the closed immersion
$j:Z\rightarrow X$
. Let
$P_{0}$
be the
$G$
-invariant subscheme of
$P$
defined by the
$G$
-invariant section
$\overline{s\oplus W}\in \unicode[STIX]{x1D6E4}(P,{\mathcal{O}}(1)(\unicode[STIX]{x1D712}))^{G}$
which is corresponding to the composition
$$\begin{eqnarray}{\mathcal{O}}_{P}\xrightarrow[{}]{\overline{q}^{\ast }(s\oplus W)}\overline{q}^{\ast }({\mathcal{E}}\oplus {\mathcal{O}}(\unicode[STIX]{x1D712}^{-1}))^{\vee }\xrightarrow[{}]{\unicode[STIX]{x1D70E}}{\mathcal{O}}_{P}(1)(\unicode[STIX]{x1D712}),\end{eqnarray}$$
where
$\unicode[STIX]{x1D70E}$
is the canonical surjection, and let
$P|_{Z_{0}}$
be the zero scheme defined by the invariant section
$\overline{i}^{\ast }(\overline{s\oplus W})\in \unicode[STIX]{x1D6E4}(P|_{Z},{\mathcal{O}}(1)(\unicode[STIX]{x1D712}))^{G}$
. Since the pull-back of
$\overline{s\oplus W}$
(respectively
$\overline{i}^{\ast }(\overline{s\oplus W})$
) by the open immersion
$l$
(respectively
$l|_{Z}$
) is equal to
$W+Q_{s}$
(respectively
$W$
), we have the following exact cartesian square.
Denote by
$\overline{p_{0}}:P|_{Z_{0}}\rightarrow Z_{0}$
the pull-back of
$\overline{p}:P|_{Z}\rightarrow Z$
by the closed immersion
$Z_{0}\rightarrow Z$
. Since the morphisms
$\overline{i_{0}}:P|_{Z_{0}}\rightarrow P_{0}$
and
$\overline{p_{0}}:P|_{Z_{0}}\rightarrow Z_{0}$
have finite Tor dimensions, the direct images
$\mathbf{R}\overline{i_{0}}_{\ast }:\text{D}^{\text{b}}(\text{coh}P|_{Z_{0}})\rightarrow \text{D}^{\text{b}}(\text{coh}P_{0})$
and
$\mathbf{R}\overline{p_{0}}_{\ast }:\text{D}^{\text{b}}(\text{coh}P|_{Z_{0}})\rightarrow \text{D}^{\text{b}}(\text{coh}Z_{0})$
induce the following exact functors (cf. [Reference Thomason and TrobaughTT90, Proposition 2.7]),
$$\begin{eqnarray}\displaystyle \overline{i_{0}}_{\circ }:\text{D}_{G}^{\text{sg}}(P|_{Z_{0}}) & \rightarrow & \displaystyle \text{D}_{G}^{\text{sg}}(P_{0}),\nonumber\\ \displaystyle \overline{p_{0}}_{\circ }:\text{D}_{G}^{\text{sg}}(P|_{Z_{0}}) & \rightarrow & \displaystyle \text{D}_{G}^{\text{sg}}(Z_{0}).\nonumber\end{eqnarray}$$
Now we have the following commutative diagram
where the vertical arrow on the left-hand side is a Verdier localization by Proposition 3.2.
Remark 4.7. If
$Z$
is smooth, the above vertical arrows are equivalences. Indeed, in that case, the singular locus
$\operatorname{Sing}(P_{0})$
is contained in
$V_{0}$
, whence
${l_{0}}^{\circ }$
is an equivalence by a similar argument in the proof of [Reference OrlovOrl04, Proposition 1.14]. The equivalence of
$\unicode[STIX]{x1D70B}$
follows from Remark 3.3.
Let
$\overline{i_{0}}^{!}:\text{D}^{\text{b}}(\text{coh}_{G}P_{0})\rightarrow \text{D}^{\text{b}}(\text{coh}_{G}P|_{Z_{0}})$
be the functor defined by
$$\begin{eqnarray}\overline{i_{0}}^{!}:=\mathbf{L}\overline{i_{0}}^{\ast }(-)\otimes \bigwedge ^{r}({\mathcal{I}}/{\mathcal{I}}^{2})^{\vee }[-r],\end{eqnarray}$$
where
${\mathcal{I}}$
is the ideal sheaf of
$\overline{i_{0}}:P|_{Z_{0}}{\hookrightarrow}P_{0}$
. The functor
$\overline{i_{0}}^{!}$
is a right adjoint functor of
$\overline{i_{0}}_{\ast }:\text{D}^{\text{b}}(\text{coh}_{G}P|_{Z_{0}})\rightarrow \text{D}^{\text{b}}(\text{coh}_{G}P_{0})$
. Indeed, these functors are adjoint when
$G$
is trivial by [Reference HartshorneHar66, III Theorem 6.7, Corollary 7.3], and the isomorphism
$$\begin{eqnarray}\operatorname{Hom}(\overline{i_{0}}_{\ast }(A),B)\cong \operatorname{Hom}(A,\overline{i_{0}}^{!}(B)),\end{eqnarray}$$
where
$A\in \text{D}^{\text{b}}(\text{coh}P|_{Z_{0}})$
and
$B\in \text{D}^{\text{b}}(\text{coh}P_{0})$
, commutes with
$G$
-actions on each vector space of morphisms by the property in [Reference HartshorneHar66, III Proposition 6.9.c]. Hence we see that
$\overline{i_{0}}^{!}$
is right adjoint to
$\overline{i_{0}}_{\ast }$
by [Reference Ballard, Favero and KatzarkovBFK12, Lemma 2.2.8]. Denote by
$$\begin{eqnarray}\overline{i_{0}}^{\flat }:\text{D}_{G}^{\text{sg}}(P_{0})\rightarrow \text{D}_{G}^{\text{sg}}(P|_{Z_{0}})\end{eqnarray}$$
the functor induced by
$\overline{i_{0}}^{!}$
. By the above argument, we have the following adjoint pair
$$\begin{eqnarray}\overline{i_{0}}_{\circ }\dashv \overline{i_{0}}^{\flat }.\end{eqnarray}$$
Similarly, we have a right adjoint functor
$$\begin{eqnarray}{i_{0}}^{\flat }:\text{D}_{G}^{\text{cosg}}(V_{0})\rightarrow \text{D}_{G}^{\text{cosg}}(V|_{Z_{0}}/\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712}))|_{Z})\end{eqnarray}$$
of
${i_{0}}_{\circ }:\text{D}_{G}^{\text{cosg}}(V|_{Z_{0}}/\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712}))|_{Z})\rightarrow \text{D}_{G}^{\text{cosg}}(V_{0})$
.
