# Closedness of the singular locus and generation for derived categories

Souvik Dey  S. Dey, Faculty of Mathematics and Physics, Department of Algebra,
Charles University, Sokolovská 83, 186 75 Praha, Czech Republic
<EMAIL_ADDRESS>and  Pat Lank  P. Lank, Department of Mathematics,
University of South Carolina, Columbia, SC 29208, U.S.A.
<EMAIL_ADDRESS>
###### Abstract.

This work is concerned with a relationship regarding the closedness of the
singular locus of a Noetherian scheme and existence of classical generators in
its category of coherent sheaves, associated bounded derived category, and
singularity category. Particularly, we extend an observation initially made by
Iyengar and Takahashi in the affine context to the global setting.
Furthermore, we furnish an example a Noetherian scheme whose bounded derived
category admits a classical generator, yet not every finite scheme over it
exhibits the same property.

###### Key words and phrases:

Triangulated categories, classical generation, bounded derived category,
singularity category, singular locus, abelian categories

###### 2020 Mathematics Subject Classification:

14F08 (primary), 18G80, 13D09, 14B05, 18E10

##  1\. Introduction

This note strengthens a relationship between the closedness of the singular
locus of a Noetherian scheme $X$ and the existence of classical generators in
the bounded derived category of coherent sheaves, denoted by
$D^{b}_{\operatorname{coh}}(X)$. By doing so, it establishes a useful
homological condition that can be used to probe whether an important
topological property concerning singularities of the scheme is satisfied.

In a triangulated category $\mathcal{T}$, an object $G$ is called a classical
generator if the smallest thick subcategory of $\mathcal{T}$ containing $G$,
denoted by $\langle G\rangle$, coincides with $\mathcal{T}$. Put differently,
any object in $\mathcal{T}$ can be built from $G$ utilizing a finite
combination of shifts, cones, and retracts of finite coproducts. This concept
was first introduced in [BvdB03].

There has been a very active front towards understanding conditions which
$D^{b}_{\operatorname{coh}}(X)$ admit a classical generator. These cases
include quasi-excellent Noetherian schemes of finite Krull dimension [Aok21],
Noetherian schemes $J\textrm{-}2$ [ELS20], Noetherian schemes admitting a
separator [Jat21], and several instances in the affine setting [DLT23, Ola23,
IT16, IT19].

An objective in this line of work is to explicitly identify classical
generators in $D^{b}_{\operatorname{coh}}(X)$. The progress made here includes
Noetherian schemes of prime characteristic [BIL+23], noncommutative techniques
[BDL24], and varieties over a field [Rou08, HHL23, HH23, FH23, Pir23, BE23,
LO24, Lan23].

Let us motivate the objective behind our note. In
$D^{b}_{\operatorname{coh}}(X)$, a perfect complex is an object which is
locally quasi-isomorphic to a bounded complex of finite locally free sheaves.
The triangulated subcategory of $D^{b}_{\operatorname{coh}}(X)$ consisting of
perfect complexes admits a classical generator [BvdB03].

A Noetherian scheme is regular if, and only if, every object in
$D^{b}_{\operatorname{coh}}(X)$ is quasi-isomorphic to a perfect; see [Nee21]
for recent developments. In other words, if there exists a perfect complex $P$
such that $\langle P\rangle=D^{b}_{\operatorname{coh}}(X)$, then $X$ is
regular, and vice versa.

On the contrary, if $X$ fails to be regular, yet
$D^{b}_{\operatorname{coh}}(X)$ still admits a classical generator, it follows
that the regular locus of $X$ must be open; see [IT19, Lemma 2.6] for the
affine case. In this regard, there are varying singularities based on the
openness of the regular locus.

The singular locus of a Noetherian scheme $X$ comprises points $p$ where
$\mathcal{O}_{X,p}$ is not a regular local ring, while its set-theoretic
complement is called the regular locus. We say $X$ is $J\textrm{-}0$ if its
regular locus contains a nonempty open subset and $J\textrm{-}1$ if the
regular locus is an open subset. For further details, refer to Remark 2.5.
This leads us to our main result.

######  Theorem 1.1.

For a Noetherian scheme $X$, the following conditions are equivalent:

  1. (1)

Every closed integral subscheme is $J\textrm{-}0$.

  2. (2)

Every closed integral subscheme is $J\textrm{-}1$.

  3. (3)

$D^{b}_{\operatorname{coh}}(Z)$ admits a classical generator for every closed
integral subscheme $Z$ of $X$

  4. (4)

$D_{\operatorname{sg}}(Z)$ admits a classical generator for every closed
integral subscheme $Z$ of $X$

  5. (5)

$\operatorname{coh}Z$ admits a classical generator for every closed integral
subscheme $Z$ of $X$.

Moreover, if any of these conditions are satisfied, both
$D^{b}_{\operatorname{coh}}(Y)$ and $D_{\operatorname{sg}}(Y)$ admit a
classical generator for any closed subscheme $Y$ of $X$.

Theorem 1.1 connects the closedness of the singular locus and the existence of
classical generators in various categories of interest that are constructed
from coherent sheaves. For background on these categories, please see Section
2. Our work is a globalization of [IT19, Theorem 1.1], which observed a
similar result in the setting of an affine scheme. However, we give an
independent proof without relying on their techniques, offering a
geometrically flavored strategy.

An important consequence to our result lies in its generality. We exhibit this
with an example satisfying the conditions of Theorem 1.1, yet does not qualify
as quasi-excellent nor $J\textrm{-}2$ scheme, ensuring the results of [Aok21]
and [ELS20] are not applicable. This example is follows a construction by
Nagata, cf. Example 3.8.

