# Survey on homological flips and homological flops

Wai-Kit Yeung  Kavli IPMU, The University of Tokyo<EMAIL_ADDRESS>
###### Abstract.

We give a survey for the results in [13, 14, 15], which attempts to relate the
derived categories under general classes of flips and flops. We indicate how
the approach fails because of what appears to be a formal problem. We give
some ideas, and record some failed attempts, to fix this problem. We also
present some new examples.

###### Contents

  1. 1 Log flips
  2. 2 Flips and flops
  3. 3 Weight truncation
  4. 4 Some examples

##  1\. Log flips

In this article, we give a survey for an approach developed in [13, 14, 15] to
the problem of relating derived categories under flips and flops. The notions
of flips and flops are sometimes a source of confusion because in the
literature (especially those on derived categories) the terminology is
sometimes used to refer to the closely related notions of $K$-dominant and
$K$-equivalent birational maps. For the sake of clarity, we will recall the
notions of flips and flops. In this section, we will start with the notion of
a log flip (which generalizes both flips and flops), and we describe an
algebraic setup associated to a log flip. In the next section, we will define
flips and flops, and elaborate more on this algebraic setup.

We do not assume knowledge of birational geometry from the reader. Most
notions from birational geometry that we use will be reminded. We will also
add several “checkpoints” in this article that summarize some essential
information of the situation, so that readers less familiar with some parts of
birational geometry or homological algebra will be able to read the rest of
the article even if (s)he does not follow the arguments that lead to that
description.

######  Definition 1.1.

A _log flip_ consists of birational projective morphisms between normal
varieties

${X^{-}}$${X^{+}}$${Y}$$\scriptstyle{\pi^{-}}$$\scriptstyle{\pi^{+}}$

such that both $\pi^{-}$ and $\pi^{+}$ are _small_ , meaning that ${\rm
Ex}(\pi^{\pm})\subset X^{\pm}$ have codimension $\geq 2$, together with Weil
divisors $D^{+},D^{-},D^{0}$ on $X^{+}$, $X^{-}$, $Y$ respectively, strict
transforms of each other, and satisfy

  1. (1)

$D^{+}$ is $\mathbb{Q}$-Cartier and $\pi^{+}$-ample.

  2. (2)

$-D^{-}$ is $\mathbb{Q}$-Cartier and $\pi^{-}$-ample.

Recall that a Weil divisor $D\in{\rm WDiv}(X)$ is said to be
$\mathbb{Q}$-Cartier if $mD$ is Cartier for some
$m\in\mathbb{Z}\setminus\\{0\\}$. This allows one to discuss (relative)
ampleness of $D$. By replacing them with a suitable multiple, one may take
$D^{\pm}$ to be Cartier111Sometimes one would like to think of $D^{\pm}$ as
part of the structure of a log flip, in which case one will have to tackle the
case when $D^{\pm}$ are not Cartier.. However, notice that $D^{0}$ is never
Cartier (unless when $\pi^{\pm}$ are the identity maps).

Since $\pi^{\pm}$ are small, we have

(1.1)
$\pi^{\pm}_{*}\mathcal{O}_{X^{\pm}}(iD^{\pm})=\mathcal{O}_{Y}(iD^{0})\qquad\text{for
any }i\in\mathbb{Z}$

Let $\mathcal{A}$ be the sheaf of $\mathbb{Z}$-graded rings over $Y$ defined
by $\mathcal{A}=\bigoplus_{i\in\mathbb{Z}}\mathcal{O}_{Y}(iD^{0})$, then by
(1.1) and by relative ampleness of $D^{+}$ and $-D^{-}$, we have

(1.2) $X^{+}=\underline{{\rm Proj}}^{+}_{Y}(\mathcal{A}):=\underline{{\rm
Proj}}_{Y}(\mathcal{A}_{\geq 0})\qquad\text{and}\qquad X^{-}=\underline{{\rm
Proj}}^{-}_{Y}(\mathcal{A}):=\underline{{\rm Proj}}_{Y}(\mathcal{A}_{\leq 0})$

Moreover, the reflexive sheaves $\mathcal{O}_{X^{\pm}}(iD^{\pm})$ also admit a
description in terms of $\mathcal{A}$. Namely, denote by ${\rm
Gr}(\mathcal{A})$ the category of quasi-coherent sheaves of graded
$\mathcal{A}$-modules. Then any $M\in{\rm Gr}(\mathcal{A})$ is also an object
in ${\rm Gr}(\mathcal{A}_{\geq 0})$, and hence determines an associated sheaf
$\widetilde{M}\in{\rm QCoh}(\underline{{\rm Proj}}^{+}_{Y}(\mathcal{A}))$. The
same is true for the negative direction. Then we have

$\mathcal{O}_{X^{\pm}}(iD^{\pm})\cong\widetilde{\mathcal{A}(i)}\qquad\text{under
the identifications }\eqref{X_Proj_1}.$

Everything we discuss here is local over $Y$, so we may assume that $Y$ is
affine, in which case we summarize the essential information in the following

######  Checkpoint 1.2.

Every log flip is locally described by a Noetherian $\mathbb{Z}$-graded ring
$A$. Namely, we have $R=A_{0}$, $Y={\rm Spec}\,R$, $X^{\pm}={\rm
Proj}^{\pm}(A)$. Let $\mathcal{O}_{X^{\pm}}(i):=\widetilde{A(i)}$, then we
have $\pi^{\pm}_{*}(\mathcal{O}_{X^{\pm}}(i))=A_{i}$.

Here we mentioned that $A$ is Noetherian. It is relevant to our situation
because of the following result (see, e.g., [2, Theorem 1.5.5]):

######  Proposition 1.3.

Let $A$ be a $\mathbb{Z}$-graded ring. The the followings are equivalent:

  1. (1)

$A$ is a Noetherian ring;

  2. (2)

every graded ideal of $A$ is finitely generated;

  3. (3)

$A_{0}$ is Noetherian, and both $A_{\geq 0}$ and $A_{\leq 0}$ are finitely
generated over $A_{0}$;

  4. (4)

$A_{0}$ is Noetherian, and $A$ is finitely generated over $A_{0}$.

We wish to study the derived category of $X^{\pm}$ in terms of the commutative
algebra of $A$. The relation is obtained by Serre’s equivalence. We have
already seen above that there is a functor $(-)^{\sim}:{\rm
Gr}(A)\rightarrow{\rm QCoh}(X^{+})$. Let ${\rm Tor}^{+}\subset{\rm Gr}(A)$ be
the subcategory consisting of graded modules $M$ such that for every $x\in M$
there exists $s\in\mathbb{N}$ such that $x\cdot(A_{>0})^{s}=0$. Then the
functor $(-)^{\sim}$ passes to the Serre quotient

(1.3) $(-)^{\sim}\,:\,{\rm Gr}(A)/{\rm Tor}^{+}\,\rightarrow\,{\rm
QCoh}(X^{+})$

The usual statement of Serre equivalence asserts that the functor (1.3) is an
equivalence if $A$ is non-negatively graded and generated in degree $0$ and
$1$. If one sharpen each step of the usual proof, one have the following
stronger version:

######  Theorem 1.4 ([15], Theorem 3.15).

Let $\mathcal{O}_{X^{+}}(i):=\widetilde{A(i)}$ as above. Assume that the
natural map

(1.4)
$\mathcal{O}_{X^{+}}(i)\otimes_{\mathcal{O}_{X^{+}}}\mathcal{O}_{X^{+}}(j)\,\rightarrow\,\mathcal{O}_{X^{+}}(i+j)$

is an isomorphism for any $i,j\in\mathbb{Z}$. Then the functor (1.3) is an
equivalence.

In the case of log flips, the assumption of this Theorem is satisfied if and
only if $D^{+}$ is Cartier. Motivated by this example, we will say that a
Noetherian $\mathbb{Z}$-graded ring $A$ is _positively Cartier_ if (1.4) is an
isomorphism for any $i,j\in\mathbb{Z}$.

At the level of derived categories, Serre’s equivalence can be described in
terms of a semi-orthogonal decomposition. More precisely, there is a local
cohomology SOD (so called because of (1.6)):

(1.5) $\mathcal{D}({\rm
Gr}(A))\,=\,\langle\,\mathcal{D}_{I^{+}\text{-triv}}({\rm
Gr}(A))\,,\,\mathcal{D}_{{\rm Tor}^{+}}({\rm Gr}(A))\,\rangle$

Here, $\mathcal{D}({\rm Gr}(A))$ is the unbounded derived category of (not
necessarily finitely generated) graded modules. $\mathcal{D}_{{\rm
Tor}^{+}}({\rm Gr}(A))\subset\mathcal{D}({\rm Gr}(A))$ is the full
triangulated subcategory consisting of objects $M\in\mathcal{D}({\rm Gr}(A))$
such that each cohomology module $H^{i}(M)$ is in the Serre subcategory ${\rm
Tor}^{+}$. The inclusion $\iota:\mathcal{D}_{{\rm Tor}^{+}}({\rm
Gr}(A))\hookrightarrow\mathcal{D}({\rm Gr}(A))$ has a right adjoint denoted as
${\bm{R}}\Gamma_{I^{+}}$, whose kernel is by definition
$\mathcal{D}_{I^{+}\text{-triv}}({\rm Gr}(A))$.

As the notation suggests, ${\bm{R}}\Gamma_{I^{+}}(M)$ is the local cohomology
complex with respect to the graded ideal $I^{+}:=A_{>0}\cdot A$. If we denote
the decomposition sequence of the SOD (1.5) by

(1.6)
$\ldots\,\rightarrow\,{\bm{R}}\Gamma_{I^{+}}(M)\,\xrightarrow[]{\epsilon}\,M\,\xrightarrow[]{\eta}\,\check{\mathcal{C}}_{I^{+}}(M)\,\rightarrow\,\ldots$

then $\check{\mathcal{C}}_{I^{+}}(M)$ can be described as a certain Čech
complex, while ${\bm{R}}\Gamma_{I^{+}}(M)$ can be described as a certain
extended Čech complex. Explicitly, if $I^{+}$ is generated by homogeneous
elements $f_{1},\ldots,f_{r}\in A_{>0}$, then they are given by

(1.7)
$\begin{split}\check{\mathcal{C}}_{I^{+}}(M)\,&:=\,\bigl{[}\,\prod_{1\leq
i_{0}\leq r}M_{f_{i_{0}}}\,\rightarrow\,\prod_{1\leq i_{0}<i_{1}\leq
r}M_{f_{i_{0}}f_{i_{1}}}\,\rightarrow\,\ldots\,\rightarrow\,M_{f_{1}\ldots
f_{r}}\,\bigr{]}\\\
{\bm{R}}\Gamma_{I^{+}}(M)\,&:=\,\bigl{[}\,M\,\rightarrow\,\prod_{1\leq
i_{0}\leq r}M_{f_{i_{0}}}\,\rightarrow\,\prod_{1\leq i_{0}<i_{1}\leq
r}M_{f_{i_{0}}f_{i_{1}}}\,\rightarrow\,\ldots\,\rightarrow\,M_{f_{1}\ldots
f_{r}}\,\bigr{]}\end{split}$

where the differentials are the alternating sums of restriction maps.