Proof of Theorem 4.2.
We have the following commutative diagram
where the vertical arrows are equivalences by Theorem 3.6. Hence, it suffices to show that the functor
${i_{0}}_{\circ }{p_{0}}^{\circ }:\text{D}_{G}^{\text{sg}}(Z_{0}/Z)\rightarrow \text{D}_{G}^{\text{sg}}(V_{0})$
is an equivalence.
First, we prove that the functor
${i_{0}}_{\circ }{p_{0}}^{\circ }:\text{D}_{G}^{\text{cosg}}(Z_{0}/Z)\rightarrow \text{D}_{G}^{\text{cosg}}(V_{0})$
is fully faithful. Let
$$\begin{eqnarray}\unicode[STIX]{x1D700}_{G}:\text{id}_{\text{D}_{G}^{\text{cosg}}(Z_{0}/Z)}\rightarrow {p_{0}}_{\circ }{i_{0}}^{\flat }\circ {i_{0}}_{\circ }{p_{0}}^{\circ }\end{eqnarray}$$
be the adjunction morphism of the adjoint pair
${i_{0}}_{\circ }{p_{0}}^{\circ }\dashv {p_{0}}_{\circ }{i_{0}}^{\flat }$
. It is enough to show that for any object
$A\in \text{D}_{G}^{\text{cosg}}(Z_{0}/Z)$
, the cone
$C_{G}(A)$
of the morphism
$\unicode[STIX]{x1D700}_{G}(A):A\rightarrow {p_{0}}_{\circ }{i_{0}}^{\flat }\circ {i_{0}}_{\circ }{p_{0}}^{\circ }(A)$
is the zero object. But the object
$\text{Res}_{G}(C_{G}(A))$
is isomorphic to the cone
$C(A)$
of the adjunction morphism of
$\unicode[STIX]{x1D700}(\text{Res}_{G}(A)):\text{Res}_{G}(A)\rightarrow {p_{0}}_{\circ }{i_{0}}^{\flat }\circ {i_{0}}_{\circ }{p_{0}}^{\circ }(\text{Res}_{G}(A))$
of the adjoint pair of functors between
$\text{D}^{\text{cosg}}(Z_{0}/Z)$
and
$\text{D}^{\text{cosg}}(V_{0})$
. Since we have the following commutative diagram
where the vertical arrows are equivalences by Theorem 3.6, the functor
${i_{0}}_{\circ }{p_{0}}^{\circ }$
is fully faithful by Lemma 4.5. This implies that the object
$C(A)$
is the zero object. Hence,
$C_{G}(A)$
is also the zero object since the restriction functor
$\text{Res}_{G}$
is faithful by Lemma 3.4. Hence,
${i_{0}}_{\circ }{p_{0}}^{\circ }:\text{D}_{G}^{\text{cosg}}(Z_{0}/Z)\rightarrow \text{D}_{G}^{\text{cosg}}(V_{0})$
is fully faithful. This implies that
${i_{0}}_{\circ }{p_{0}}^{\circ }:\text{D}_{G}^{\text{sg}}(Z_{0}/Z)\rightarrow \text{D}_{G}^{\text{sg}}(V_{0})$
is also fully faithful, since the natural inclusions
$\text{D}_{G}^{\text{sg}}(Z_{0}/Z)\rightarrow \text{D}_{G}^{\text{cosg}}(Z_{0}/Z)$
and
$\text{D}_{G}^{\text{sg}}(V_{0})\rightarrow \text{D}_{G}^{\text{cosg}}(V_{0})$
are fully faithful by Theorem 3.6 and Proposition2.25(1).
It only remains to show that the functor
${i_{0}}_{\circ }{p_{0}}^{\circ }:\text{D}_{G}^{\text{sg}}(Z_{0}/Z)\rightarrow \text{D}_{G}^{\text{sg}}(V_{0})$
is essentially surjective. Consider the following commutative diagram.
By a similar argument as in the proof of [Reference OrlovOrl04, Lemma 1.11], we see that every object in
$\text{D}_{G}^{\text{sg}}(V_{0})$
is isomorphic to an object
$F[k]$
for some
$G$
-equivariant coherent sheaf
$F$
and for some integer
$k\in \mathbb{Z}$
. Hence the vertical arrow on the right-hand side in the above diagram is essentially surjective, since for every object
$E$
in
$\text{coh}_{G}V_{0}$
there exists an object
$\overline{E}$
in
$\text{coh}_{G}P_{0}$
such that
${l_{0}}^{\ast }(\overline{E})\cong E$
. Thus, we only need to prove that
$\overline{i_{0}}_{\circ }\overline{p_{0}}^{\circ }:\text{D}_{G}^{\text{sg}}(Z_{0})\rightarrow \text{D}_{G}^{\text{sg}}(P_{0})$
is essentially surjective. To prove that, it is enough to show that the right adjoint functor
$\overline{p_{0}}_{\circ }\overline{i_{0}}^{\flat }:\text{D}_{G}^{\text{sg}}(P_{0})\rightarrow \text{D}_{G}^{\text{sg}}(Z_{0})$
is fully faithful. Since the restriction functor
$\text{Res}_{G}:\text{D}_{G}^{\text{sg}}(P_{0})\rightarrow \text{D}^{\text{sg}}(P_{0})$
is faithful by Lemma 3.4 and [Reference Polishchuk and VaintrobPV11, Proposition 3.8], it follows from [Reference OrlovOrl06, Theorem 2.1] that the adjunction
$\overline{i_{0}}_{\circ }\overline{p_{0}}^{\circ }\circ \overline{p_{0}}_{\circ }\overline{i_{0}}^{\flat }\rightarrow \text{id}_{\text{D}_{G}^{\text{sg}}(P_{0})}$
is an isomorphism of functors by a similar argument as in the proof of the fully faithfulness of
${i_{0}}_{\circ }{p_{0}}^{\circ }:\text{D}_{G}^{\text{cosg}}(Z_{0}/Z)\rightarrow \text{D}_{G}^{\text{cosg}}(V_{0})$
in the previous paragraph.◻
4.3 Cases when
$W=0$
In the previous section, we prove the main result assuming that
$W|_{Z}:Z\rightarrow \mathbb{A}^{1}$
is flat. In this section, we consider the cases when
$W=0$
. In these cases, using results in [Reference ShipmanShi12], we can show the following.