######  Acknowledgements.

We thank Srikanth Iyengar and Josh Pollitz for comments on an earlier draft.
Moreover, the second author would like to thank Takumi Murayama for references
and correspondence in regards to Example 3.8. The first author was partially
supported by the Charles University Research Center program No.
UNCE/24/SCI/022 and a grant GA CR 23-05148S from the Czech Science Foundation.

######  Notation 1.2.

Let $X$ be a Noetherian scheme. We will consider the following triangulated
categories:

  1. (1)

$D(X)$ is the unbounded derived category of complexes of
$\mathcal{O}_{X}$-modules

  2. (2)

$D_{\operatorname{Qcoh}}(X)$ is the unbounded derived category of complexes of
$\mathcal{O}_{X}$-modules with quasi-coherent cohomology

  3. (3)

$D^{b}_{\operatorname{coh}}(X)$ is the derived category of bounded complexes
of $\mathcal{O}_{X}$-modules with coherent cohomology

  4. (4)

$\operatorname{perf}X$ is category of perfect complexes on $X$, i.e. those
objects in $D^{b}_{\operatorname{coh}}(X)$ which locally are quasi-isomorphic
to a bounded complex of locally free sheaves of finite rank

  5. (5)

$D_{\operatorname{sg}}(X)$ is the Verdier quotient of
$D^{b}_{\operatorname{coh}}(X)$ by $\operatorname{perf}X$, see Remark 2.7 for
details.

##  2\. Generation

This section briefly discusses notions of generation for both triangulated and
abelian categories. The primary sources of references are respectively
[BvdB03, Nee21, Rou08] and [DT14, DLT23, IT16].

###  2.1. Triangulated categories

Let $\mathcal{T}$ be a triangulated category with shift functor
$[1]\colon\mathcal{T}\to\mathcal{T}$ and $\mathcal{S}$ be a subcategory of
$\mathcal{T}$.

######  Definition 2.1.

  1. (1)

A triangulated subcategory of $\mathcal{T}$ is thick if it is closed under
direct summands. The smallest thick subcategory of $\mathcal{T}$ containing
$\mathcal{S}$ is denoted $\langle\mathcal{S}\rangle$.

  2. (2)

Consider the following additive subcategories of $\mathcal{T}$:

    1. (a)

$\operatorname{add}(\mathcal{S})$ is the strictly full subcategory of retracts
of finite coproducts of shifts of objects in $\mathcal{S}$

    2. (b)

$\langle\mathcal{S}\rangle_{0}$ consists of all objects in $\mathcal{T}$
isomorphic to the zero object

    3. (c)

$\langle\mathcal{S}\rangle_{1}:=\operatorname{add}(\mathcal{S})$

    4. (d)

$\langle\mathcal{S}\rangle_{n}:=\operatorname{add}\\{\operatorname{cone}(\phi):\phi\in\operatorname{Hom}_{\mathcal{T}}(\langle\mathcal{S}\rangle_{n-1},\langle\mathcal{S}\rangle_{1})\\}$
if $n\geq 2$.

######  Remark 2.2.

In the notation of Definition 2.1, there exists an exhaustive ascending chain
of (additive) subcategories for the smallest thick subcategory containing
$\mathcal{S}$. That is, $\langle\mathcal{S}\rangle_{i}$ is contained in
$\langle\mathcal{S}\rangle_{i+1}$ for all $i$ and $\langle\mathcal{S}\rangle$
coincides with $\bigcup^{\infty}_{n=0}\langle\mathcal{S}\rangle_{n}$.

######  Definition 2.3.

An object $G$ of $\mathcal{T}$ is called a classical generator if $\langle
G\rangle=\mathcal{T}$. Additionally, if there exists $n\geq 0$ such that
$\langle G\rangle_{n}=\mathcal{T}$, we say $G$ is a strong generator.

######  Example 2.4.

  1. (1)

If $X$ is a regular Noetherian scheme, then $D^{b}_{\operatorname{coh}}(X)$
admits a classical generator, cf. [BvdB03, Theorem 3.1.1]. Additionally, if
$X$ is separated and of finite Krull dimension, then there exists a strong
generator for $D^{b}_{\operatorname{coh}}(X)$, cf. [Nee21, Theorem 0.5]. In
the special case $X$ is quasi-affine or a quasi-projective variety over a
field, one can explicitly identify such objects, cf. respectively [Ola23,
Corollary 5] and [Orl09, Theorem 4].

  2. (2)

Let $X$ be a Noetherian $F$-finite scheme (i.e. whose Frobenius morphism is
finite). If $G$ is a classical generator for $\operatorname{perf}X$, then
$F_{\ast}^{e}G$ is a classical generator for $D^{b}_{\operatorname{coh}}(X)$
whenever $e\gg 0$. See [BIL+23, Theorem A] for details.

  3. (3)

$D^{b}_{\operatorname{coh}}(X)$ admits a strong generator for any quasi-
excellent separated Noetherian scheme $X$ of finite Krull dimension, cf.
[Aok21, Main Theorem].

  4. (4)

Suppose $\pi\colon\widetilde{X}\to X$ is an alteration of varieties over a
field where $\widetilde{X}$ is a smooth projective variety. If $\mathcal{L}$
is an ample line bundle on $\widetilde{X}$, then
$\mathbb{R}\pi_{\ast}(\bigoplus^{\dim\widetilde{X}}_{i=1}\mathcal{L}^{\otimes
i})$ is a strong generator for $D^{b}_{\operatorname{coh}}(X)$. See [DL24,
Example 3.8] for details. If $X$ is a variety over a perfect field, then such
alterations exist, cf. [dJ96, Theorem 4.1].

######  Remark 2.5.

Let $X$ be a Noetherian scheme. For further background on the following,
please refer to [Sta23, Tag 07P6] and [Sta23, Tag 07R2].