Here and below, if $X$ is a scheme or stack, then $\mathcal{D}(X)$ will denote
the unbounded derived category of quasi-coherent sheaves
$\mathcal{D}(X):=\mathcal{D}({\rm QCoh}(X))$.

If $A$ is positively Cartier, then it is a formal consequence of Serre’s
equivalence that the functor $(-)^{\sim}:\mathcal{D}({\rm
Gr}(A))\rightarrow\mathcal{D}(X^{+})$ restricts to an equivalence
$\mathcal{D}_{I^{+}\text{-triv}}({\rm Gr}(A))\rightarrow\mathcal{D}(X^{+})$.
This is the basic setting we wish to exploit to pass from homological algebra
on ${\rm Gr}(A)$ to the derived category of $X^{+}$. We summarize the
situation in the following

######  Checkpoint 1.5.

For any Noetherian $\mathbb{Z}$-graded ring $A$, there is a local cohomology
SOD described by (1.5), (1.6), (1.7). If $A$ is positively Cartier then the
semi-orthogonal component $\mathcal{D}_{I^{+}\text{-triv}}({\rm Gr}(A))$ is
equivalent to $\mathcal{D}(X^{+}):=\mathcal{D}({\rm QCoh}(X^{+}))$. The same
is true for the negative side.

There is also a closely related stacky picture. While it will not be used in
any technical way in this work, it will serve as a useful viewpoint.

Let $\mathfrak{X}$ be the stack $\mathfrak{X}=[{\rm Spec}(A)/\mathbb{G}_{m}]$,
so that ${\rm Gr}(A)\simeq{\rm QCoh}(\mathfrak{X})$. Let
$\mathfrak{X}^{\pm}\subset\mathfrak{X}$ be the open substacks given by the
complements of the closed subsets $V(I^{\pm})\subset{\rm Spec}\,A$. Then
$\mathfrak{X}^{\pm}$ are Deligne-Mumford, whose coarse moduli space are
$X^{\pm}$. Serre’s equivalence always holds with $X^{\pm}$ replaced by
$\mathfrak{X}^{\pm}$. Moreover, $A$ is positively Cartier if and only if the
map $\mathfrak{X}^{+}\rightarrow X^{+}$ is an isomorphism. Under this stacky
viewpoint, the varieties $X^{\pm}$ (or rather their stacky versions
$\mathfrak{X}^{\pm}$) are related by a variation of GIT quotients.

######  Checkpoint 1.6.

The terms in the local cohomology sequence (1.6) (at least for $M=A$) admit
descriptions from two viewpoints

| $\pi^{+}:X^{+}\rightarrow Y$ | $j:\mathfrak{X}^{+}\subset\mathfrak{X}$
---|---|---
$A$ | $A_{i}=\pi^{+}_{*}(\mathcal{O}_{X^{+}}(i))$ | $A=\mathcal{O}_{\mathfrak{X}}$
$\check{\mathcal{C}}_{I^{+}}(A)$ | $\check{\mathcal{C}}_{I^{+}}(A)_{i}={\bm{R}}\pi^{+}_{*}(\mathcal{O}_{X^{+}}(i))$ | $\check{\mathcal{C}}_{I^{+}}(A)={\bm{R}}j_{*}j^{*}(\mathcal{O}_{\mathfrak{X}})$
${\bm{R}}\Gamma_{I^{+}}(A)$ | ${\bm{R}}\Gamma_{I^{+}}(A)_{i}[1]={\bm{R}}^{>0}\pi^{+}_{*}(\mathcal{O}_{X^{+}}(i))$ |  local cohomology along the unstable locus $V(I^{+})$

Here and below, a subscript $(-)_{i}$ will always mean its weight grading $i$
part. Thus, given a complex $M$ of graded $A$-modules, then $M_{i}$ is a
complex of $R$-module, where $R=A_{0}$. This gives a functor
$(-)_{i}:\mathcal{D}({\rm Gr}(A))\rightarrow\mathcal{D}(R)$. This explains the
meaning of the left column of this table. For example, the middle term means
that $\check{\mathcal{C}}_{I^{+}}(A)_{i}\in\mathcal{D}(R)$ and
${\bm{R}}\pi^{+}_{*}(\mathcal{O}_{X^{+}}(i))\in\mathcal{D}(Y)$ are canonically
identified under the equivalence $\mathcal{D}(Y)\simeq\mathcal{D}(R)$. Indeed,
this follows from the explicit Čech complex description (1.7), whose weight
$i$ part is exactly the same Čech complex222Since $X^{+}$ is defined as ${\rm
Proj}(A_{\geq 0})$, the Čech complex that computes
${\bm{R}}\pi^{+}_{*}(\mathcal{O}_{X^{+}}(i))$ is
$\check{\mathcal{C}}_{I^{+}}(A_{\geq 0})_{i}$. However, for any homogeneous
element $f$ with $\deg(f)>0$, we have $(A_{\geq 0})_{f}=A_{f}$, so that these
two Čech complexes are the same. This is an instance of the fact that,
although $X^{+}$ is defined as ${\rm Proj}(A_{\geq 0})$, it is really about
$A$. Namely, it is covered by the affine open subschemes ${\rm
Spec}(A_{(f)})={\rm Spec}((A_{\geq 0})_{(f)})$ for $\deg(f)>0$. that computes
${\bm{R}}\pi^{+}_{*}(\mathcal{O}_{X^{+}}(i))$. Notice that we have already
seen in (1.2) the first item of the left column. The third term is then a
formal consequence of the first two (here, the notation
${\bm{R}}^{>0}\pi^{+}_{*}$ is a shorthand for the good truncation
$\tau^{>0}{\bm{R}}\pi^{+}_{*}$).

The equalities of the column on the right means that they are canonically
identified under the equivalence $\mathcal{D}({\rm
Gr}(A))\simeq\mathcal{D}(\mathfrak{X})$. Indeed, under this equivalence, the
description (1.7) is standard.

Generally speaking, when we discuss flips and flops in the next section, we
will use the column on the left to extract homological information on $A$ from
the situation of flips/flops, and then use the column on the right (as a
viewpoint) to relate the derived categories of $X^{-}$ and $X^{+}$.

##  2\. Flips and flops

We consider flips and flops in this section. Before getting to that, we recall
some results in algebraic geometry. The first is the following result, which
may be found in [7, Proposition 5.75]:

######  Theorem 2.1.

Let $X$ be a projective normal variety333In this article, we will assume that
$k$ is a perfect field. Hence a variety is smooth if and only if it is
regular.. Then there is an isomorphism $\mathcal{O}(K_{X})\cong\omega_{X}$,
where $K_{X}$ is the canonical divisor, and $\omega_{X}$ is the dualizing
sheaf.

######  Remark 2.2.

This statement is still true if $X$ is not projective. Namely, for any variety
$X$, take the canonical dualizing complex444Recall that for a morphism
$f:X\rightarrow Y$ between separated schemes of finite type over $k$, the
functor $f^{!}$ is defined to be $f^{!}=j^{*}\circ g^{!}$, where
$X\xrightarrow[]{j}\overline{X}\xrightarrow[]{g}Y$ is a factorization of $f$
such that $g$ is proper and $j$ is a Zariski open inclusion, and the functor
$g^{!}$ is the right adjoint to ${\bm{R}}g_{*}$. All of these functors are
taken between $\mathcal{D}_{{\rm qcoh}}(-)\simeq\mathcal{D}({\rm QCoh}(-))$.
$\omega_{X}^{\bullet}:=p^{!}(k)$, where $p:X\rightarrow{\rm Spec}\,k$ is the
projection to a point, and define
$\omega_{X}:=\mathcal{H}^{-n}(\omega_{X}^{\bullet})$ where $n=\dim(X)$. If $X$
is proper (over $k$), then $\omega_{X}$ can be characterized as the dualizing
sheaf. But even if $X$ is not proper, $\omega_{X}\in{\rm Coh}(X)$ is still
canonically defined. Moreover, if $X$ is normal then we have
$\mathcal{O}(K_{X})\cong\omega_{X}$, where $K_{X}$ is the canonical divisor.
For example, if $X$ is quasi-projective then this follows from Theorem 2.1 by
taking a compactification. This implies that $\omega_{X}$ is reflexive
whenever $X$ is a normal variety (since we can argue on each affine open
subscheme, which is quasi-projective). Since $\omega_{X}$ and
$\mathcal{O}(K_{X})$ are both reflexive and coincide on the smooth locus, they
must be isomorphic.

We will also need a version555See [9, Theorem 1-2-3] for the case when $X$ is
smooth. This implies our present case by taking a resolution of singularities
$f:\widetilde{X}\rightarrow X$. Namely, since $X$ has rational singularities,
we have ${\bm{R}}f_{*}(\omega_{\widetilde{X}})\cong\omega_{X}$, so that we
have ${\bm{R}}f_{*}(\omega_{\widetilde{X}}\otimes
f^{*}\mathscr{L})\cong\omega_{X}\otimes\mathscr{L}$ by the projection formula.
Then apply ${\bm{R}}(\pi\circ f)_{*}\cong{\bm{R}}\pi_{*}\circ{\bm{R}}f_{*}$ to
$\omega_{\widetilde{X}}\otimes f^{*}\mathscr{L}$. of the Kawamata-Viehweg
vanishing theorem:

######  Theorem 2.3.

Assume ${\rm char}(k)=0$. Let $\pi:X\rightarrow Y$ be a projective birational
morphism between varieties, where $X$ has at most rational singularities. Let
$\mathscr{L}\in{\rm Pic}(X)$ be $\pi$-nef, then we have
${\bm{R}}^{i}\pi_{*}(\omega_{X}\otimes\mathscr{L})=0$ for all $i>0$.

Now we recall the notion of flips and flops, which are two mutually exclusive
subclasses of log flips. We will say that a log flip is trivial if both
$\pi^{\pm}$ are the identity maps. Otherwise, we say that the log flip is non-
trivial.

######  Definition 2.4.

A _flip_ is a non-trivial log flip $(X^{\pm},Y,\pi^{\pm},D^{\pm})$ in which
$D^{\pm}=K_{X^{\pm}}$.

A _flop_ is a non-trivial log flip in which $K_{X^{\pm}}$ is numerically
$\pi^{\pm}$-trivial.

A _Gorenstein flop_ is a non-trivial log flip in which $K_{Y}$ is Cartier. By
smallness, we then have $(\pi^{\pm})^{*}K_{Y}=K_{X^{\pm}}$, and therefore it
is a flop.

We will consider flips and Gorenstein flops. In fact, we will impose one more
condition:

(2.1) Assume that $X^{\pm}$ have at most rational Gorenstein singularities.