With notation as above, consider
$\mathbb{G}_{m}\times G$
-action on
$X$
induced by the projection
$\mathbb{G}_{m}\times G\rightarrow G$
. Let
$\unicode[STIX]{x1D703}:\mathbb{G}_{m}\times G\rightarrow \mathbb{G}_{m}$
be the character defined as the projection. Since the first factor of
$\mathbb{G}_{m}\times G$
trivially acts on
$X$
, the
$G$
-equivariant locally free sheaf
${\mathcal{E}}$
has a natural
$\mathbb{G}_{m}\times G$
-equivariant structure.
Proposition 4.8. We have an equivalence
$$\begin{eqnarray}\text{D}^{\text{b}}(\text{coh}_{G}Z)\xrightarrow[{}]{{\sim}}\text{Dcoh}_{\mathbb{G}_{m}\times G}(\text{V}({\mathcal{E}}(\unicode[STIX]{x1D703})),\unicode[STIX]{x1D703},Q_{s}).\end{eqnarray}$$
Proof. By a similar argument as in § 2.2, we obtain an equivalence
$$\begin{eqnarray}\text{D}^{\text{b}}(\text{coh}_{G}Z)\xrightarrow[{}]{{\sim}}\text{Dcoh}_{\mathbb{G}_{m}\times G}(Z,\unicode[STIX]{x1D703},0).\end{eqnarray}$$
Hence, it is enough to show the functor
$$\begin{eqnarray}i_{\ast }p^{\ast }:\text{Dcoh}_{\mathbb{ G}_{m}\times G}(Z,\unicode[STIX]{x1D703},0)\rightarrow \text{Dcoh}_{\mathbb{G}_{m}\times G}(\text{V}({\mathcal{E}}(\unicode[STIX]{x1D703})),\unicode[STIX]{x1D703},Q_{s})\end{eqnarray}$$
is an equivalence.
By Lemma 4.5, it follows that
$$\begin{eqnarray}i_{\ast }p^{\ast }:{\text{D}^{\text{co}}\text{Qcoh}}_{\mathbb{ G}_{m}}(Z,\unicode[STIX]{x1D712}_{1},0)\rightarrow {\text{D}^{\text{co}}\text{Qcoh}}_{\mathbb{ G}_{m}}(\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712}_{1})),\unicode[STIX]{x1D712}_{1},Q_{s})\end{eqnarray}$$
is fully faithful since the forgetful functor
${\text{D}^{\text{co}}\text{Qcoh}}_{\mathbb{G}_{m}}(Z,\unicode[STIX]{x1D712}_{1},0)\rightarrow \text{D}^{\text{co}}\text{Qcoh}(Z,0)$
is faithful. Furthermore, the above functor
$i_{\ast }p^{\ast }$
is an equivalence since the right orthogonal of the image of the restricted functor
$i_{\ast }p^{\ast }:\text{Dcoh}_{\mathbb{G}_{m}}(Z,\unicode[STIX]{x1D712}_{1},0)\rightarrow {\text{D}^{\text{co}}\text{Qcoh}}_{\mathbb{G}_{m}}(\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712}_{1})),\unicode[STIX]{x1D712}_{1},Q_{s})$
vanishes by the argument in [Reference ShipmanShi12, Theorem 3.4]. In particular, the right adjoint functor
$$\begin{eqnarray}p_{\ast }i^{!}:{\text{D}^{\text{co}}\text{Qcoh}}_{\mathbb{ G}_{m}}(\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712}_{1})),\unicode[STIX]{x1D712}_{1},Q_{s})\rightarrow {\text{D}^{\text{co}}\text{Qcoh}}_{\mathbb{ G}_{m}}(Z,\unicode[STIX]{x1D712}_{1},0)\end{eqnarray}$$
of
$i_{\ast }p^{\ast }$
is also fully faithful.