  1. (1)

The regular locus of $X$, denoted $\operatorname{Reg}(X)$, is the collection
of points $p$ of $X$ such that $\mathcal{O}_{X,p}$ is a regular local ring.
The singular locus of $X$ is the collection
$\operatorname{Sing}(X):=X\setminus\operatorname{Reg}(X)$.

  2. (2)

We say $X$ is $J\textrm{-}0$ if the regular locus of $X$ contains an nonempty
open subset, and is $J\textrm{-}1$ if the regular locus of $X$ is an open
subset. If $D_{\textrm{sg}}(X)$ admits a classical generator, then $X$ is
$J\textrm{-}1$, cf. [IT19, Lemma 2.9] for the affine case.

  3. (3)

A Noetherian scheme $X$ is said to be $J\textrm{-}2$ if for every morphism
$Y\to X$ which is locally of finite type the regular locus
$\operatorname{Reg}(Y)$ is open in $Y$. If $X$ is a $J\textrm{-}2$ scheme,
then $D^{b}_{\operatorname{coh}}(X)$ admits a classical generator, cf. [ELS20,
Theorem 4.15].

  4. (4)

Any quasi-excellent Noetherian scheme is $J\textrm{-}2$, i.e. proper schemes
over a complete local ring.

######  Remark 2.6.

Let $X$ be a Noetherian scheme and $E$ an object of
$D^{b}_{\operatorname{coh}}(X)$.

  1. (1)

$\operatorname{Supp}(E):=\cup_{n=-\infty}^{\infty}\operatorname{Supp}(\mathcal{H}^{n}(E))$

  2. (2)

$E$ is supported on a closed subscheme $Z$ of $X$ if $\operatorname{Supp}(E)$
is contained in $Z$

  3. (3)

$E$ is scheme-theoretically supported on a closed subscheme $Z$ of $X$ if
there is an object $E^{\prime}$ of $D^{b}_{\operatorname{coh}}(Z)$ such that
$i_{\ast}E^{\prime}\cong E$ where $i$ is the associated closed immersion.

  4. (4)

Any object $E$ is supported on a closed subscheme is scheme-theoretically
supported on a nilpotent thickening of the closed subscheme, cf. [Rou08, Lemma
7.40].

Given a closed subscheme $Z$ of $X$, then we consider the following thick
subcategories of $D^{b}_{\operatorname{coh}}(X)$:

  1. (1)

$D^{b}_{\operatorname{coh,Z}}(X)$ is the strictly full subcategory of objects
in $D^{b}_{\operatorname{coh}}(X)$ whose cohomology is supported in $Z$.

  2. (2)

$\operatorname{perf}_{Z}X$ is the strictly full subcategory of perfect
complexes on $X$ whose cohomology is supported in $Z$.

######  Remark 2.7.

Let $X$ be a Noetherian scheme. The singularity category of $X$, denoted
$D_{\operatorname{sg}}(X)$, is the Verdier quotient of
$D^{b}_{\operatorname{coh}}(X)$ by $\operatorname{perf}X$. Note that $X$ is
regular if, and only if, $D_{\operatorname{sg}}(X)$ is trivial. Please see
[Orl04, Buc21] for further background in both geometric and algebraic
contexts. The thick subcategory of objects $E$ in $D_{\operatorname{sg}}(X)$
which are isomorphic to the image of an object $E$ in
$D^{b}_{\operatorname{coh},Z}(X)$ under the quotient functor
$D^{b}_{\operatorname{coh}}(X)\to D_{\operatorname{sg}}(X)$ is denoted
$D_{\operatorname{sg,Z}}(X)$.

######  Remark 2.8.

  1. (1)

Let $X$ be a Noetherian scheme. If $j\colon U\to X$ denotes an open immersion
and $Z=X\setminus U$, then there exists Verdier localization sequences:

    1. (a)

$D^{b}_{\operatorname{coh},Z}(X)\to
D^{b}_{\operatorname{coh}}(X)\xrightarrow{\mathbb{L}j^{\ast}}D^{b}_{\operatorname{coh}}(U)$

    2. (b)

$\operatorname{perf}_{Z}X\to\operatorname{perf}X\xrightarrow{\mathbb{L}j^{\ast}}\operatorname{perf}U$

    3. (c)

$D_{\operatorname{sg},Z}(X)\to
D_{\operatorname{sg}}(X)\xrightarrow{\mathbb{L}j^{\ast}}D_{\operatorname{sg}}(U)$

  2. (2)

If there exists a Verdier localization sequence:

$\mathcal{K}\to\mathcal{T}\to\mathcal{T}/\mathcal{S}$

where $\mathcal{K}$ and $\mathcal{T}/\mathcal{K}$ admit classical generators,
then so does $\mathcal{T}$, cf. [ELS20, Proposition 4.3].

######  Definition 2.9.

Suppose $\mathcal{T}$ admits all small coproducts. Consider the following
additive subcategories of $\mathcal{T}$:

  1. (1)

$\operatorname{Add}(\mathcal{S})$ is the strictly full subcategory of retracts
of small coproducts of shifts of objects in $\mathcal{S}$

  2. (2)

$\overline{\langle\mathcal{S}\rangle}_{0}$ consists of all objects in
$\mathcal{T}$ isomorphic to the zero object

  3. (3)

$\overline{\langle\mathcal{S}\rangle}_{1}:=\operatorname{Add}(\mathcal{S})$

  4. (4)

$\overline{\langle\mathcal{S}\rangle}_{n}:=\operatorname{Add}\\{\operatorname{cone}(\phi):\phi\in\operatorname{Hom}_{\mathcal{T}}(\overline{\langle\mathcal{S}\rangle}_{n-1},\overline{\langle\mathcal{S}\rangle}_{1})\\}$
if $n\geq 2$.