In the case of flips or Gorenstein flops, by applying Theorem 2.3 to
$\pi^{-}:X^{-}\rightarrow Y$, the assumption (2.1) implies that $Y$ has at
most rational singularities, and hence is Cohen-Macaulay. In this case, since
$X^{\pm}$ and $Y$ are Cohen-Macaualay, their canonical dualizing complexes
(see Remark 2.2) is concentrated in degree $-n$, so that
$\omega_{X}^{\bullet}\cong\omega_{X}[n]$ for $X$ being $X^{\pm}$ or $Y$.

Thus, in the case of flips satisfying (2.1), we have

$(\pi^{\pm})^{!}(\omega_{Y})\,=\,\omega_{X^{\pm}}\,\stackrel{{\scriptstyle\eqref{OKX_omega}}}{{\cong}}\,\mathcal{O}(K_{X^{\pm}})\,=\,\mathcal{O}_{X^{\pm}}(1)$

(the reader is cautioned that $\mathcal{O}_{X^{-}}(1)$ is $\pi^{-}$-anti-ample
in our convention).

In the case of Gorenstein flops satisfying (2.1), we have

$(\pi^{\pm})^{!}(\mathcal{O}_{Y})\,\cong\,\mathcal{O}_{X^{\pm}}$

Indeed, if $f:X\rightarrow Y$ is a proper birational morphism between
Gorenstein normal varieties such that $f^{*}K_{Y}=K_{X}$, then we have
$f^{*}(\omega_{Y})\cong\omega_{X}$ by (2.1). Also, we again have
$f^{!}(\omega_{Y})\cong\omega_{X}$ by virtue of $X$ and $Y$ being Cohen-
Macaulay. Since $f^{!}(\mathcal{F})\cong f^{*}(\mathcal{F})\otimes
f^{!}(\mathcal{O}_{Y})$ for perfect complexes $\mathcal{F}\in\mathcal{D}_{{\rm
perf}}(Y)$, we concludes that $f^{!}(\mathcal{O}_{Y})\cong\mathcal{O}_{X}$.

Thus, in both cases, we have the following

######  Checkpoint 2.5.

For flips and Gorenstein flops satisfying (2.1), there is a dualizing complex
$\omega_{Y}^{\bullet}$ on $Y$ such that

$(\pi^{\pm})^{!}(\omega_{Y}^{\bullet})\,\cong\,\mathcal{O}_{X^{\pm}}(a)$

where $a=0$ for Gorenstein flops and $a=1$ for flips.

Our goal now is to translate this condition into $\mathcal{D}({\rm Gr}(A))$,
in the spirit of the last section. First, we introduce the following
degreewise dualizing functor666Of course, the weight-by-weight description
(2.2) of $\mathbb{D}_{Y}(M)$ is not enough to determine $\mathbb{D}_{Y}(M)$ as
an object in $\mathcal{D}({\rm Gr}(A))$. To actually define $\mathbb{D}_{Y}$,
one could start with $\underline{{\rm Hom}}_{R}(-,-):{\rm Gr}(A)^{{\rm
op}}\times{\rm Mod}(R)\rightarrow{\rm Gr}(A)$ and take its derived functor.
(recall that $R=A_{0}$ and $Y={\rm Spec}\,R$).

(2.2) $\mathbb{D}_{Y}:\mathcal{D}({\rm Gr}(A))^{{\rm
op}}\rightarrow\mathcal{D}({\rm
Gr}(A))\,,\quad\qquad\mathbb{D}_{Y}(M)_{i}={\bm{R}}{\rm
Hom}_{R}(M_{-i},\omega_{Y}^{\bullet})$

We postulate the following as an imitation of (2.5):

######  Definition 2.6.

A _homological flip/flop_ consists of a pair of isomorphisms in
$\mathcal{D}({\rm Gr}(A))$:

(2.3)
$\begin{split}\Phi^{+}&:\check{\mathcal{C}}_{I^{+}}(A)(a)\xrightarrow[]{\cong}\mathbb{D}_{Y}(\check{\mathcal{C}}_{I^{+}}(A))\\\
\Phi^{-}&:\check{\mathcal{C}}_{I^{-}}(A)(a)\xrightarrow[]{\cong}\mathbb{D}_{Y}(\check{\mathcal{C}}_{I^{-}}(A))\end{split}$

such that they are compatible, in the sense that the following diagram
commutes:

(2.4)
${\check{\mathcal{C}}_{I^{+}}(A)(a)}$${\mathbb{D}_{Y}(\check{\mathcal{C}}_{I^{+}}(A))}$${A(a)}$${\mathbb{D}_{Y}(A)}$${\check{\mathcal{C}}_{I^{-}}(A)(a)}$${\mathbb{D}_{Y}(\check{\mathcal{C}}_{I^{-}}(A))}$$\scriptstyle{\Phi^{+}}$$\scriptstyle{\cong}$$\scriptstyle{\mathbb{D}_{Y}(\eta^{+})}$$\scriptstyle{\eta^{+}}$$\scriptstyle{\eta^{-}}$$\scriptstyle{\Phi^{-}}$$\scriptstyle{\cong}$$\scriptstyle{\mathbb{D}_{Y}(\eta^{-})}$

In the case $a=0$, we call it a homological flop. In the case $a=1$, we call
it a homological flip777One may call the general notion a homological flap, so
that it becomes a fl0p when $a=0$, and a fl1p when $a=1$..

The existence of the isomorphisms $\Phi^{\pm}$ should be quite believable if
we look at both sides of (2.3) weight by weight. For example, by Grothendieck
duality, we have

(2.5) ${\bm{R}}\mathscr{H}\text{\kern-3.0pt{\calligra\large
om}}\,_{\mathcal{O}_{Y}}({\bm{R}}\pi^{+}_{*}(\mathcal{O}_{X^{+}}(-i)),\omega_{Y}^{\bullet})\,\cong\,{\bm{R}}\pi^{+}_{*}{\bm{R}}\mathscr{H}\text{\kern-3.0pt{\calligra\large
om}}\,_{\mathcal{O}_{X^{+}}}(\mathcal{O}_{X^{+}}(-i),\pi^{!}(\omega_{Y}^{\bullet}))\stackrel{{\scriptstyle\eqref{chkpt_4}}}{{\cong}}{\bm{R}}\pi^{+}_{*}(\mathcal{O}_{X^{+}}(i+a))$

In view of (1.6), the left and right hand side are
$\mathbb{D}_{Y}(\check{\mathcal{C}}_{I^{+}}(A))_{i}$ and
$\check{\mathcal{C}}_{I^{+}}(A)_{i+a}$ respectively.

The meaning of the commutative diagram (2.4) is harder to describe. Let us
just say that it boils down to the fact that the isomorphism
$\Phi_{X}:\mathcal{O}(K_{X})\xrightarrow[]{\cong}\omega_{X}$ in Theorem 2.1
can be appropriately chosen so that they satisfy a certain compatibility
condition across the maps $\pi^{\pm}$. More precisely, we may require the
following diagram to commute:

(2.6)
${\pi^{-}_{*}\mathcal{O}(K_{X^{-}})}$${\mathcal{O}(K_{Y})}$${\pi^{+}_{*}\mathcal{O}(K_{X^{+}})}$${\pi^{-}_{*}\omega_{X^{-}}}$${\omega_{Y}}$${\pi^{+}_{*}\omega_{X^{+}}}$$\scriptstyle{\pi^{-}_{*}(\Phi_{X^{-}})}$$\scriptstyle{\Phi_{Y}}$$\scriptstyle{\pi^{+}_{*}(\Phi_{X^{+}})}$$\scriptstyle{{\rm
Tr}_{X^{-}/Y}}$$\scriptstyle{{\rm Tr}_{X^{+}/Y}}$

where $\Phi_{X^{\pm}}$ and $\Phi_{Y}$ are the isomorphisms in Theorem 2.1, and
${\rm Tr}_{X^{\pm}/Y}$ are induced from certain adjunction morphisms on
dualizing complexes. Here, we have written equalities in the first row because
they are equal as subsheaves of the sheaf of rational functions on $Y$.

In fact, to pass from the weight-by-weight description (2.5) to an actual
isomorphism (2.3) in $\mathcal{D}({\rm Gr}(A))$, we had to develop from
scratch the effect of Grothendieck duality in $\mathcal{D}({\rm Gr}(A))$ (this
is performed in [15]). Then, we had to show that, under this formalism, (2.6)
indeed translates into (2.4). For this, we had to check many commutative
diagrams. All these checkings are lengthy and tedious (see [13] for details),
but doesn’t seem to offer any additional insight, so we will skip it in this
article. We summarize this discussion by the following

######  Theorem 2.7.

For flips and Gorenstein flops satisfying (2.1), the corresponding888For
flips, there is no ambiguity in choosing $D^{\pm}$, so that $A$ is canonically
defined. For flops, we will assume that $D^{\pm}$ are chosen to be Cartier,
possibly by replacing them by their multiple. Since we work under the
assumption (2.1), we see that $D^{\pm}$ are Cartier in both cases. In fact,
this was already implicitly used in the second isomorphism in (2.5), which
holds because $\mathcal{O}_{X^{+}}(-i)$ is locally free. Notice that it also
holds if $\mathcal{O}_{X^{+}}(-i)=\mathcal{O}_{X^{+}}(-iD^{+})$ is maximal
Cohen-Macaulay, which is satisfied whenever $X^{\pm}$ have at most log
terminal singularities (see [7, Corollary 5.25]). Accordingly, the assumption
(2.1) can be weakened. graded ring $A$ admits the structure of a homological
flip/flop.

We will formulate one more property of $A$ that imitates the Kawamata-Viehweg
vanishing theorem. We will write $\mathcal{D}_{<w}({\rm Gr}(A))$ as the full
subcategory of $\mathcal{D}({\rm Gr}(A))$ consisting of $M\in\mathcal{D}({\rm
Gr}(A))$ such that $M_{i}=0$ for all $i\geq w$. Again, we emphasize here that
$M_{i}$ refers to the weight degree, not the homological degree. In
particular, $\mathcal{D}_{<w}({\rm Gr}(A))$ is a triangulated subcategory. The
subcategory $\mathcal{D}_{>w}({\rm Gr}(A))$ is also defined in a similar way.

######  Definition 2.8.

A $\mathbb{Z}$-graded ring $A$ is said to satisfy _canonical vanishing_ at
$a\in\mathbb{Z}$ if we have ${\bm{R}}\Gamma_{I^{+}}(A)\in\mathcal{D}_{<a}({\rm
Gr}(A))$ and ${\bm{R}}\Gamma_{I^{-}}(A)\in\mathcal{D}_{>a}({\rm Gr}(A))$.