Next we will show that the functor
$$\begin{eqnarray}i_{\ast }p^{\ast }:{\text{D}^{\text{co}}\text{Qcoh}}_{\mathbb{ G}_{m}\times G}(Z,\unicode[STIX]{x1D703},0)\rightarrow {\text{D}^{\text{co}}\text{Qcoh}}_{\mathbb{ G}_{m}\times G}(\text{V}({\mathcal{E}}(\unicode[STIX]{x1D703})),\unicode[STIX]{x1D703},Q_{s})\end{eqnarray}$$
is an equivalence. Let
$$\begin{eqnarray}\unicode[STIX]{x1D700}_{\mathbb{G}_{m}\times G}:\text{id}_{{\text{D}^{\text{co}}\text{Qcoh}}_{\mathbb{ G}_{m}\times G}(Z,\unicode[STIX]{x1D703},0)}\rightarrow p_{\ast }i^{!}\circ i_{\ast }p^{\ast }\end{eqnarray}$$
be the adjunction morphism. To show that the functor
$i_{\ast }p^{\ast }:{\text{D}^{\text{co}}\text{Qcoh}}_{\mathbb{G}_{m}\times G}(Z,\unicode[STIX]{x1D703},0)\rightarrow {\text{D}^{\text{co}}\text{Qcoh}}_{\mathbb{G}_{m}\times G}(\text{V}({\mathcal{E}}(\unicode[STIX]{x1D703})),\unicode[STIX]{x1D703},Q_{s})$
is fully faithful, we will prove that the adjunction morphism
$\unicode[STIX]{x1D700}_{\mathbb{G}_{m}\times G}$
is an isomorphism of functors. For this, it suffices to show that for any object
$F\in {\text{D}^{\text{co}}\text{Qcoh}}_{\mathbb{G}_{m}\times G}(Z,\unicode[STIX]{x1D703},0)$
the cone
$C_{\mathbb{G}_{m}\times G}(F)$
of the morphism
$\unicode[STIX]{x1D700}_{\mathbb{G}_{m}\times G}(F):F\rightarrow p_{\ast }i^{!}\circ i_{\ast }p^{\ast }(F)$
is the zero object. Recall that the categories
$\text{Qcoh}_{\mathbb{G}_{m}}Z$
and
$\text{Qcoh}_{\mathbb{G}_{m}\times G}Z$
are equivalent to the categories
$\text{Qcoh}[Z/\mathbb{G}_{m}]$
and
$\text{Qcoh}_{G}[Z/\mathbb{G}_{m}]$
, respectively, where
$[Z/\mathbb{G}_{m}]$
denotes the quotient stack, and we can consider the restriction and the induction functors for algebraic stacks as in § 2.6. Let
$\unicode[STIX]{x1D70B}_{G}:\text{Qcoh}_{\mathbb{G}_{m}\times G}Z\rightarrow \text{Qcoh}_{\mathbb{G}_{m}}Z$
be the functor corresponding to the restriction functor
$\text{Res}_{G}:\text{Qcoh}_{G}[Z/\mathbb{G}_{m}]\rightarrow \text{Qcoh}[Z/\mathbb{G}_{m}]$
via the equivalences
$\text{Qcoh}_{\mathbb{G}_{m}}Z\cong \text{Qcoh}[Z/\mathbb{G}_{m}]$
and
$\text{Qcoh}_{\mathbb{G}_{m}\times G}Z\cong \text{Qcoh}_{G}[Z/\mathbb{G}_{m}]$
. Then
$\unicode[STIX]{x1D70B}_{G}$
naturally induces the following exact functor
$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{G}:{\text{D}^{\text{co}}\text{Qcoh}}_{\mathbb{G}_{m}\times G}(Z,\unicode[STIX]{x1D703},0)\rightarrow {\text{D}^{\text{co}}\text{Qcoh}}_{\mathbb{G}_{m}}(Z,\unicode[STIX]{x1D712}_{1},0),\end{eqnarray}$$
and
$\unicode[STIX]{x1D70B}_{G}$
has the right adjoint functor
$\unicode[STIX]{x1D70E}_{G}:{\text{D}^{\text{co}}\text{Qcoh}}_{\mathbb{G}_{m}}(Z,\unicode[STIX]{x1D712}_{1},0)\rightarrow {\text{D}^{\text{co}}\text{Qcoh}}_{\mathbb{G}_{m}\times G}(Z,\unicode[STIX]{x1D703},0)$
induced by the induction functor. Since the argument in the proof of Lemma 2.33 works for algebraic stacks, the adjunction morphism
$\text{id}\rightarrow \unicode[STIX]{x1D70E}_{G}\circ \unicode[STIX]{x1D70B}_{G}$
is a split mono. Hence,
$\unicode[STIX]{x1D70B}_{G}$
is faithful. The object
$\unicode[STIX]{x1D70B}_{G}(C_{\mathbb{G}_{m}\times G}(F))$
is isomorphic to the cone
$C_{\mathbb{G}_{m}}(F)$
of the adjunction morphism
$\unicode[STIX]{x1D700}_{\mathbb{G}_{m}}(\unicode[STIX]{x1D70B}_{G}(F)):\unicode[STIX]{x1D70B}_{G}(F)\rightarrow p_{\ast }i^{!}\circ i_{\ast }p^{\ast }(\unicode[STIX]{x1D70B}_{G}(F))$
, and
$C_{\mathbb{G}_{m}}(F)$
is the zero object since the functor
$i_{\ast }p^{\ast }:{\text{D}^{\text{co}}\text{Qcoh}}_{\mathbb{G}_{m}}(Z,\unicode[STIX]{x1D712}_{1},0)\rightarrow {\text{D}^{\text{co}}\text{Qcoh}}_{\mathbb{G}_{m}}(\text{V}({\mathcal{E}}(\unicode[STIX]{x1D712}_{1})),\unicode[STIX]{x1D712}_{1},Q_{s})$
is fully faithful. Hence we see that the object
$C_{\mathbb{G}_{m}\times G}(F)$
is also the zero object since
$\unicode[STIX]{x1D70B}_{G}$
is faithful. By an identical argument, we see that the right adjoint functor
$$\begin{eqnarray}p_{\ast }i^{!}:{\text{D}^{\text{co}}\text{Qcoh}}_{\mathbb{ G}_{m}\times G}(\text{V}({\mathcal{E}}(\unicode[STIX]{x1D703})),\unicode[STIX]{x1D703},Q_{s})\rightarrow {\text{D}^{\text{co}}\text{Qcoh}}_{\mathbb{ G}_{m}\times G}(Z,\unicode[STIX]{x1D703},0)\end{eqnarray}$$
is also fully faithful. Hence, the functor
$$\begin{eqnarray}i_{\ast }p^{\ast }:{\text{D}^{\text{co}}\text{Qcoh}}_{\mathbb{ G}_{m}\times G}(Z,\unicode[STIX]{x1D703},0)\rightarrow {\text{D}^{\text{co}}\text{Qcoh}}_{\mathbb{ G}_{m}\times G}(\text{V}({\mathcal{E}}(\unicode[STIX]{x1D703})),\unicode[STIX]{x1D703},Q_{s})\end{eqnarray}$$
is an equivalence.