###  2.2. Abelian categories

Let $\mathcal{A}$ be an abelian category and $\mathcal{S}$ be a subcategory of
$\mathcal{A}$.

######  Definition 2.10.

  1. (1)

$\operatorname{add}(\mathcal{S})$ is the smallest strictly full subcategory of
$\mathcal{A}$ closed under direct summands of finite coproducts of
$\mathcal{S}$

  2. (2)

A strictly full additive subcategory $\mathcal{T}$ of $\mathcal{A}$ is said to
be thick if it is closed under direct summands and if given any object
appearing in a short exact sequence

$0\to A\to B\to C\to 0$

where the other two objects belong to $\mathcal{T}$, so does the third.

  3. (3)

The smallest thick subcategory of $\mathcal{A}$ containing $\mathcal{S}$ is
denoted $\operatorname{thick}(\mathcal{S})$.

  4. (4)

Consider the following additive subcategories:

    1. (a)

$\operatorname{th}^{0}(\mathcal{S})$ consists of all objects in $\mathcal{A}$
isomorphic to the zero object

    2. (b)

$\operatorname{th}^{1}(\mathcal{S}):=\operatorname{add}(\mathcal{S})$

    3. (c)

$\operatorname{th}^{n}(\mathcal{S})$ is defined as the strictly full
subcategory consisting of direct summands of objects where appear in a short
exact sequence

$0\to A\to B\to C\to 0$

where the other two objects belong to $\operatorname{th}^{n-1}(\mathcal{S})$.

The following is an easy generalization of [DLT23, Lemma 4.2.2].

######  Lemma 2.11.

For any subcategory $\mathcal{S}$ of $\mathcal{A}$, we have
$\operatorname{thick}(\mathcal{S})=\bigcup^{\infty}_{i=0}\operatorname{th}(\mathcal{S})$.

###### Proof.

First, we show by induction that $\operatorname{th}^{n}(\mathcal{S})$ is
contained in $\operatorname{thick}(\mathcal{S})$ for each $n$. Let $E$ belong
to $\operatorname{th}^{1}(\mathcal{S})$. Then $E$ is a direct summand of a
finite direct sum of objects in $\mathcal{S}$, but any such object must belong
to $\operatorname{thick}(\mathcal{S})$. Assume there exists $n$ such that
$\operatorname{th}^{k}(\mathcal{S})$ is contained in
$\operatorname{thick}(\mathcal{S})$ for each $1\leq k\leq n$. Let $E$ be in
$\operatorname{th}^{n+1}(\mathcal{S})$. There exists a short exact sequence in
$\mathcal{A}$:

$0\to A\to B\to C\to 0$

where $E$ is a direct summand of one of the objects and the other two are
objects of $\operatorname{th}^{n}(\mathcal{S})$. We have three cases to check,
and they can all be shown similarly to one another, so assume that $A,B$ both
belong to $\operatorname{th}^{n}(\mathcal{S})$. However, the definition of a
thick subcategory tells us $C$ belongs to $\operatorname{thick}(\mathcal{S})$
via the two-out-of-three property for short exact sequences. Hence, we have
shown that $\cup^{\infty}_{n=1}\operatorname{th}^{n}(\mathcal{S})$ is
contained in $\operatorname{thick}(\mathcal{S})$.

Lastly, we show $\bigcup^{\infty}_{i=0}\operatorname{th}(\mathcal{S})$ is a
thick subcategory of $\mathcal{A}$. For each $N\geq n$, we know that
$\operatorname{th}^{n}(\mathcal{S})$ is contained in
$\operatorname{th}^{N}(\mathcal{S})$. Suppose there is a short exact sequence
where two of the three objects belong to
$\bigcup^{\infty}_{i=0}\operatorname{th}(\mathcal{S})$:

$0\to A\to B\to C\to 0.$

There are three cases to check, and as above, it suffices to check one of
them. Assume that $A$ is an object in $\operatorname{th}^{s}(\mathcal{S})$ and
$B$ is an object in $\operatorname{th}^{t}(\mathcal{A})$. Set
$v=\max\\{s,t\\}$. Then $A,B$ are objects of
$\operatorname{th}^{v}(\mathcal{A})$, and so $C$ belongs to
$\operatorname{th}^{v+1}(\mathcal{S})$. Moreover, the definition of each
$\bigcup^{\infty}_{i=0}\operatorname{th}(\mathcal{S})$ ensures it is closed
under direct summands, and so this shows the desired claim. ∎

######  Lemma 2.12.

If $\operatorname{thick}(\mathcal{S})=\mathcal{A}$, then
$\langle\mathcal{S}\rangle=D^{b}(\mathcal{A})$.

###### Proof.

We know from Lemma 2.11 that
$\operatorname{thick}(\mathcal{S})=\bigcup^{\infty}_{i=0}\operatorname{th}(\mathcal{S})$.
Choose an object $E$ in $D^{b}(\mathcal{A})$. There exists a distinguished
triangle in $D^{b}(\mathcal{A})$:

$Z(E)\to E\to B(E)[-1]\to Z(E)[1]$

where $Z(E)$ is the complex of cycles and $B(E)$ is the complex of boundaries.
Note that the differentials of both $Z(E),B(E)[-1]$ is zero, so these are
complexes of shifts of objects in $\mathcal{A}$. If one can show that
$Z(E),B(E)$ belongs to $\langle\mathcal{S}\rangle$, then the desired claim
holds as one would have $E$ is in $\langle\mathcal{S}\rangle$. It suffices to
check that each object of $A$ belongs to $\langle\mathcal{S}\rangle$. However,
an induction argument tells us $\operatorname{th}^{n}(\mathcal{S})$ is
contained in $\langle\mathcal{S}\rangle_{2^{n}}$ for all $n$. The case $n=1$
is clear, so assume there exists $n$ such that
$\operatorname{th}^{k}(\mathcal{S})$ is contained in
$\langle\mathcal{S}\rangle_{2^{k-1}}$ for all $1\leq k\leq n$. Let $E$ belong
to $\operatorname{th}^{n+1}(\mathcal{S})$. There exists a short exact sequence
in $\mathcal{A}$:

$0\to A\to B\to C\to 0$

where $E$ is a direct summand of one of the objects and the other two are
objects of $\operatorname{th}^{n}(\mathcal{A})$. This gives us a distinguished
triangle in $D^{b}(\mathcal{A})$:

$A\to B\to C\to A[1].$

There are three cases, lets prove it for the case where $A,B$ are in
$\operatorname{th}^{n}(\mathcal{S})$ as the others follow similarly. The
inductive hypothesis tells us $A,B$ belong to
$\langle\mathcal{S}\rangle_{2^{n-1}}$. Hence, it follows that $B$ belongs to
$\langle\mathcal{S}\rangle_{2^{n}}$. This completes the proof. ∎

##  3\. Results

This section establishes Theorem 1.1. The following lemma might be known to
experts, but we spell it out for the sake of convenience. For details on Serre
subcategories, please refer to [Sta23, Tag 02MN].

######  Lemma 3.1.

Let $\mathcal{A}$ be an abelian category, $\mathcal{B}$ be a Serre
subcategory, and $\pi\colon\mathcal{A}\to\mathcal{A}/\mathcal{B}$ be the
associated quotient functor. Let $\mathcal{D}$ be a thick subcategory of
$\mathcal{A}$ containing $\mathcal{B}$. If there exists objects $D$ in
$\mathcal{D}$ and $A$ in $\mathcal{A}$ such that $\pi(D)\cong\pi(A)$ in
$\mathcal{A}/\mathcal{B}$, then $A$ is an object of $\mathcal{D}$.

###### Proof.

Let $f\colon\pi(D)\to\pi(A)$ be an isomorphism in $\mathcal{A}/\mathcal{B}$.
There exists a short exact sequence in $\mathcal{A}/\mathcal{B}$:

$0\to\pi(D)\xrightarrow{f}\pi(A)\to 0\to 0.$

The map $f$ is the image of a map $f^{\prime}\colon D^{\prime}\to
A/A^{\prime}$ where $D^{\prime},A^{\prime}$ are respectively subobjects of
$D,A$ and $D/D^{\prime},A^{\prime}$ belong to $\mathcal{B}$. We have a short
exact sequence in $\mathcal{A}$:

$0\to D^{\prime}\to D\to D/D^{\prime}\to 0.$

If both $D$ and $D/D^{\prime}$ belong to $\mathcal{D}$ (i.e.
$\mathcal{B}\subseteq\mathcal{D}$), then $D^{\prime}$ is in $\mathcal{D}$. Set
$D_{1}:=D^{\prime}/\ker f^{\prime}$. Since $\pi(f^{\prime})$ is a
monomorphism, we know that $\ker f^{\prime}$ is in $\mathcal{B}$, and hence,
$\ker f^{\prime}$ is in $\mathcal{D}$. There exists a short exact sequence in
$\mathcal{A}$:

$0\to\ker f^{\prime}\to D^{\prime}\to D_{1}\to 0.$

As both $\ker f^{\prime}$ and $D^{\prime}$ belong to $\mathcal{D}$, we have
$D_{1}$ belongs to $\mathcal{D}$. Note that
$D_{1}\cong\operatorname{coim}f^{\prime}$ in $\mathcal{A}$. Let
$f_{1}\colon\operatorname{coim}f^{\prime}\to A/A^{\prime}$ be the natural map
induced by $f^{\prime}$. There exists a short exact sequence in $\mathcal{A}$:

$0\to\operatorname{coim}f^{\prime}\to
A/A^{\prime}\to\operatorname{coker}f_{1}\to 0.$

Tying our work so far together, there exists a commutative diagram in
$\mathcal{A}/\mathcal{B}$ (cf. [Gab62, $\S 3.1$ Corollaire 1]):

${0}$${{\pi(D)}}$${{\pi(A)}}$${0}$${0}$${0}$${{\pi(\operatorname{coim}f^{\prime})}}$${{\pi(A/A^{\prime})}}$${{\pi(\operatorname{coker}f_{1})}}$${0}$$\scriptstyle{\pi(f_{1})}$$\scriptstyle{w}$$\scriptstyle{v}$$\scriptstyle{f}$$\scriptstyle{u}$

where $u,v,w$ are isomorphisms in $\mathcal{A}/\mathcal{B}$. This tells us
that $\pi(f_{1})$ is an isomorphism, and so both $\ker f_{1}$ and
$\operatorname{coker}f_{1}$ belong to $\mathcal{D}$ as these belong to
$\mathcal{B}$ (cf. [Gab62, $\S 3.1$ Lemme 4]). If both
$\operatorname{coim}f^{\prime}$ and $\operatorname{coker}f_{1}$ are in
$\mathcal{D}$, then $A/A^{\prime}$ belongs to $\mathcal{D}$. Once more, there
is the short exact sequence in $\mathcal{A}$:

$0\to A^{\prime}\to A\to A/A^{\prime}\to 0.$

If $A^{\prime}$ and $A/A^{\prime}$ are in $\mathcal{D}$, then so is $A$, which
shows the desired claim. ∎

######  Lemma 3.2.

Let $\mathcal{A}$ be an abelian category, $\mathcal{B}$ be a Serre
subcategory, and $\pi\colon\mathcal{A}\to\mathcal{A}/\mathcal{B}$ be the
associated quotient functor. There exists a bijective correspondence of thick
subcategories $\mathcal{C}$ of $\mathcal{A}/\mathcal{B}$ and thick
subcategories of $\mathcal{A}$ containing $\mathcal{B}$.