In view of the table of (1.6), we see that Theorem 2.3 immediately implies the
following

######  Proposition 2.9.

For flips and Gorenstein flops satisfying (2.1), the corresponding graded ring
$A$ satisfies canonical vanishing at $a$ (where $a=0$ for flops and $a=1$ for
flips).

Next, we explore some consequences of Theorem 2.7 and Proposition 2.9. The
following Theorem, as well as its proof, may be regarded as the main result of
this work:

######  Theorem 2.10.

Suppose $(A,\Phi^{+},\Phi^{-})$ is a homological flip/flop that satisfies
canonical vanishing at $a$ (where $a=0$ for flops and $a=1$ for flips), then
there is an isomorphism in $\mathcal{D}({\rm Gr}(A))$:

$\Psi\,:\,{\bm{R}}\Gamma_{I^{+}}(A)(a)[1]\,\xrightarrow[]{\cong}\,\mathbb{D}_{Y}({\bm{R}}\Gamma_{I^{-}}(A))$

Moreover, $A$ is Gorenstein.

###### Proof.

Consider the commutative diagram (2.4). Since the maps $\Phi^{\pm}$ are
isomorphisms, we may regard it as a commutative square. As such, the $3\times
3$-lemma (see, e.g., [1, Proposition 1.1.11] or [10, Lemma 2.6]) asserts that
it can be extended to a $3\times 3$ square. More precisely, the object $Z$ as
well as the maps in the dotted lines of the followng diagram exists, making
each row and column part of a distinguished triangle:

(2.7)
${A(a)}$${\check{\mathcal{C}}_{I^{+}}(A)(a)}$${{\bm{R}}\Gamma_{I^{+}}(A)(a)[1]}$${\check{\mathcal{C}}_{I^{-}}(A)(a)}$${\mathbb{D}_{Y}(A)}$${\mathbb{D}_{Y}({\bm{R}}\Gamma_{I^{-}}(A))}$${{\bm{R}}\Gamma_{I^{-}}(A)(a)[1]}$${\mathbb{D}_{Y}({\bm{R}}\Gamma_{I^{+}}(A))}$${Z}$$\scriptstyle{\eta^{+}}$$\scriptstyle{\eta^{-}}$$\scriptstyle{\mathbb{D}_{Y}(\eta^{+})\circ\Phi^{+}}$$\scriptstyle{\delta^{+}}$$\scriptstyle{\Psi}$$\scriptstyle{\mathbb{D}_{Y}(\eta^{-})\circ\Phi^{-}}$$\scriptstyle{\delta^{-}}$$\scriptstyle{\mathbb{D}_{Y}(\epsilon^{-})}$$\scriptstyle{\mathbb{D}_{Y}(\epsilon^{+})}$$\scriptstyle{\Psi^{\prime}}$

By the assumption of canonical vanishing, we have
${\bm{R}}\Gamma_{I^{+}}(\mathcal{A})(a)[1]\in\mathcal{D}_{<0}({\rm Gr}(A))$
and
$\mathbb{D}_{Y}({\bm{R}}\Gamma_{I^{-}}(\mathcal{A}))\in\mathcal{D}_{<-a}({\rm
Gr}(A))$, so that, as the cone of $\Psi$, we have $Z\in\mathcal{D}_{<0}({\rm
Gr}(A))$. Likewise, we have
${\bm{R}}\Gamma_{I^{-}}(\mathcal{A})(a)[1]\in\mathcal{D}_{>0}({\rm Gr}(A))$
and
$\mathbb{D}_{Y}({\bm{R}}\Gamma_{I^{+}}(\mathcal{A}))\in\mathcal{D}_{>-a}({\rm
Gr}(A))$, so that as the cone of $\Psi^{\prime}$, we have
$Z\in\mathcal{D}_{>-a}({\rm Gr}(A))$. Since $a=0$ or $a=1$, we must have
$Z=0$. i.e., $\Psi$ is an isomorphism.

To prove that $A$ is Gorenstein, we will show that it has finite injective
dimension999Given a graded module $M\in{\rm Gr}(A)$, then it has finite
injective dimension in the abelian category ${\rm Gr}(A)$ if and only if it
has finite injective dimension in the abelian category ${\rm Mod}(A)$. In
fact, we have ${\rm inj\,dim}_{{\rm Gr}(A)}(M)\,\leq\,{\rm inj\,dim}_{{\rm
Mod}(A)}(M)\,\leq\,{\rm inj\,dim}_{{\rm Gr}(A)}(M)+1$ (see, e.g. [2,
Proposition 3.6.6]). Hence there is no ambiguity in our discussion.. Take the
local cohomology sequence

$\ldots\,\rightarrow\,{\bm{R}}\Gamma_{I^{+}}(A)\,\rightarrow\,A\,\rightarrow\,\check{\mathcal{C}}_{I^{+}}(A)\,\rightarrow\,\ldots$

Then the terms can be rewritten as
${\bm{R}}\Gamma_{I^{+}}(A)\cong\mathbb{D}_{Y}({\bm{R}}\Gamma_{I^{-}}(A))(-a)[-1]$
and
$\check{\mathcal{C}}_{I^{+}}(A)\cong\mathbb{D}_{Y}(\check{\mathcal{C}}_{I^{+}}(A))(-a)$.
Notice that both ${\bm{R}}\Gamma_{I^{-}}(A)$ and
$\check{\mathcal{C}}_{I^{+}}(A)$ have finite Tor dimension because of the
explicit presentation (1.7). Therefore their $\mathbb{D}_{Y}$-dual have finite
injective dimension, hence so does $A$. ∎

For general classes of flips and flops, it seems quite difficult to describe
this duality between local cohomology groups explicitly. There is however one
simple example where such a description is possible:

######  Example 2.11.

Let $A=k[x_{1},\ldots,x_{p},y_{1},\ldots,y_{q}]$, where $\deg(x_{i})=1$ and
$\deg(y_{i})=-1$. This corresponds to the standard flip/flop, and we have
$a=q-p$. By the explicit presentation (1.7), it is easy to see that

${\bm{R}}\Gamma_{I^{+}}(A)\,=\,k[y_{1},\ldots,y_{q}]\otimes{\bm{R}}\Gamma_{I^{+}}(k[x_{1},\ldots,x_{p}])$

Also, in view of the table in (1.6) again, we see that
${\bm{R}}\Gamma_{I^{+}}(k[x_{1},\ldots,x_{p}])_{i}$ is simply the higher
cohomology of $\mathbb{P}^{p-1}$ (shifted by $1$) of $\mathcal{O}(i)$. Such a
computation is standard (see, e.g., [5, Section III.5]), and we have

${\bm{R}}\Gamma_{I^{+}}(A)\,=\,k[y_{1},\ldots,y_{q}]\otimes
k[x_{1},\ldots,x_{p}]^{*}(p)[-p]$

where we denote $M^{*}$ to be the $k$-linear dual of $M$. Similarly, we have

${\bm{R}}\Gamma_{I^{-}}(A)\,=\,k[x_{1},\ldots,x_{p}]\otimes
k[y_{1},\ldots,y_{q}]^{*}(-q)[-q]$

We want to claim that they are $\mathbb{D}_{Y}$-dual to each other. While the
description of $\mathbb{D}_{Y}$ on $R=A_{0}$ seems to be quite complicated in
general, we will only need a special case. Namely, since
${\bm{R}}\Gamma_{I^{+}}(A)_{i}$ corresponds to higher pushforwards of
$\mathcal{O}_{X^{+}}(i)$, it must be supported on the image of the exceptional
locus, which is a point. In this case, $\mathbb{D}_{Y}$ is easy to describe:

######  Lemma 2.12.

Suppose that $M\in\mathcal{D}^{b}_{{\rm coh}}(R)$ is (set-theoretically)
supported on ${\rm Spec}\,k={\rm Spec}\,R/J$, and let
$\omega_{Y}^{\bullet}\in\mathcal{D}^{b}_{{\rm coh}}(R)$ be a dualizing
complex, normalized so that its cohomology sheaf is concentrated101010$R$ is
Cohen-Macaulay in this example, so that we may assume that
$\mathcal{H}^{\bullet}(\omega_{Y}^{\bullet})$ is concentrated in degree $0$.
An analogous statement holds for more general $R$ (at least for domains
finitely generated over $k$), for which we may assume that
$\mathcal{H}^{\bullet}(\omega_{Y}^{\bullet})$ is concentrated in degree $0$ on
the smooth part. in degree $0$. Then we have $\mathbb{D}_{Y}(M)\cong
M^{*}[-n]$, where $n=\dim R=p+q-1$.

From this, we see that

$\mathbb{D}_{Y}({\bm{R}}\Gamma_{I^{-}}(A))\,=\,k[x_{1},\ldots,x_{p}]^{*}\otimes
k[y_{1},\ldots,y_{q}](q)[q][-p-q+1)]\,=\,{\bm{R}}\Gamma_{I^{+}}(A)(a)[1]$

######  Remark 2.13.

This example can be directly generalized to the case
$A=k[x_{1},\ldots,x_{p},y_{1},\ldots,y_{q}]$, where $\deg(x_{i})>0$ and
$\deg(y_{i})<0$. In this case, let $\eta^{+}=\sum_{i=1}^{p}\deg(x_{i})$ and
$\eta^{-}=-\sum_{j=1}^{p}\deg(y_{j})$, and take $a=\eta^{-}-\eta^{+}$. Notice
that ${\bm{R}}\Gamma_{I^{+}}(A)$ is still the same extended Čech complex as in
Example 2.11, but with generators in different degrees. As such, the
computation in [5, Section III.5] carries through verbatim, and we have

$\begin{split}{\bm{R}}\Gamma_{I^{+}}(A)\,&=\,k[y_{1},\ldots,y_{q}]\otimes
k[x_{1},\ldots,x_{p}]^{*}(\eta^{+})[-p]\\\
{\bm{R}}\Gamma_{I^{-}}(A)\,&=\,k[x_{1},\ldots,x_{p}]\otimes
k[y_{1},\ldots,y_{q}]^{*}(-\eta^{-})[-q]\end{split}$

and hence we still have
$\mathbb{D}_{Y}({\bm{R}}\Gamma_{I^{-}}(A))\cong{\bm{R}}\Gamma_{I^{+}}(A)(a)[1]$.

We wish to apply the results in Theorem 2.7, Proposition 2.9 and Theorem 2.10
to relate the derived categories of $X^{-}$ and $X^{+}$. We may summarize the
situation as follows

${\mathcal{D}({\rm
Gr}(A))}$${\mathcal{D}(X^{-})}$${\mathcal{D}(X^{+})}$$\scriptstyle{\text{kill
}{\bm{R}}\Gamma_{I^{-}}}$$\scriptstyle{\text{kill }{\bm{R}}\Gamma_{I^{+}}}$

where the derived categories $\mathcal{D}(X^{\pm})$ are obtained by “killing”
the endofunctors
${\bm{R}}\Gamma_{I^{\pm}}(M)=M\otimes_{A}^{{\bm{L}}}{\bm{R}}\Gamma_{I^{\pm}}(A)$.
The duality of local cohomology groups in Theorem 2.10 then suggests a way to
relate these two derived categories. However, an implementation of this idea
seems to be not so straightforward. We present one attempt in the next
section. This approach is successful in some useful cases (see Section 4), but
fall short in general because of what seems to be a formal problem.