By Proposition 2.25(1), we see that the equivalence
$i_{\ast }p^{\ast }:{\text{D}^{\text{co}}\text{Qcoh}}_{\mathbb{G}_{m}\times G}(Z,\unicode[STIX]{x1D703},0)\rightarrow {\text{D}^{\text{co}}\text{Qcoh}}_{\mathbb{G}_{m}\times G}(\text{V}({\mathcal{E}}(\unicode[STIX]{x1D703})),\unicode[STIX]{x1D703},Q_{s})$
induces an equivalence of the compact objects
$$\begin{eqnarray}i_{\ast }p^{\ast }:\overline{\text{Dcoh}_{\mathbb{ G}_{m}\times G}(Z,\unicode[STIX]{x1D703},0)}\rightarrow \overline{\text{Dcoh}_{\mathbb{G}_{m}\times G}(\text{V}({\mathcal{E}}(\unicode[STIX]{x1D703})),\unicode[STIX]{x1D703},Q_{s})},\end{eqnarray}$$
where
$\overline{(-)}$
denotes the idempotent completion of
$(-)$
. But
$\text{Dcoh}_{\mathbb{G}_{m}\times G}(Z,\unicode[STIX]{x1D703},0)$
on the left-hand side is already idempotent complete since it is equivalent to
$\text{D}^{\text{b}}(\text{coh}_{G}Z)$
. Hence, the functor
$$\begin{eqnarray}i_{\ast }p^{\ast }:\text{Dcoh}_{\mathbb{ G}_{m}\times G}(Z,\unicode[STIX]{x1D703},0)\rightarrow \text{Dcoh}_{\mathbb{G}_{m}\times G}(\text{V}({\mathcal{E}}(\unicode[STIX]{x1D703})),\unicode[STIX]{x1D703},Q_{s})\end{eqnarray}$$
is an equivalence. ◻
5 Orlov’s theorem for gauged LG models
In this section, we obtain a gauged LG version of the following theorem of Orlov.
Theorem 5.1 [Reference OrlovOrl09, Theorem 40].
Let
$X\subset \mathbb{P}_{k}^{N-1}$
be the hypersurface defined by a section
$f\in \unicode[STIX]{x1D6E4}(\mathbb{P}_{k}^{N-1},{\mathcal{O}}(d))$
. Denote by
$F$
the corresponding homogeneous polynomial.
-
(1) If
$d<N$
, there is a semi-orthogonal decomposition
$$\begin{eqnarray}\text{D}^{\text{b}}(\text{coh}X)=\langle {\mathcal{O}}_{X}(d-N+1),\ldots ,{\mathcal{O}}_{X},\text{Dcoh}_{\mathbb{G}_{m}}(\mathbb{A}_{k}^{N},\unicode[STIX]{x1D712}_{d},F)\rangle .\end{eqnarray}$$
-
(2) If
$d=N$
, there is an equivalence
$$\begin{eqnarray}\text{D}^{\text{b}}(\text{coh}X)\cong \text{Dcoh}_{\mathbb{G}_{m}}(\mathbb{A}_{k}^{N},\unicode[STIX]{x1D712}_{d},F).\end{eqnarray}$$
-
(3) If
$d>N$
, there is a semi-orthogonal decomposition
$$\begin{eqnarray}\text{Dcoh}_{\mathbb{G}_{m}}(\mathbb{A}_{k}^{N},\unicode[STIX]{x1D712}_{d},F)=\langle k,\ldots ,k(N-d+1),\text{D}^{\text{b}}(\text{coh}X)\rangle .\end{eqnarray}$$
We combine the main result with the theory of variations of GIT quotients to obtain a gauged LG version of the above theorem. For the theory of variations of GIT quotients, see [Reference Ballard, Favero and KatzarkovBFK12] or [Reference Ballard, Deliu, Favero, Isik and KatzarkovBDFIK14, §2]. This kind of approach to Orlov’s theorem appeared in [Reference ShipmanShi12, Reference Ballard, Favero and KatzarkovBFK12], and [Reference Ballard, Deliu, Favero, Isik and KatzarkovBDFIK14], and our argument is similar to that in [Reference Ballard, Deliu, Favero, Isik and KatzarkovBDFIK14, §3].
Let
$S$
be a smooth quasi-projective variety with
$\mathbb{G}_{m}$
-action, and set
$$\begin{eqnarray}Q:=S\times \mathbb{A}^{N}\times \mathbb{A}^{1}.\end{eqnarray}$$
For
$i=1,2$
, set
$G_{i}:=\mathbb{G}_{m}$
, and let
$G:=G_{1}\times G_{2}$
. For a positive integer
$d>1$
, we define a
$G$
-action on
$Q$
as follows:
$$\begin{eqnarray}G\times Q\ni (g_{1},g_{2})\times (s,v_{1},\ldots ,v_{N},u)\mapsto (g_{2}\cdot s,g_{1}v_{1},\ldots ,g_{1}v_{N},g_{1}^{-d}g_{2}u)\in Q,\end{eqnarray}$$
where the action
$\cdot$
is the original
$\mathbb{G}_{m}$
-action on
$S$
. Let
$\unicode[STIX]{x1D706}:\mathbb{G}_{m}\rightarrow G$
be the character defined by
$\unicode[STIX]{x1D706}(a):=(a,1)$
. Denote by
$Z_{\unicode[STIX]{x1D706}}$
the fixed locus of
$\unicode[STIX]{x1D706}$
-action on
$Q$
. Then
$Z_{\unicode[STIX]{x1D706}}$
coincides with the zero section
$S\times 0\times 0\subset Q$
. Furthermore, set
$S_{+}:=\{q\in Q\mid \lim _{a\rightarrow 0}\unicode[STIX]{x1D706}(a)q\in Z_{\unicode[STIX]{x1D706}}\}$
and
$S_{-}:=\{q\in Q\mid \lim _{a\rightarrow 0}\unicode[STIX]{x1D706}(a)^{-1}q\in Z_{\unicode[STIX]{x1D706}}\}$
. Then
$$\begin{eqnarray}S_{+}=S\times \mathbb{A}^{N}\times 0\quad \text{and}\quad S_{-}=S\times 0\times \mathbb{A}^{1}.\end{eqnarray}$$
Denote by
$Q_{+}$
(respectively
$Q_{-}$
) be the complement of
$S_{+}$
(respectively
$S_{-}$
) in
$Q$
. Then the stratifications
$$\begin{eqnarray}Q=Q_{+}\sqcup S_{+}\quad \text{and}\quad Q=Q_{-}\sqcup S_{-}\end{eqnarray}$$
are elementary wall crossings in the sense of [Reference Ballard, Favero and KatzarkovBFK12].