###### Proof.

Let $\mathcal{C}$ be a strictly full subcategory of $\mathcal{A}/\mathcal{B}$.
Let $\pi\colon\mathcal{A}\to\mathcal{A}/\mathcal{B}$ be the quotient functor.
Set $\pi^{-1}(\mathcal{C})$ to be the collection of objects $C$ in
$\mathcal{A}$ such that $\pi(C)$ belongs to $\mathcal{C}$.

Suppose $\mathcal{C}$ is thick. We show that $\pi^{-1}(\mathcal{C})$ is a
thick subcategory of $\mathcal{A}$ containing $\mathcal{B}$. It is clear that
$\mathcal{B}$ is contained in $\pi^{-1}(\mathcal{C})$ as
$\mathcal{B}=\ker\pi$. Suppose we have a short exact sequence in $\mathcal{A}$
where two of the three objects belong to $\pi^{-1}(\mathcal{C})$. Applying
$\pi$ gives short exact in $\mathcal{A}/\mathcal{B}$, and so the third object
must belong to $\mathcal{C}$. Hence, the third object belongs to
$\pi^{-1}(\mathcal{C})$. A similar argument tells us that
$\pi^{-1}(\mathcal{C})$ is closed under direct summands. It is evident that
$\pi(\pi^{-1}(\mathcal{C}))=\mathcal{C}$. Let $\mathcal{D}$ be a thick
subcategory of $\mathcal{A}$ containing $\mathcal{B}$. Hence, we have shown
that each thick subcategory of $\mathcal{A}/\mathcal{B}$ is the essential
image of a thick subcategory of $\mathcal{A}$ containing $\mathcal{B}$.

It suffices to check that every thick subcategory $\mathcal{D}$ of
$\mathcal{A}$ containing $\mathcal{B}$ is of the form
$\pi^{-1}(\mathcal{D}^{\prime})$ where $\mathcal{D}^{\prime}$ is a thick
subcategory of $\mathcal{A}/\mathcal{B}$. Let $\mathcal{D}^{\prime}$ be the
thick subcategory of $\mathcal{A}/\mathcal{B}$ generated by the essential
image of $\pi(\mathcal{D})$. We show that
$\mathcal{D}=\pi^{-1}(\mathcal{D}^{\prime})$. If $\pi(\mathcal{D})$ is
contained in $\mathcal{D}^{\prime}$, then $\mathcal{D}$ is contained in
$\pi^{-1}(\mathcal{D}^{\prime})$. It suffices to check the reverse inclusion.
Let $E$ belong to $\pi^{-1}(\mathcal{D})$. Then $\pi(E)$ is in
$\mathcal{D}^{\prime}$. By Lemma 2.11, there exists an $n\geq 0$ such that
$\pi(E)$ belongs to $\operatorname{th}^{n}(\pi(\mathcal{D}))$. We induct on
$n$ to show that if $\pi(E)$ is in $\operatorname{th}^{n}(\pi(\mathcal{D}))$,
then $E$ belongs to $\mathcal{D}$. If $n=0$, then we already know that
$\mathcal{B}$ is contained in $\pi^{-1}(\mathcal{D}^{\prime})$. Suppose
$\pi(E)$ is in $\operatorname{th}^{1}(\pi(\mathcal{D}))$. There exists
$E^{\prime}$ in $\mathcal{A}/\mathcal{B}$ such that $\pi(E)\oplus E^{\prime}$
is a finite coproduct of objects in $\pi(\mathcal{D})$. That is, $\pi(E)\oplus
E^{\prime}\cong\oplus^{n}_{i=1}\pi(D_{i})$ for $\pi(D_{i})$ in
$\pi(\mathcal{D})$. As the quotient functor
$\pi\colon\mathcal{A}\to\mathcal{A}/\mathcal{B}$ is exact (cf. [Gab62, $\S
3.1$ Proposition 1]), it follows that $\pi(E)\oplus
E^{\prime}\cong\pi(\oplus^{n}_{i=1}D_{i})$ in $\mathcal{A}/\mathcal{B}$. By
Lemma 3.1, we see that $\oplus^{n}_{i=1}D_{i}$ is in $\mathcal{D}$. Choose
$E^{\prime\prime}$ in $\mathcal{A}$ such that $\pi(E^{\prime\prime})\cong
E^{\prime}$. Once more, it follows that $\pi(E\oplus
E^{\prime\prime})\cong\pi(\oplus^{n}_{i=1}D_{i})$. Hence, $E\oplus
E^{\prime\prime}$ is in $\mathcal{D}$. As $\mathcal{D}$ is thick, it follows
that $E$ is in $\mathcal{D}$. This shows the inductive step.