##  3\. Weight truncation

Before describing the actual construction, we would like to suggest a
parallelism with a paper [11] of Orlov. Namely, we may summarize our desired
picture as follows:

(3.1) $\begin{split}&\text{Flip or Gorenstein flop (with good
singularities)}\\\ \Longrightarrow&\,A\text{ is Gorenstein, and
}{\bm{R}}\Gamma_{I^{+}}(A)(a)[1]\cong\mathbb{D}_{Y}({\bm{R}}\Gamma_{I^{-}}(A))\\\
\stackrel{{\scriptstyle(?)}}{{\Longrightarrow}}&\,\begin{cases}\mathcal{D}_{{\rm
perf}}(X^{-})\,\hookleftarrow\,\mathcal{D}_{{\rm perf}}(X^{+})&\text{ if
}a>0,\\\ \mathcal{D}_{{\rm perf}}(X^{-})\,\simeq\,\mathcal{D}_{{\rm
perf}}(X^{+})&\text{ if }a=0,\\\ \mathcal{D}_{{\rm
perf}}(X^{-})\,\hookrightarrow\,\mathcal{D}_{{\rm perf}}(X^{+})&\text{ if
}a<0\end{cases}\end{split}$

On the other hand, for a Gorenstein projective variety $X$ of dimension $n$,
Orlov studied the relation between $\mathcal{D}^{b}_{{\rm coh}}(X)$ and the
triangulated category of singularities of the projective cone of $X$. More
precisely, assume that $\mathcal{O}(1)$ is very ample, and let
$A=\bigoplus_{i\geq 0}H^{0}(X,\mathcal{O}(i))$. Assume that
$H^{j}(X,\mathcal{O}(i))=0$ for all $j\neq 0,n$ for all $i\in\mathbb{Z}$. Then
Orlov established in [11] the following implications

(3.2) $\begin{split}&\omega_{X}\cong\mathcal{O}(-a)\\\
\Longrightarrow&\,A\text{ is Gorenstein, and }{\bm{R}}\underline{{\rm
Hom}}_{A}(k,A)\cong k(a)[-n]\\\
\stackrel{{\scriptstyle(*)}}{{\Longrightarrow}}&\,\begin{cases}\mathcal{D}_{{\rm
sg}}({\rm gr}(A))\,\hookrightarrow\,\mathcal{D}^{b}_{{\rm coh}}(X)&\text{ if
}a>0,\\\ \mathcal{D}_{{\rm sg}}({\rm gr}(A))\,\simeq\,\mathcal{D}^{b}_{{\rm
coh}}(X)&\text{ if }a=0,\\\ \mathcal{D}_{{\rm sg}}({\rm
gr}(A))\,\hookleftarrow\,\mathcal{D}^{b}_{{\rm coh}}(X)&\text{ if
}a<0\end{cases}\end{split}$

Our goal in this section is to imitate the construction of the step $(*)$ in
(3.2) and try to establish the step $(?)$ in (3.1). Our construction is also
influenced and motivated by the papers [6, 3].

Fix an integer $w\in\mathbb{Z}$ once and for all. Let $\mathcal{D}_{[\geq
w]}({\rm Gr}(A))$ be the smallest cocomplete (i.e., closed under arbitrary
direct sums) triangulated subcategory of $\mathcal{D}({\rm Gr}(A))$ containing
the objects $\\{A(-i)\\}_{i\geq w}$, and let $\mathcal{D}_{<w}({\rm Gr}(A))$
be defined as above. i.e., it consists of $M\in\mathcal{D}({\rm Gr}(A))$ such
that $M_{i}=0$ for all $i\geq w$. By Neeman-Brown representability, we have an
SOD

(3.3) $\mathcal{D}({\rm Gr}(A))\,=\,\langle\,\mathcal{D}_{<w}({\rm
Gr}(A))\,,\,\mathcal{D}_{[\geq w]}({\rm Gr}(A))\,\rangle$

The terminology $\mathcal{D}_{[\geq w]}({\rm Gr}(A))$ refers to those objects
that are generated in weight $\geq w$, while $\mathcal{D}_{<w}({\rm Gr}(A))$
refers to those that are concentrated in weight $<w$. In the setting of [11],
$A$ is non-negatively graded, so that $\mathcal{D}_{[\geq w]}({\rm
Gr}(A))=\mathcal{D}_{\geq w}({\rm Gr}(A))$. But in our setting of
$\mathbb{Z}$-graded rings, these two are very different. In particular, none
contain the other.

The decomposition sequence associated to (3.3) will be denoted as

(3.4) $\ldots\,\rightarrow\,\mathcal{L}_{[\geq
w]}M\,\rightarrow\,M\,\rightarrow\,\mathcal{L}_{<w}M\,\rightarrow\,\ldots$

We will put together the SODs (1.5) and (3.3). Notice that
$\mathcal{D}_{<w}({\rm Gr}(A))$ consists of those $M\in\mathcal{D}({\rm
Gr}(A))$ such that each $H^{i}(M)$ has weight concentrated in degree $<w$, and
hence we have $\mathcal{D}_{<w}({\rm Gr}(A))\subset\mathcal{D}_{{\rm
Tor}^{+}}({\rm Gr}(A))$. Thus, if we compare (1.5) and (3.3), we would expect
that $\mathcal{D}_{[\geq w]}({\rm Gr}(A))$ would be “bigger” than
$\mathcal{D}_{I^{+}\text{-triv}}({\rm Gr}(A))$. In fact, one can show that
$\mathcal{D}_{I^{+}\text{-triv}}({\rm Gr}(A))$ embeds (via a non-identity
functor) as a semi-orthogonal summand of $\mathcal{D}_{[\geq w]}({\rm
Gr}(A))$. Combined with (3.3), it then gives us a $3$-term semi-orthogonal
decomposition of $\mathcal{D}({\rm Gr}(A))$. This is summarized in the
following result, whose proof is simple once it is formulated precisely as
below:

######  Theorem 3.1.

The restriction of $\mathcal{L}_{[\geq w]}$ to
$\mathcal{D}_{I^{+}\text{-triv}}({\rm Gr}(A))$ gives a fully faithful functor
$\mathcal{L}_{[\geq w]}:\mathcal{D}_{I^{+}\text{-triv}}({\rm
Gr}(A))\rightarrow\mathcal{D}_{[\geq w]}({\rm Gr}(A))$, with a left adjoint
given by $\check{\mathcal{C}}_{I^{+}}$. Hence there is a semi-orthogonal
decomposition

$\mathcal{D}_{[\geq w]}({\rm Gr}(A))\,=\,\langle\,\mathcal{L}_{[\geq
w]}\mathcal{D}_{I^{+}\text{-triv}}({\rm Gr}(A))\,,\,\mathcal{D}_{[\geq w],{\rm
Tor}^{+}}({\rm Gr}(A))\,\rangle$

Combined with (3.3), there is therefore a $3$-term semi-orthogonal
decomposition

(3.5) $\mathcal{D}({\rm
Gr}(A))\,=\,\langle\,\UOLunderbrace{\mathcal{D}_{<w}({\rm
Gr}(A))\,,\,}[\mathcal{L}_{[\geq w]}\mathcal{D}_{I^{+}\text{-triv}}({\rm
Gr}(A))]_{=:\,\mathcal{D}_{(<w)}({\rm
Gr}(A))}\UOLoverbrace{\,,\,\mathcal{D}_{[\geq w],{\rm Tor}^{+}}({\rm
Gr}(A))}^{\mathcal{D}_{[\geq w]}({\rm Gr}(A))}\,\rangle$

where the subcategory $\mathcal{D}_{(<w)}({\rm Gr}(A))\subset\mathcal{D}({\rm
Gr}(A))$ can be characterized more precisely as

$\mathcal{D}_{(<w)}({\rm Gr}(A))\,=\,\\{\,M\in\mathcal{D}({\rm
Gr}(A))\,|\,{\bm{R}}\Gamma_{I^{+}}(M)\in\mathcal{D}_{<w}({\rm Gr}(A))\,\\}$

The middle component, called the window subcategory can therefore be
characterized as

$\mathcal{L}_{[\geq w]}\mathcal{D}_{I^{+}\text{-triv}}({\rm
Gr}(A))\,=\,\mathcal{D}_{[\geq w],(<w)}({\rm Gr}(A))\,=\,\mathcal{D}_{[\geq
w]}({\rm Gr}(A))\cap\mathcal{D}_{(<w)}({\rm Gr}(A))$

which is equivalent to $\mathcal{D}_{I^{+}\text{-triv}}({\rm Gr}(A))$ (and
hence to $\mathcal{D}(\mathfrak{X}^{+})$) via the functors
$\check{\mathcal{C}}_{I^{+}}$ and $\mathcal{L}_{[\geq w]}$.

Notice that, in this Theorem, we have embedded $\mathcal{D}(\mathfrak{X}^{+})$
into $\mathcal{D}({\rm Gr}(A))$ in a non-standard way, i.e., as the window
subcategory $\mathcal{D}_{[\geq w],(<w)}({\rm Gr}(A))$, instead of the more
straightforward $\mathcal{D}_{I^{+}\text{-triv}}({\rm Gr}(A))$. Notice that
the straightforward embedding to $\mathcal{D}_{I^{+}\text{-triv}}({\rm
Gr}(A))$ has the disadvantage that $\check{\mathcal{C}}_{I^{+}}(M)$ almost
never have coherent cohomology even if $M$ does (it has finite cohomological
dimension though!). In contrast, this non-standard “window embedding” have the
following advantage:

######  Theorem 3.2.

The $3$-term SOD (3.5) restrict to a $3$-term SOD on $\mathcal{D}^{-}_{{\rm
coh}}({\rm Gr}(A))$:

$\mathcal{D}^{-}_{{\rm coh}}({\rm Gr}(A))\,=\,\langle\,\mathcal{D}^{-}_{{\rm
coh},<w}({\rm Gr}(A))\,,\,\mathcal{D}^{-}_{{\rm coh},[\geq w],(<w)}({\rm
Gr}(A))\,,\,\mathcal{D}^{-}_{{\rm coh},[\geq w],{\rm Tor}^{+}}({\rm
Gr}(A))\,\rangle$

where the middle component is equivalent to $\mathcal{D}^{-}_{{\rm
coh}}(\mathfrak{X}^{+})$.

A proof may be found in [14] (a revised version is under preparation where the
arguments will be simplified and put in a broader scope), and is based on the
tensor form (4.2) below. We will skip the proof in this article.