Let
$W:S\rightarrow \mathbb{A}^{1}$
be a
$\unicode[STIX]{x1D712}_{1}$
-semi-invariant function which is flat. Let
$f\in \unicode[STIX]{x1D6E4}(\mathbb{P}_{S}^{N-1},{\mathcal{O}}(d))^{\mathbb{G}_{m}}$
be a non-zero
$\mathbb{G}_{m}$
-invariant section, and denote by
$F:\mathbb{A}_{S}^{N}\rightarrow \mathbb{A}^{1}$
the corresponding regular function. Since
$Q$
is the trivial line bundle over
$\mathbb{A}_{S}^{N}$
, the function
$F$
induces a regular function
$\widetilde{F}:Q\rightarrow \mathbb{A}^{1}$
. Then the function
$$\begin{eqnarray}W+\widetilde{F}:Q\rightarrow \mathbb{A}^{1}\end{eqnarray}$$
is a
$\unicode[STIX]{x1D712}_{0,1}$
-semi-invariant regular function, where
$W$
is the pull-back of
$W:S\rightarrow \mathbb{A}^{1}$
by the projection
$Q\rightarrow S$
, and
$\unicode[STIX]{x1D712}_{0,1}:G\rightarrow \mathbb{G}_{m}$
is the character defined by
$\unicode[STIX]{x1D712}_{0,1}(g_{1},g_{2}):=g_{2}$
. By [Reference Ballard, Favero and KatzarkovBFK12, Lemma 3.4.4] and [Reference Ballard, Favero and KatzarkovBFK12, Theorem 3.5.2], we have the following proposition.
Proposition 5.2. Let
$t_{\pm }$
be the
$\unicode[STIX]{x1D706}$
-weight of the restriction of relative canonical bundle
$\unicode[STIX]{x1D714}_{S_{\pm }/Q}$
to
$Z_{\unicode[STIX]{x1D706}}$
, and set
$\unicode[STIX]{x1D707}:=-t_{+}+t_{-}$
. Let
$\unicode[STIX]{x1D712}:G\rightarrow \mathbb{G}_{m}$
be the character defined by
$\unicode[STIX]{x1D712}(g_{1},g_{2}):=g_{1}g_{2}$
.
-
(1) If
$\unicode[STIX]{x1D707}<0$
, there exist fully faithful functors and we have the following semi-orthogonal decomposition
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F6}_{-}:\text{Dcoh}_{G/\unicode[STIX]{x1D706}}(Z_{\unicode[STIX]{x1D706}},\unicode[STIX]{x1D712}_{1},W+\widetilde{F}) & \rightarrow & \displaystyle \text{Dcoh}_{G}(Q_{-},\unicode[STIX]{x1D712}_{0,1},W+\widetilde{F}),\nonumber\\ \displaystyle \unicode[STIX]{x1D6F7}_{-}:\text{Dcoh}_{G}(Q_{+},\unicode[STIX]{x1D712}_{0,1},W+\widetilde{F}) & \rightarrow & \displaystyle \text{Dcoh}_{G}(Q_{-},\unicode[STIX]{x1D712}_{0,1},W+\widetilde{F}),\nonumber\end{eqnarray}$$
where we denote by
$$\begin{eqnarray}\text{Dcoh}_{G}(Q_{-},\unicode[STIX]{x1D712}_{0,1},W+\widetilde{F})=\langle \unicode[STIX]{x1D6F6}_{-}(\unicode[STIX]{x1D707}+1),\ldots ,\unicode[STIX]{x1D6F6}_{-},\unicode[STIX]{x1D6F7}_{-}(\text{Dcoh}_{G}(Q_{+},\unicode[STIX]{x1D712}_{0,1},W+\widetilde{F}))\rangle ,\end{eqnarray}$$
$\unicode[STIX]{x1D6F6}_{-}(n)$
the essential image of the composition
$(-)\otimes {\mathcal{O}}(\unicode[STIX]{x1D712}^{n})\circ \unicode[STIX]{x1D6F6}_{-}$
.
-
(2) If
$\unicode[STIX]{x1D707}=0$
, we have an equivalence
$$\begin{eqnarray}\text{Dcoh}_{G}(Q_{-},\unicode[STIX]{x1D712}_{0,1},W+\widetilde{F})\cong \text{Dcoh}_{G}(Q_{+},\unicode[STIX]{x1D712}_{0,1},W+\widetilde{F}).\end{eqnarray}$$
-
(3) If
$\unicode[STIX]{x1D707}>0$
, there exist fully faithful functors and we have the following semi-orthogonal decomposition
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F6}_{+}:\text{Dcoh}_{G/\unicode[STIX]{x1D706}}(Z_{\unicode[STIX]{x1D706}},\unicode[STIX]{x1D712}_{1},W+\widetilde{F}) & \rightarrow & \displaystyle \text{Dcoh}_{G}(Q_{+},\unicode[STIX]{x1D712}_{0,1},W+\widetilde{F}),\nonumber\\ \displaystyle \unicode[STIX]{x1D6F7}_{+}:\text{Dcoh}_{G}(Q_{-},\unicode[STIX]{x1D712}_{0,1},W+\widetilde{F}) & \rightarrow & \displaystyle \text{Dcoh}_{G}(Q_{+},\unicode[STIX]{x1D712}_{0,1},W+\widetilde{F}),\nonumber\end{eqnarray}$$
where we denote by
$$\begin{eqnarray}\text{Dcoh}_{G}(Q_{+},\unicode[STIX]{x1D712}_{0,1},W+\widetilde{F})=\langle \unicode[STIX]{x1D6F6}_{+},\ldots ,\unicode[STIX]{x1D6F6}_{+}(-\unicode[STIX]{x1D707}+1),\unicode[STIX]{x1D6F7}_{+}(\text{Dcoh}_{G}(Q_{-},\unicode[STIX]{x1D712}_{0,1},W+\widetilde{F}))\rangle ,\end{eqnarray}$$
$\unicode[STIX]{x1D6F6}_{+}(n)$
the essential image of the composition
$(-)\otimes {\mathcal{O}}(\unicode[STIX]{x1D712}^{n})\circ \unicode[STIX]{x1D6F6}_{+}$
.