Assume there exists $n\geq 0$ such that for all $0\leq k\leq n$ and all
objects $A$ of $\mathcal{A}$, if $\pi(A)$ is in
$\operatorname{th}^{k}(\pi(\mathcal{D}))$, then $A$ belongs to $\mathcal{D}$.
Suppose $E$ is an object of $\mathcal{A}$ such that $\pi(E)$ is in
$\operatorname{th}^{n+1}(\pi(\mathcal{D}))$. There exists a short exact
sequence

$0\to M\to N\to L\to 0$

where two of the three objects belong to
$\operatorname{th}^{n}(\pi(\mathcal{D}))$ and $\pi(E)$ is a direct summand of
the third object. Without loss of generality, we can assume $\pi(E)$ is a
direct summand of $L$. There exists a commutative diagram in
$\mathcal{A}/\mathcal{B}$ (cf. [Gab62, $\S 3.1$ Corollaire 1]):

${0}$${{M}}$${{N}}$${{\pi(E)\oplus\pi(E^{\prime})}}$${0}$${0}$${{\pi(M^{\prime})}}$${{\pi(N^{\prime})}}$${{\pi(L^{\prime})}}$${0}$$\scriptstyle{w}$$\scriptstyle{v}$$\scriptstyle{u}$

where $u,v,w$ are isomorphisms in $\mathcal{A}/\mathcal{B}$. The last row
above is the direct image of the short exact sequence in $\mathcal{A}$:

$0\to M^{\prime}\to N^{\prime}\to L^{\prime}\to 0.$

Applying Lemma 3.1, we see that $M^{\prime},N^{\prime}$ are in $\mathcal{D}$.
Hence, $L^{\prime}$ is in $\mathcal{D}$. As $\pi(E\oplus
E^{\prime}),\pi(L^{\prime})$ are isomorphic in $\mathcal{A}/\mathcal{B}$,
Lemma 3.1 tells us once more that $E\oplus E^{\prime}$ belongs to
$\mathcal{D}$. This completes the proof by induction. ∎

The following has a special case for the category of coherent sheaves on a
Noetherian scheme, cf. [ELS20, Remark 4.5].

######  Lemma 3.3.

(Elagin-Lunts-Schnürer) Let $\mathcal{A}$ be an abelian category,
$\mathcal{S}$ be a Serre subcategory, and
$\pi\colon\mathcal{A}\to\mathcal{A}/\mathcal{B}$ be the associated quotient
functor. If there exists $B$ in $\mathcal{B}$ and $G$ in
$\mathcal{A}/\mathcal{B}$ such that $\operatorname{thick}(B)=\mathcal{B}$ and
$\operatorname{thick}(G)=\mathcal{A}/\mathcal{B}$, then
$\operatorname{thick}(B\oplus G)=\mathcal{A}$.

###### Proof.

By Lemma 3.2, the smallest thick subcategory generated by $G\oplus B$
coincides with $\mathcal{A}$. ∎

######  Lemma 3.4.

Let $X$ be a Noetherian scheme. If $i\colon Z\to X$ is a closed immersion with
$Z$ reduced and there exists $G$ in $\operatorname{coh}Z$ such that
$\operatorname{thick}(G)=\operatorname{coh}Z$, then
$\operatorname{thick}(i_{\ast}G)=\operatorname{coh}_{Z}X$.

###### Proof.

Note that $\operatorname{coh}_{Z}X$ is a thick subcategory of
$\operatorname{coh}X$, and so $\operatorname{thick}(i_{\ast}G)$ is contained
in $\operatorname{coh}_{Z}X$. We check the converse. Let $E$ be an object of
$\operatorname{coh}_{Z}X$. Let $\mathcal{I}$ be the ideal sheaf corresponding
to $Z$. There is an $n\geq 0$ such that $\mathcal{I}^{n}E=(0)$. This gives us
a descending chain in $\operatorname{coh}X$:

$(0)=\mathcal{I}^{n}E\subseteq\mathcal{I}^{n-1}E\subseteq\cdots\subseteq\mathcal{I}E\subseteq
E.$

Each quotient $\mathcal{I}^{j}E/\mathcal{I}^{j+1}E$ are annihilated by
$\mathcal{I}$, and so these belong to $\operatorname{thick}(i_{\ast}G)$. Thus,
$E$ belongs to $\operatorname{thick}(i_{\ast}G)$. ∎

######  Lemma 3.5.

Let $X$ be a Noetherian scheme. Let $\pi_{i}\colon Z_{i}\to X$ denote the
closed immersions from the irreducible components of $X$. If for each $1\leq
i\leq n$ there exists $G_{i}$ in $\operatorname{coh}Z_{i}$ such that
$\operatorname{thick}(G_{i})=\operatorname{coh}Z_{i}$, then
$\operatorname{thick}(\oplus^{n}_{i=1}\pi_{i,\ast}G_{i})=\operatorname{coh}X$.

###### Proof.

Let $E$ be in $\operatorname{coh}X$. An induction on number of irreducible
components for $\operatorname{Supp}(E)$ to reduce to case where $E$ is
supported on a closed integral subscheme, cf. [Sta23, Tag 01YD]. The desired
claims follows from Lemma 3.4. ∎

###### Proof of Theorem 1.1.

We prove the desired claim by Noetherian induction on $X$. It is evident that
the claim holds for the empty scheme, so it may be assumed that $X$ is
nonempty. Without loss of generality, we may further impose that $X$ is
integral as it suffices to verify the claim on each irreducible component of
$X$, cf. Lemma 3.5

It is clear that $(3)$ and $(4)$ are equivalent.

Let $Z$ be a closed integral subscheme of $X$. If $(4)$ holds, then
$D_{\operatorname{sg}}(U)$ admits a classical generator for each open affine
subscheme $U$ of $Z$. By Remark 2.5, we observe that any such $U$ must be
$J\textrm{-}1$. This shows $(4)$ implies $(2)$.

Next, we verify $(2)$ implies $(3)$. If $X$ is regular, there is nothing to
check, so we may assume additionally that $X$ is not regular. If $X$ is
$J\textrm{-1}$ and integral, then there exists a nonempty open affine $U$ in
the regular locus of $X$. In particular, $D^{b}_{\operatorname{coh}}(U)$
admits a classical generator. Note that $\dim Z<\dim X$ where $Z:=X\setminus
U$. Hence, our inductive hypothesis promises that
$D^{b}_{\operatorname{coh}}(Z)$ admits a classical generator. Therefore, an
application of Remark 2.8 tells us $D^{b}_{\operatorname{coh}}(X)$ admits a
classical generator.