Now we are ready to describe the functor that relate the derived categories
under flips/flops. First, we introduce the functor

$\mathbb{D}_{A}\,:\,\mathcal{D}({\rm Gr}(A))^{{\rm
op}}\,\rightarrow\,\mathcal{D}({\rm
Gr}(A))\,,\qquad\mathbb{D}_{A}(M):={\bm{R}}\underline{{\rm Hom}}_{A}(M,A)$

We postulate the following idea:

(3.6) The functor $\mathbb{D}_{A}$ “should” send the window
$\mathcal{D}_{[\geq w],(<w)}({\rm Gr}(A))$ in the positive direction to the
window $\mathcal{D}_{[\leq-w],(>-w)}({\rm Gr}(A))$ in the negative direction.

The idea does not work as it is stated. But let’s press on with the idea,
which may be broken down into two parts:

(3.7) (1) The functor $\mathbb{D}_{A}$ “should” send $\mathcal{D}_{[\geq
w]}({\rm Gr}(A))$ to $\mathcal{D}_{[\leq-w]}({\rm Gr}(A))$ (2) The functor
$\mathbb{D}_{A}$ “should” send $\mathcal{D}_{(<w)}({\rm Gr}(A))$ to
$\mathcal{D}_{(>-w)}({\rm Gr}(A))$

The first item seems quite reasonable, because $\mathbb{D}_{A}$ sends the
generators $\\{A(-i)\\}_{i\geq w}$ of $\mathcal{D}_{[\geq w]}({\rm Gr}(A))$ to
the generators $\\{A(-i)\\}_{i\leq-w}$ of $\mathcal{D}_{[\leq-w]}({\rm
Gr}(A))$. However, notice that $\mathbb{D}_{A}$ sends infinite direct sums to
infinite direct products, and $\mathcal{D}_{[\leq-w]}({\rm Gr}(A))$ seems to
be not closed under infinite direct products in general, so that (1) seems to
be not true in general. In fact, it seems to be not true even if one restrict
to $\mathcal{D}^{-}_{{\rm coh},[\geq w]}({\rm Gr}(A))$. More precisely, one
may encounter the following problem:

(3.8) Suppose $M$ is a bounded below complex $M=[0\rightarrow M^{n}\rightarrow
M^{n+1}\rightarrow\ldots]$ such that each $M^{j}$ is a finite direct sums of
the free graded modules $A(-i)$ for $i\leq-w$, then it is not clear whether
$M$ is in $\mathcal{D}_{[\leq-w]}({\rm Gr}(A))$.

In contrast, one can actually prove the item (2), when restricted to
$\mathcal{D}^{-}_{{\rm coh}}({\rm Gr}(A))$:

######  Proposition 3.3.

Suppose there is a dualizing complex $\omega_{Y}^{\bullet}$ on $R=A_{0}$ such
that there is an isomorphism
${\bm{R}}\Gamma_{I^{+}}(A)(a)[1]\cong\mathbb{D}_{Y}({\bm{R}}\Gamma_{I^{-}}(A))$
in $\mathcal{D}({\rm Gr}(A))$. Then $\mathbb{D}_{A}$ sends
$\mathcal{D}^{-}_{{\rm coh},(<w)}({\rm Gr}(A))$ to $\mathcal{D}^{+}_{{\rm
coh},(>-w+a)}({\rm Gr}(A))$.

###### Proof.

Suppose $M\in\mathcal{D}^{-}_{{\rm coh}}({\rm Gr}(A))$ satisfies
${\bm{R}}\Gamma_{I^{+}}(M)\in\mathcal{D}_{<w}({\rm Gr}(A))$, then we have

$\begin{split}{\bm{R}}\Gamma_{I^{-}}(\mathbb{D}_{A}(M))\,&=\,{\bm{R}}\underline{{\rm
Hom}}_{A}(M,A)\otimes_{A}^{{\bm{L}}}{\bm{R}}\Gamma_{I^{-}}(A)\\\
\,&\stackrel{{\scriptstyle(*)}}{{=}}\,{\bm{R}}\underline{{\rm
Hom}}_{A}(M,{\bm{R}}\Gamma_{I^{-}}(A))\\\
\,&\cong\,\mathbb{D}_{Y}(M\otimes_{A}^{{\bm{L}}}{\bm{R}}\Gamma_{I^{+}}(A))(-a)[-1]\end{split}$

where $(*)$ holds because ${\bm{R}}\Gamma_{I^{-}}(A)$ has finite Tor dimension
and $M\in\mathcal{D}^{-}_{{\rm coh}}({\rm Gr}(A))$. ∎

By the results of the last section, we may assume that $A$ is Gorenstein, so
that $\mathbb{D}_{A}$ is in some sense well-behaved. Thus, if (3.7)(1) holds
in some good context, we would have obtained a relation between the derived
categories of $\mathfrak{X}^{+}$ and $\mathfrak{X}^{-}$ similar to the
expected relation (3.1). While this picture of using $\mathbb{D}_{A}$ to
relate the two windows was the first argument along this line that the author
discovered (and is still very attractive to the author), it seems to be not
the most efficient one. We now seek to reformulate it without using the
duality $\mathbb{D}_{A}$.

In our above argument, the functor that we want to use to relate the derived
categories of $\mathfrak{X}^{-}$ and $\mathfrak{X}^{+}$ can be written as
follows:

(3.9) $\mathcal{D}^{-}_{{\rm
coh}}(\mathfrak{X}^{+})\,\xleftarrow[\simeq]{(j^{+})^{*}}\,\mathcal{D}^{-}_{{\rm
coh},[\geq w],(<w)}({\rm
Gr}(A))\,\xrightarrow[(?)]{\mathbb{D}_{A}}\,\mathcal{D}^{+}_{{\rm
coh},[\leq-w],(>-w)}({\rm Gr}(A))^{{\rm
op}}\,\xrightarrow[]{(j^{-})^{*}}\,\mathcal{D}^{+}_{{\rm
coh}}(\mathfrak{X}^{-})^{{\rm op}}$

Here, we have marked the middle arrow with a question mark to indicate that it
is not clear whether the functor $\mathbb{D}_{A}$ really sends
$\mathcal{D}^{-}_{{\rm coh},[\geq w],(<w)}({\rm Gr}(A))$ to
$\mathcal{D}^{+}_{{\rm coh},[\leq-w],(>-w)}({\rm Gr}(A))$. Thus, the question
mark in (3.9) is a version of (3.6) where we restrict to
$\mathcal{D}^{-}_{{\rm coh}}$. If it holds, then we see that (3.9) is a
composition of three fully faithful functors, and is therefore fully faithful,
giving a result along the lines of (3.1).

Whether or not the question mark in (3.9) holds, the functor (3.9) makes sense
if we consider

$\mathcal{D}^{-}_{{\rm
coh}}(\mathfrak{X}^{+})\,\xleftarrow[\simeq]{(j^{+})^{*}}\,\mathcal{D}^{-}_{{\rm
coh},[\geq w],(<w)}({\rm
Gr}(A))\,\xrightarrow[]{(j^{-})^{*}\circ\mathbb{D}_{A}}\,\mathcal{D}^{+}_{{\rm
coh}}(\mathfrak{X}^{-})^{{\rm op}}$

Since duality is local on $\mathcal{D}^{-}_{{\rm coh}}(\mathfrak{X})$, the
last functor may be rewritten as
$(j^{-})^{*}\circ\mathbb{D}_{A}=\mathbb{D}_{\mathfrak{X}^{-}}\circ(j^{-})^{*}$.
But $\mathfrak{X}^{-}$ is Gorenstein, so that $\mathbb{D}_{\mathfrak{X}^{-}}$
is a contravariant equivalence on $\mathcal{D}_{{\rm coh}}(\mathfrak{X}^{-})$,
and we may skip that functor. In other words, the desired functor is in fact

(3.10) ${\mathcal{D}^{-}_{{\rm
coh}}(\mathfrak{X}^{+})}$${\mathcal{D}^{-}_{{\rm coh},[\geq w],(<w)}({\rm
Gr}(A))}$${\mathcal{D}^{-}_{{\rm
coh}}(\mathfrak{X}^{-})}$$\scriptstyle{\simeq}$$\scriptstyle{\mathcal{L}_{[\geq
w]}\circ{\bm{R}}j^{+}_{*}}$$\scriptstyle{(j^{+})^{*}}$$\scriptstyle{(j^{-})^{*}}$

Since the duality $\mathbb{D}_{A}$ is not involved, we expect that our
argument will not need to use the Gorenstein property. In fact, we will show
that the duality
${\bm{R}}\Gamma_{I^{+}}(A)(a)[1]\cong\mathbb{D}_{Y}({\bm{R}}\Gamma_{I^{-}}(A))$
“almost implies” that this functor is fully faithful, except that a similar
formal problem arise. To describe the problem, we start with the following
straightforward

######  Lemma 3.4.

Given a full triangulated subcategory $\mathcal{E}\subset\mathcal{D}({\rm
Gr}(A))$, then the functor
$\check{\mathcal{C}}_{I^{-}}:\mathcal{E}\rightarrow\mathcal{D}_{I^{-}\text{-triv}}({\rm
Gr}(A))$ is fully faithful if and only if

${\bm{R}}{\rm Hom}_{A}(M,{\bm{R}}\Gamma_{I^{-}}(N)):={\bm{R}}\underline{{\rm
Hom}}_{A}(M,{\bm{R}}\Gamma_{I^{-}}(N))_{0}\simeq 0\qquad\text{ for all
}M,N\in\mathcal{E}$

We wish to show that the second functor in (3.10) is fully faithful by
verifying this condition. Thus, let $M\in\mathcal{D}^{-}_{{\rm coh},(<w)}({\rm
Gr}(A))$ and $N\in\mathcal{D}^{-}_{{\rm coh},[\geq w]}({\rm Gr}(A))$, we
compute

${\bm{R}}\underline{{\rm
Hom}}_{A}(M,{\bm{R}}\Gamma_{I^{-}}(N))\stackrel{{\scriptstyle(?)}}{{\simeq}}{\bm{R}}\underline{{\rm
Hom}}_{A}(M,{\bm{R}}\Gamma_{I^{-}}(A))\otimes_{A}^{{\bm{L}}}N\simeq\mathbb{D}_{Y}({\bm{R}}\Gamma_{I^{+}}(M))(-a)[-1]\otimes_{A}^{{\bm{L}}}N$

Notice that $\mathbb{D}_{Y}({\bm{R}}\Gamma_{I^{+}}(M))\otimes_{A}^{{\bm{L}}}N$
is in $\mathcal{D}_{>-w}\otimes_{A}^{{\bm{L}}}\mathcal{D}_{[\geq
w]}\subset\mathcal{D}_{>0}$. Therefore, its shift by $a\geq 0$ is zero at
weight zero.