Since
$Z_{\unicode[STIX]{x1D706}}=S\times 0\times 0$
, the function
$\widetilde{F}$
vanishes on
$Z_{\unicode[STIX]{x1D706}}\subset Q$
. Hence, we have
$$\begin{eqnarray}\text{Dcoh}_{G/\unicode[STIX]{x1D706}}(Z_{\unicode[STIX]{x1D706}},\unicode[STIX]{x1D712}_{1},W+\widetilde{F})\cong \text{Dcoh}_{\mathbb{G}_{m}}(S,\unicode[STIX]{x1D712}_{1},W).\end{eqnarray}$$
Next, we have
$$\begin{eqnarray}Q_{-}=S\times \mathbb{A}^{N}\setminus 0\times \mathbb{A}^{1}.\end{eqnarray}$$
Since
$F|_{S\times \mathbb{A}^{N}\setminus 0}\in \unicode[STIX]{x1D6E4}(S\times \mathbb{A}^{N}\setminus 0,{\mathcal{O}}(\unicode[STIX]{x1D712}_{-d,0})^{\vee })^{G}$
and
$Q_{-}=\text{V}({\mathcal{O}}(\unicode[STIX]{x1D712}_{-d,1}))$
, Theorem 4.2 implies the following equivalence;
$$\begin{eqnarray}\text{Dcoh}_{G}(Q_{-},\unicode[STIX]{x1D712}_{0,1},W+\widetilde{F})\cong \text{Dcoh}_{G}(Z,\unicode[STIX]{x1D712}_{0,1},W),\end{eqnarray}$$
where
$Z\subset S\times \mathbb{A}^{N}\setminus 0$
is the zero scheme of
$F$
. Moreover, the quotient stack
$[Z/G_{1}]$
is isomorphic to the hypersurface
$X$
in the projective space bundle
$\mathbb{P}_{S}^{N-1}$
over
$S$
defined by the invariant section
$f\in \unicode[STIX]{x1D6E4}(\mathbb{P}_{S}^{N-1},{\mathcal{O}}(d))^{G_{2}}$
. Hence, we have an equivalence
$$\begin{eqnarray}\text{Dcoh}_{G}(Z,\unicode[STIX]{x1D712}_{0,1},W)\cong \text{Dcoh}_{G_{2}}(X,\unicode[STIX]{x1D712}_{1},W).\end{eqnarray}$$
On the other hand, we have
$$\begin{eqnarray}Q_{+}=S\times \mathbb{A}^{N}\times \mathbb{A}^{1}\setminus 0.\end{eqnarray}$$
We consider another action of
$G$
on
$Q_{+}$
as follows:
$$\begin{eqnarray}G\times Q_{+}\ni (g_{1},g_{2})\times (s,v,u)\mapsto (g_{1}^{d}\cdot s,g_{1}v,g_{1}^{-d}g_{2}u)\in Q_{+}.\end{eqnarray}$$
We denote by
$\widetilde{Q_{+}}$
the new
$G$
-variety. Then we have a
$G$
-equivariant isomorphism
$$\begin{eqnarray}\unicode[STIX]{x1D711}:\widetilde{Q_{+}}\xrightarrow[{}]{{\sim}}Q_{+},\end{eqnarray}$$
given by
$\unicode[STIX]{x1D711}(s,v,u):=(u\cdot s,v,u),$
where
$u\in \mathbb{A}^{1}\setminus 0$
is considered as a point in
$\mathbb{G}_{m}$
. Since
$G_{2}$
trivially acts on the first two components
$S\times \mathbb{A}^{N}$
of
$\widetilde{Q_{+}}$
, we have
$$\begin{eqnarray}[\widetilde{Q_{+}}/G_{2}]\cong S\times \mathbb{A}^{N}\times [\mathbb{A}^{1}\setminus 0/G_{2}]\cong \mathbb{A}_{S}^{N}.\end{eqnarray}$$
Hence, we have the following equivalence
$$\begin{eqnarray}\text{Dcoh}_{G}(Q_{+},\unicode[STIX]{x1D712}_{0,1},W+\widetilde{F})\cong \text{Dcoh}_{G_{1}}(\mathbb{A}_{S}^{N},\unicode[STIX]{x1D712}_{d},W+F),\end{eqnarray}$$
where, on the right-hand side, the
$G_{1}$
-action is given by the following
$$\begin{eqnarray}G_{1}\times S\times \mathbb{A}^{N}\ni g_{1}\times (s,v)\mapsto (g_{1}^{d}\cdot s,g_{1}v).\end{eqnarray}$$
Finally, note that
$\unicode[STIX]{x1D707}=d-N$
and that the twisting by the
$G$
-equivariant invertible sheaf
${\mathcal{O}}(\unicode[STIX]{x1D712})$
corresponds to the twisting, in
$\text{Dcoh}_{G_{2}}(X,\unicode[STIX]{x1D712}_{1},W)$
, by the
$G_{2}$
-equivariant invertible sheaf
${\mathcal{O}}(1)$
on
$X$
which is the pull-back of the tautological
$G_{2}$
-equivariant invertible sheaf on
$\mathbb{P}_{S}^{N-1}$
. Combining Proposition 5.2 and the above argument, we obtain the following gauged LG version of Orlov’s theorem.
Let
$S$
be a smooth quasi-projective variety with a
$\mathbb{G}_{m}$
-action, and let
$W:S\rightarrow \mathbb{A}^{1}$
be a
$\unicode[STIX]{x1D712}_{1}$
-semi-invariant regular function which is flat. Consider
$\mathbb{G}_{m}$
-actions on
$\mathbb{A}_{S}^{N}$
and on
$\mathbb{P}_{S}^{N-1}$
given by
$$\begin{eqnarray}\displaystyle & \displaystyle \mathbb{G}_{m}\times \mathbb{A}_{S}^{N}\ni t\times (s,v_{1},\ldots ,v_{N})\mapsto (t^{d}\cdot s,tv_{1},\ldots ,tv_{N})\in \mathbb{A}_{S}^{N}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle \mathbb{G}_{m}\times \mathbb{P}_{S}^{N-1}\ni t\times (s,v_{1}:\cdots :v_{N})\mapsto (t\cdot s,v_{1}:\cdots :v_{N})\in \mathbb{P}_{S}^{N-1}. & \displaystyle \nonumber\end{eqnarray}$$
Theorem 5.3. For
$d>1$
, let
$f\in \unicode[STIX]{x1D6E4}(\mathbb{P}_{S}^{N-1},{\mathcal{O}}(d))^{\mathbb{G}_{m}}$
be a non-zero invariant section, and let
$F:\mathbb{A}_{S}^{N}\rightarrow \mathbb{A}^{1}$
be the corresponding
$\unicode[STIX]{x1D712}_{d}$
-semi-invariant regular function. Let
$X\subset \mathbb{P}_{S}^{N-1}$
be the hypersurface defined by
$f$
, and assume that the morphism
$W|_{X}$
is flat.