Now we verify $(1)\iff(2)$. Clearly, $(2)\implies(1)$, so we verify
$(1)\implies(2)$. If $X$ is regular, there is nothing to check, so we may
assume additionally that $X$ is not regular. If $X$ is $J\textrm{-}0$ and
integral, then there exists a nonempty open affine $U$ in the regular locus of
$X$. In particular, $D^{b}_{\operatorname{coh}}(U)$ admits a classical
generator. The closed subscheme $Z:=(X\setminus U)_{\operatorname{red}}$ is
properly contained in $X$, so the claim holds on $Z$. Note that every closed
integral subscheme of $Z$ is $J\textrm{-}0$. Hence, our inductive hypothesis
promises that $D^{b}_{\operatorname{coh}}(Z)$ admits a classical generator.
Therefore, an application of Remark 2.8 tells us
$D^{b}_{\operatorname{coh}}(X)$ admits a classical generator.

Lastly, we show that $(1)\iff(5)$. Assume $(5)$ holds. There exists $G$ in
$\operatorname{coh}X$ such that $\operatorname{G}=\operatorname{coh}X$. By
Lemma 2.12, it follows that $\langle G\rangle=D^{b}_{\operatorname{coh}}(X)$,
so $(5)\implies(1)$.

Suppose $(1)$ holds. There exists a nonempty open affine $j\colon U\to X$
contained in the regular locus, and hence, a sequence of abelian categories

$\operatorname{coh}_{Z}X\to\operatorname{coh}X\to\operatorname{coh}U.$

As $Z$ is properly contained, the induction hypothesis tells us there exists
$G$ in $\operatorname{coh}Z$. Hence, we know that $i_{\ast}G$ is a classical
generator for $\operatorname{coh}_{Z}X$ via Lemma 3.4. The desired claim
follows from Lemma 3.3. ∎

The following examples are new instances where Theorem 1.1 can be applied.

######  Example 3.6.

Any Nagata schemes of Krull dimension one is $J\textrm{-}2$. This is checked
locally in light of [Sta23, Tag 07PJ]. Appealing to [ELS20, Theorem 4.15] or
Theorem 1.1, it can be seen that $D^{b}_{\operatorname{coh}}(X)$ admits a
classical generator.

######  Example 3.7.

Let $X=\operatorname{Spec}(R)$ where $R$ is a Noetherian semi-local integral
domain of Krull dimension two with open regular locus. We can verify that
$D^{b}_{\operatorname{coh}}(X)$ admits a classical generator. Since $X$ has
finitely many closed points, it suffices to check the claim at each closed
point. Indeed, if for each maximal ideal $p$ in $X$ one has
$D^{b}_{\operatorname{coh}}(\mathcal{O}_{X,p})$ admitting a classical
generator, then it is the image of an object in
$D^{b}_{\operatorname{coh}}(X)$ under the Verdier localization functor
$D^{b}_{\operatorname{coh}}(X)\to
D^{b}_{\operatorname{coh}}(\mathcal{O}_{X,p})$. For each closed point $p$ in
$X$, choose an object $G_{p}$ of $D^{b}_{\operatorname{coh}}(X)$ satisfying
this condition. By [Let21, Theorem 3.6], we see that
$G:=\oplus_{p\in\operatorname{mSpec}(R)}G_{p}$ is a classical generator for
$D^{b}_{\operatorname{coh}}(X)$.

Consider the case where $R$ is local. If $Z$ is a properly contained closed
subscheme of $X$, then $Z$ is the affine spectrum of Noetherian local ring of
Krull dimension at most one. By [Sta23, Tag 07PJ], any such closed subscheme
is $J\textrm{-}2$, and hence, $J\textrm{-}0$. Therefore, Theorem 1.1 tells us
$D^{b}_{\operatorname{coh}}(X)$ admits a classical generator.

######  Example 3.8 (Murayama).

There is an example of a Noetherian scheme which satisfies condition $(2)$ of
Theorem 1.1, and yet is not $J\textrm{-}2$. That is, there exists a Noetherian
scheme whose closed integral subschemes are $J\textrm{-}1$, but $X$ is not
$J\textrm{-}2$. We follow the construction made by Nagata in [Nag59, $\S 4$].
Let $k$ be a perfect field of characteristic $p\not=0$ and
$v_{1},\ldots,v_{n},\ldots$ be infinitely many algebraically independent
elements over $k_{0}$. Set $k$ to be the field
$k_{0}(v_{1},\ldots,v_{n},\ldots)$. Choose analytically independent elements
$x_{1},x_{2}$ over $k$. Set $A=k^{p}\\{x_{1},x_{2}\\}[k]$. Then $A$ is a
regular local ring. Let $p_{1},\ldots,p_{n},\ldots$ be infinitely many prime
elements in $A$ such that $p_{1}A\not=p_{j}A$ if $i\not=j$. For $n\geq 0$, let
$q_{n}=p_{1}\cdots p_{n}$ and $c=\sum_{i}v_{i}q_{i}$. The singular locus of
$\operatorname{Spec}(A[c])$ is not contained in any proper closed subset of
$\operatorname{Spec}(A[c])$. Let $I=A[\frac{1}{x_{1}}]$. Then $I$ is a
Dedekind domain and $B[\frac{1}{x_{1}}]=I[c]$ is a finite $I$-algebra.
However, the singular locus of $I[c]$ is not closed. Since $I$ is a Dedekind
domain, it satisfies the property that every closed integral subscheme is
$J\operatorname{-}1$.

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