We see once again that the argument falls short because of what seems like a
formal problem (more precisely, the equality marked with (?) may not hold).
However, notice that our arguments in either the formulations (3.6) or (3.10)
work if the $3$-term SOD restricts to $\mathcal{D}_{{\rm perf}}({\rm Gr}(A))$.
In the next section, we will see two examples in which it holds.

######  Remark 3.5.

One can “solve” the problem (3.7)(1) by formulating the duality
$\mathbb{D}_{A}$ in terms of a perfect pairing of DG categories

$\mathcal{D}_{[\geq w]}({\rm Gr}(A))\times\mathcal{D}_{[\leq-w]}({\rm
Gr}(A))\,\rightarrow\,\mathcal{D}(k)\,,\qquad(M,N)\mapsto(M\otimes^{{\bm{L}}}_{A}N)_{0}$

which then restricts to a pairing on the window subcategories
$\mathcal{D}_{[\geq w],(<w)}({\rm Gr}(A))$ and
$\mathcal{D}_{[\leq-w],(>-w)}({\rm Gr}(A))$. One can then try to show that it
is a perfect pairing. Unravelling the definition (at least in the way the
author formulated it), it corresponds to the functor (3.10), except that we do
not restrict to $\mathcal{D}^{-}_{{\rm coh}}$.

######  Remark 3.6.

The functor (3.10) is $\mathcal{O}_{Y}$-linear.

######  Remark 3.7.

By a result [8] of Kawamata, any two birational projective Calabi-Yau
varieties with at most $\mathbb{Q}$-factorial terminal singularities are
connected by a finite number of flops, each of which is obviously a Gorenstein
flop, and hence falls into our present setting.

######  Remark 3.8.

One might try to repeat the arguments in this section for a different version
of $\mathcal{D}({\rm Gr}(A))$. Some potential candidates are: (1) ${\rm
Ind}\,\mathcal{D}^{b}_{{\rm coh}}({\rm Gr}(A))$, (2) co-derived or contra-
derived categories, (3) homotopy category of (unbounded) complexes of finitely
generated projectives. For example, if we work with (3), then problems such as
(3.8) seem to not arise (although other problems might arise then).
Accordingly, the theory of relative homological algebra (e.g. [4]) might be
relevant here, although the author is not very familiar with this part of the
literature. If one changes the setting, one should also guarantee that results
of Section 2 can be carried over. It is not clear to us whether endeavors in
these formal directions will be useful (the author has spent a lot of time
trying out various formal modifications, but such effort has not been fruitful
so far).

##  4\. Some examples

We will present two examples in which the $3$-term SOD (3.5) restricts to
$\mathcal{D}_{{\rm perf}}({\rm Gr}(A))$, so that our argument does work. For
this to hold, it is certainly necessary that the functor $\mathcal{L}_{[\geq
w]}$ in (3.4) preserves $\mathcal{D}_{{\rm perf}}({\rm Gr}(A))$. The following
result says that this condition is also sufficient. A proof may be found in
[14] (a revised version is under preparation, for which the arguments will be
simplified).

######  Proposition 4.1.

Suppose that the functor $\mathcal{L}_{[\geq w]}$ in (3.4) preserves
$\mathcal{D}_{{\rm perf}}({\rm Gr}(A))$, then the the $3$-term SOD (3.5)
restricts to $\mathcal{D}_{{\rm perf}}({\rm Gr}(A))$.

In this case, the functor (3.10) restricts to

(4.1) ${\mathcal{D}_{{\rm perf}}(\mathfrak{X}^{+})}$${\mathcal{D}_{{\rm
perf},[\geq w],(<w)}({\rm Gr}(A))}$${\mathcal{D}_{{\rm
perf}}(\mathfrak{X}^{-})}$$\scriptstyle{\simeq}$$\scriptstyle{\mathcal{L}_{[\geq
w]}\circ{\bm{R}}j^{+}_{*}}$$\scriptstyle{(j^{+})^{*}}$$\scriptstyle{(j^{-})^{*}}$

Moreover, the arguments in the previous section then guarantees that, if the
duality
$\mathbb{D}_{Y}({\bm{R}}\Gamma_{I^{-}}(A))\cong{\bm{R}}\Gamma_{I^{+}}(A)(a)[1]$
holds for some $a\geq 0$, then (4.1) is fully faithful.

In general, it seems quite difficult to describe $\mathcal{L}_{[\geq w]}$
explicitly, as it is defined in terms of the Neeman-Brown representability
theorem. The closest to an explicit formula that we managed to get is the
following tensor form

(4.2) $\mathcal{L}_{[\geq w]}(M)\,=\,M_{\geq
w}\otimes^{{\bm{L}}}_{\mathcal{F}_{\geq w}}\mathcal{F}$

Here, $\mathcal{F}$ is the small pre-additive (i.e., ${\rm Ab}$-enriched)
category given by $\rm{Ob}(\mathcal{F})=\mathbb{Z}$ and
$\mathcal{F}(i,j)=A_{i-j}$. Then a graded $A$-module is the same as a right
module over $\mathcal{F}$ (recall that a right module over a small pre-
additive category $\mathcal{C}$ means an additive functor $\mathcal{C}^{{\rm
op}}\rightarrow{\rm Ab}$). Let $\mathcal{F}_{\geq w}\subset\mathcal{F}$ be the
full subcategory on the object set $\mathbb{Z}_{\geq w}\subset\mathbb{Z}$.
Then the inclusion functor $\mathcal{F}_{\geq w}\rightarrow\mathcal{F}$ gives
rise to restriction and induction functors between the module categories. In
particular, for any $M\in{\rm Mod}(\mathcal{F})$, we may restrict it to
$\mathcal{F}_{\geq w}$ to obtain $M_{\geq w}\in{\rm Mod}(\mathcal{F}_{\geq
w})$, and then tensor it back to $\mathcal{F}$ to obtain $M_{\geq
w}\otimes_{\mathcal{F}_{\geq w}}\mathcal{F}\in{\rm Mod}(\mathcal{F})$. The
right hand side of (4.2) is its derived functor.

The description (4.2) seems to be of little use in the two examples that we
present below. In both cases, the functor $\mathcal{L}_{[\geq w]}$ is
described by some ad-hoc arguments specific to the form of the graded ring $A$
in question.

The first example is the polynomial ring

(4.3)
$A=k[x_{1},\ldots,x_{p},y_{1},\ldots,y_{q}]\,,\qquad\deg(x_{i})>0,\,\,\deg(y_{j})<0$

Write $A=A^{+}\otimes A^{-}$, where $A^{+}=k[x_{1},\ldots,x_{p}]$ and
$A^{-}=k[y_{1},\ldots,y_{q}]$. Then we have

$\mathcal{D}_{[\geq w]}({\rm Gr}(A^{+}))\otimes A^{-}\subset\mathcal{D}_{[\geq
w]}({\rm Gr}(A))\qquad\text{and}\qquad\mathcal{D}_{<w}({\rm Gr}(A^{+}))\otimes
A^{-}\subset\mathcal{D}_{<w}({\rm Gr}(A))$

This allows us to compute the weight truncation of objects of the form
$M=M^{+}\otimes A^{-}$ for $M^{+}\in\mathcal{D}({\rm Gr}(A^{+}))$. Namely, we
have

$\mathcal{L}_{[\geq w]}^{A}(M^{+}\otimes A^{-})\,=\,\mathcal{L}_{[\geq
w]}^{A^{+}}(M^{+})\otimes A^{-}$

To show that $\mathcal{L}_{[\geq w]}^{A}$ preserves $\mathcal{D}_{{\rm
perf}}({\rm Gr}(A))$, it suffices to verify it on the split-generating objects
$\\{A(-i)\\}_{i\in\mathbb{Z}}$, each of which is of the form $M^{+}\otimes
A^{-}$ for $M^{+}=A^{+}(-i)$. Hence it suffices to compute $\mathcal{L}_{[\geq
w]}^{A^{+}}(A^{+}(-i))$. Since $A^{+}$ is non-negatively graded, we in fact
have $\mathcal{D}_{[\geq w]}({\rm Gr}(A^{+}))=\mathcal{D}_{\geq w}({\rm
Gr}(A^{+}))$, so that the weight truncation is easy to describe:

######  Lemma 4.2.

For any $M\in{\rm Gr}(A^{+})$, we have

$\mathcal{L}_{[\geq w]}^{A^{+}}(M)=M_{\geq
w}\,,\qquad\text{and}\qquad\mathcal{L}_{<w}^{A^{+}}(M)=M/M_{\geq w}$

As a result, we have

$\mathcal{L}_{[\geq w]}^{A}(A(-i))=((A^{+})_{\geq w-i}\otimes A^{-})(-i)$

Since $A^{+}$ is smooth, we have $(A^{+})_{\geq w-i}\in\mathcal{D}_{{\rm
perf}}({\rm Gr}(A^{+}))$. This shows the following

######  Proposition 4.3.

For the polynomial ring (4.3), weight truncation $\mathcal{L}_{[\geq w]}$
preserves $\mathcal{D}_{{\rm perf}}({\rm Gr}(A))$.

######  Remark 4.4.

Our argument here works whenever a graded ring can be written as
$A=A^{+}\otimes A^{-}$ where $A^{+}$ is non-negatively graded and $A^{-}$ is
non-positively graded. In this case, if $A^{+}$ is smooth, then
$\mathcal{L}_{[\geq w]}$ preserves $\mathcal{D}_{{\rm perf}}({\rm Gr}(A))$.
Interestingly, if $A^{-}$ is smooth, then by the arguments of [6], one can
show that $\mathcal{L}_{[\geq w]}$ preserves $\mathcal{D}^{b}_{{\rm coh}}({\rm
Gr}(A))$.

Proposition 4.3 shows that our approach in the previous section works
perfectly for the polynomial ring (4.3) (notice that we have established the
duality
$\mathbb{D}_{Y}({\bm{R}}\Gamma_{I^{-}}(A))\cong{\bm{R}}\Gamma_{I^{+}}(A)(a)[1]$
directly in Remark 2.13 without applying the main Theorems of Section 2). We
will now compute the corresponding functor (4.1). For simplicity, we will now
assume $w=0$. Let $\eta^{+}=\sum_{i=1}^{p}\deg(x_{i})$ and
$\eta^{-}=-\sum_{j=1}^{p}\deg(y_{j})$, then $a=\eta^{-}-\eta^{+}$.

By the computation in Remark 2.13, we see that

${\bm{R}}\Gamma_{I^{+}}(A)\in\mathcal{D}_{\leq-\eta^{+}}({\rm
Gr}(A))\qquad\text{and}\qquad{\bm{R}}\Gamma_{I^{-}}(A)\in\mathcal{D}_{\geq\eta^{-}}({\rm
Gr}(A))$

Thus, in particular, for $0\leq i<\eta^{+}$, the objects $A(-i)$ are in the
window subcategory $\mathcal{D}_{[\geq 0],(<0)}({\rm Gr}(A))$. This implies
that

(4.4) For $0\leq i<\eta^{+}$, the functor (4.1) for $w=0$ sends
$\mathcal{O}_{\mathfrak{X}^{+}}(-i)$ to $\mathcal{O}_{\mathfrak{X}^{-}}(-i)$.