-
(1) If
$d<N$
, there are fully faithful functors and there is a semi-orthogonal decomposition
$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F7}:\text{Dcoh}_{\mathbb{G}_{m}}(\mathbb{A}_{S}^{N},\unicode[STIX]{x1D712}_{d},W+F)\rightarrow \text{Dcoh}_{\mathbb{G}_{m}}(X,\unicode[STIX]{x1D712}_{1},W), & \displaystyle \nonumber\\ \displaystyle & \displaystyle \unicode[STIX]{x1D6F6}:\text{Dcoh}_{\mathbb{G}_{m}}(S,\unicode[STIX]{x1D712}_{1},W)\rightarrow \text{Dcoh}_{\mathbb{G}_{m}}(X,\unicode[STIX]{x1D712}_{1},W), & \displaystyle \nonumber\end{eqnarray}$$
where
$$\begin{eqnarray}\text{Dcoh}_{\mathbb{G}_{m}}(X,\unicode[STIX]{x1D712}_{1},W)=\langle \unicode[STIX]{x1D6F6}_{d-N+1},\ldots ,\unicode[STIX]{x1D6F6}_{0},\unicode[STIX]{x1D6F7}(\text{Dcoh}_{\mathbb{G}_{m}}(\mathbb{A}_{S}^{N},\unicode[STIX]{x1D712}_{d},W+F))\rangle ,\end{eqnarray}$$
$\unicode[STIX]{x1D6F6}_{i}$
denotes the essential image of the composition
$(-)\otimes {\mathcal{O}}(i)\circ \unicode[STIX]{x1D6F6}$
.
-
(2) If
$d=N$
, we have an equivalence
$$\begin{eqnarray}\text{Dcoh}_{\mathbb{G}_{m}}(X,\unicode[STIX]{x1D712}_{1},W)\cong \text{Dcoh}_{\mathbb{G}_{m}}(\mathbb{A}_{S}^{N},\unicode[STIX]{x1D712}_{d},W+F).\end{eqnarray}$$
-
(3) If
$d>N$
, there are fully faithful functors and there is a semi-orthogonal decomposition
$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F9}:\text{Dcoh}_{\mathbb{G}_{m}}(X,\unicode[STIX]{x1D712}_{1},W)\rightarrow \text{Dcoh}_{\mathbb{G}_{m}}(\mathbb{A}_{S}^{N},\unicode[STIX]{x1D712}_{d},W+F), & \displaystyle \nonumber\\ \displaystyle & \displaystyle \unicode[STIX]{x1D6F6}:\text{Dcoh}_{\mathbb{G}_{m}}(S,\unicode[STIX]{x1D712}_{1},W)\rightarrow \text{Dcoh}_{\mathbb{G}_{m}}(\mathbb{A}_{S}^{N},\unicode[STIX]{x1D712}_{d},W+F), & \displaystyle \nonumber\end{eqnarray}$$
where
$$\begin{eqnarray}\text{Dcoh}_{\mathbb{G}_{m}}(\mathbb{A}_{S}^{N},\unicode[STIX]{x1D712}_{d},W+F)=\langle \unicode[STIX]{x1D6F6}_{0},\ldots ,\unicode[STIX]{x1D6F6}_{N-d+1},\unicode[STIX]{x1D6F9}(\text{Dcoh}_{\mathbb{G}_{m}}(X,\unicode[STIX]{x1D712}_{1},W))\rangle ,\end{eqnarray}$$
$\unicode[STIX]{x1D6F6}_{i}$
denotes the essential image of the composition
$(-)\otimes {\mathcal{O}}(\unicode[STIX]{x1D712}_{i})\circ \unicode[STIX]{x1D6F6}$
.
Remark 5.4. (1) We can view Orlov’s theorem (Theorem 5.1) as the case when
$S=\operatorname{Spec}k$
and
$W=0$
in the above theorem.
(2) If
$N>1$
, the assumption that
$W|_{X}$
is flat is satisfied whenever
$W:S\rightarrow \mathbb{A}^{1}$
is flat.
(3) For positive integers
$a_{1},\ldots ,a_{N}$
, applying the similar argument to the
$G$
-action on
$Q$
defined by
$$\begin{eqnarray}G\times Q\ni (g_{1},g_{2})\times (s,v_{1},\ldots ,v_{N},u)\mapsto (g_{2}\cdot s,g_{1}^{a_{1}}v_{1},\ldots ,g_{1}^{a_{N}}v_{N},g_{1}^{-d}g_{2}u)\in Q,\end{eqnarray}$$
we can obtain the similar result for the hypersurface
$X$
in weighted projective stack bundle
$\mathbb{P}_{S}^{N-1}(a_{1},\ldots ,a_{N}):=[S\times \mathbb{A}^{N}\setminus 0/G_{1}]$
over
$S$
defined by the section corresponding to a
$G_{1}$
-invariant section
$F\in \unicode[STIX]{x1D6E4}(\mathbb{A}_{S}^{N},{\mathcal{O}}(\unicode[STIX]{x1D712}_{d}))^{G_{1}}$
.
(4) Of course, Orlov’s theorem in [Reference OrlovOrl09] is much more general. It covers non-commutative situations unlike our setting.
Acknowledgements
The main part of this article was written whilst the author was visiting the School of Mathematics at the University of Edinburgh during the period from October 2015 to March 2016. He appreciates the hospitality, and he would like to express his deep gratitude to Michael Wemyss for fruitful discussions and many suggestions to improve this paper. The author is also grateful to his supervisor Hokuto Uehara, Timothy Logvinenko and Ed Segal for their valuable advices. Finally, I would like to thank an anonymous referee for valuable suggestions which helped improve the paper. The author is a Research Fellow of the Japan Society for the Promotion of Science. He is partially supported by Grant-in-Aid for JSPS Fellows no. 26-6240.