In fact, this uniquely characterize the functor (4.1). Namely, the arguments
of [6] can be employed to give the following characterization of the window
subcategory by the “window length” $\eta^{+}$, which then implies that
$\\{\mathcal{O}_{\mathfrak{X}^{+}}(-i)\\}_{0\leq i<\eta^{+}}$ split generates
$\mathcal{D}_{{\rm perf}}(\mathfrak{X}^{+})$.

######  Proposition 4.5.

For the polynomial ring (4.3), we have $\mathcal{D}_{{\rm perf},(<w)}({\rm
Gr}(A))=\mathcal{D}_{{\rm perf},[<w+\eta^{+}]}({\rm Gr}(A))$. As a result, the
subcategory $\mathcal{D}_{{\rm perf},[\geq 0],(<0)}({\rm Gr}(A))$ is split
generated by $\\{A,A(-1),\ldots,A(-\eta^{+}+1)\\}$.

Similarly, we have $\mathcal{D}_{{\rm perf},(>-w)}({\rm
Gr}(A))=\mathcal{D}_{{\rm perf},[>-w-\eta^{-}]}({\rm Gr}(A))$, so that
$\mathcal{D}_{{\rm perf},[\leq 0],(>0)}({\rm Gr}(A))$ is split generated by
$\\{A,A(1),\ldots,A(\eta^{-}-1)\\}$.

By our above description of $\mathcal{L}_{[\geq 0]}$ in Lemma 4.2, we also
have

(4.5) For $i>0$, the functor (4.1) for $w=0$ sends
$\mathcal{O}_{\mathfrak{X}^{+}}(i)$ to the associated sheaf on
$\mathfrak{X}^{-}$ of the graded module $((A^{+})_{\geq i}\otimes A^{-})(i)$

The image of $\mathcal{O}_{\mathfrak{X}^{+}}(i)$ for $i\leq-\eta^{+}$ seems to
be more difficult to explicate, since $\mathcal{L}_{[\geq
0]}({\bm{R}}\Gamma_{I^{+}}(A(i)))$ does not vanish in this case, so the answer
invovles a certain cone. We will skip the details.

######  Remark 4.6.

The case $\deg(x_{i})=1$ and $\deg(y_{j})=-1$ corresponds to a standard
flip/flop (we then have $\mathfrak{X}^{\pm}=X^{\pm}$ and $\eta^{+}=p$,
$\eta^{-}=q$). In this case, the functor (4.1) is isomorphic the Fourier-Mukai
functor ${\bm{R}}q_{*}\circ{\bm{L}}p^{*}$ with respect to the standard span
$X^{-}\xleftarrow[]{q}\widetilde{X}\xrightarrow[]{p}X^{+}$. Indeed, one can
simply verify this on the split generators
$\\{\mathcal{O}_{X^{+}}(-i)\\}_{0\leq i<p}$ of $\mathcal{D}_{{\rm
perf}}(X^{+})$. The author thanks Alexander Kuznetsov for help with this
remark.

We now move to our second class of examples, which are the graded rings
associated to a class of $3$-fold flips of type A, worked out by Brown and
Reid (see [12, Section 11.2]).

Fix positive integers $d,e,\alpha,\beta,\lambda,\mu$ with ${\rm
gcd}(\lambda,\mu)=1$. Take111111We have renamed the generators of [12, Section
11.2] according to
$(t,u,x_{0},x_{1},y_{0},y_{1})\rightarrow(y_{3},z,y_{1},x_{1},y_{2},x_{2})$,
in order to conform with our standing convention that $\deg(x_{i})>0$,
$\deg(y_{i})<0$ and $\deg(z_{i})=0$.

(4.6) $A=k[x_{1},x_{2},y_{0},y_{1},y_{2},z]/(f_{1},f_{2})$

with degrees and relations given by

$\displaystyle\deg(x_{1})=\lambda\,,\quad\deg(x_{2})=\mu\,,\quad\deg(y_{1})=-\mu\,,\quad\deg(y_{2})=-\lambda-\mu
e\,,\quad\deg(y_{3})=-1\,,\quad\deg(z)=0$ $\displaystyle
f_{1}=x_{1}y_{2}-y_{1}^{e}z^{\alpha}-y_{3}^{\mu e}\,,\quad
f_{2}=y_{1}x_{2}-z^{\beta}-x_{1}^{d}y_{3}^{\lambda d}$

This is an example of a complete intersection, so we will work in that more
general context below.

(4.7)
$A=k[x_{1},\ldots,x_{p},y_{1},\ldots,y_{q},z_{1},\ldots,z_{r}]/(f_{1},\ldots,f_{s})\,,\qquad\deg(x_{i})>0,\,\,\deg(y_{i})<0,\,\,\deg(z_{i})=0$

Our method will work under the following assumptions (which is satisfied by
the example (4.6) of Brown and Reid):

(4.8) $A$ is a complete intersection (i.e., $\dim(A)=p+q+r-s$) and
$\deg(f_{i})\leq 0$ for each $1\leq i\leq s$.

If we write $C=k[x_{1},\ldots,x_{p},y_{1},\ldots,y_{q},z_{1},\ldots,z_{r}]$,
then this condition guarantees that the Koszul resolution
$K_{\bullet}(C,f_{1},\ldots,f_{s})\simeq A$ gives a resolution of $A$ by free
graded $C$-modules, all of which are generated in non-positive degrees. Thus,
it gives a non-positive presentation of $A$, in the sense of the following

######  Definition 4.7.

A Noetherian graded algebra $A$ is said to _have a non-positive presentation_
if there exists a map $C\rightarrow A$ from a polynomial graded algebra $C$
such that, as an object in $\mathcal{D}({\rm Gr}(C))$, we have
$A\in\mathcal{D}_{[\leq 0]}({\rm Gr}(C))$.

######  Proposition 4.8.

If $A$ has a non-positive presentation, then $\mathcal{L}_{[\geq w]}$
preserves $\mathcal{D}_{{\rm perf}}({\rm Gr}(A))$.

###### Proof.

In general, we have $\mathcal{D}_{[\geq w]}({\rm
Gr}(C))\otimes^{{\bm{L}}}_{C}A\subset\mathcal{D}_{[\geq w]}({\rm Gr}(A))$. The
condition on non-positive presentation also guarantees that
$\mathcal{D}_{<w}({\rm
Gr}(C))\otimes^{{\bm{L}}}_{C}A\subset\mathcal{D}_{<w}({\rm Gr}(A))$. Hence we
have $\mathcal{L}_{[\geq
w]}^{A}(M\otimes^{{\bm{L}}}_{C}A)\cong\mathcal{L}_{[\geq
w]}^{C}(M)\otimes^{{\bm{L}}}_{C}A$ for all $M\in\mathcal{D}({\rm Gr}(C))$.
Apply this to $M=C(-i)$ for $i\in\mathbb{Z}$ to conclude the proof. ∎

######  Remark 4.9.

With the help of computer programs, given any finite map $C\rightarrow A$ from
a polynomial graded algebra, one should be able to compute a free resolution
of $A$ as a graded $C$-module, from which one should be able to see if the
given map $C\rightarrow A$ gives a non-positive presentation. In fact, it
suffices to compute the weights of ${\rm
Tor}^{C}_{\bullet}(A,k[z_{1},\ldots,z_{r}])$ to verify that $C\rightarrow A$
gives a non-positive presentation.

Combining with the previous results, we then have the following

######  Theorem 4.10.

For the class of 3-fold flips described by (4.6), there is a fully faithful
functor $\mathcal{D}_{{\rm
perf}}(\mathfrak{X}^{+})\rightarrow\mathcal{D}_{{\rm perf}}(\mathfrak{X}^{-})$
given by (4.1).

######  Remark 4.11.

Let $X$ be a $\mathbb{Q}$-Gorenstein variety such that
$\mathcal{O}_{X}(iK_{X})$ is maximal Cohen-Macaulay for each $i\in\mathbb{Z}$
(this is satisfied, e.g., if $X$ has at most log terminal singularities, see
[7, Corollary 5.25]), one may define its _Gorenstein root_ to be the Deligne-
Mumford stack $\mathfrak{X}:=[\,\underline{{\rm
Spec}}_{X}(\bigoplus_{i\in\mathbb{Z}}\mathcal{O}_{X}(iK_{X}))\,/\,\mathbb{G}_{m}\,]$.
Theorem 4.10 then relates the derived categories of the Gorenstein roots of
the two sides of the flip.

One can also compute the functor (4.1) in the present setting. We will work
with (4.7) satisfying (4.8). Let
$C=k[x_{1},\ldots,x_{p},y_{1},\ldots,y_{q},z_{1},\ldots,z_{r}]$, and let
$C^{+}=k[x_{1},\ldots,x_{p}]$ and
$C^{-}=k[y_{1},\ldots,y_{q},z_{1},\ldots,z_{r}]$, so that $C=C^{+}\otimes
C^{-}$. Let $\eta^{+}=\sum_{i=1}^{p}\deg(x_{i})$. Then we have

(4.9) For $0\leq i<\eta^{+}$, the functor (4.1) for $w=0$ sends
$\mathcal{O}_{\mathfrak{X}^{+}}(-i)$ to $\mathcal{O}_{\mathfrak{X}^{-}}(-i)$.
(4.10) For $i>0$, the functor (4.1) for $w=0$ sends
$\mathcal{O}_{\mathfrak{X}^{+}}(i)$ to the associated sheaf on
$\mathfrak{X}^{-}$ of the complex of graded modules $((C^{+})_{\geq i}\otimes
C^{-})\otimes^{{\bm{L}}}_{C}A(i)$

We may also want an analogue of Proposition 4.5. For that, we will need to
replace the assumption (4.8) with the following stronger one (which is
satisfied by (4.6)):

(4.11) $\dim(A)=p+q+r-s$, $\dim(A/I^{+})=q+r-s$, and $\deg(f_{i})\leq 0$ for
each $1\leq i\leq s$.

Then we have the following

######  Proposition 4.12.

For $A$ in (4.7) satisfying (4.11), we have $\mathcal{D}_{{\rm
perf},(<w)}({\rm Gr}(A))=\mathcal{D}_{{\rm perf},[<w+\eta^{+}]}({\rm Gr}(A))$.
As a result, the subcategory $\mathcal{D}_{{\rm perf},[\geq 0],(<0)}({\rm
Gr}(A))$ is split generated by $\\{A,A(-1),\ldots,A(-\eta^{+}+1)\\}$.

Acknowledgement. The author thanks Daniel Halpern-Leistner and Valery Lunts
for helpful discussions. He thanks Alexander Kuznetsov for careful reading and
helpful comments and suggestions. He thanks Chen Jiang for help with Theorem
2.3.

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