# Dévissage for generation in 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 exhibits that the essential image of the derived pushforward along a
proper surjective morphism of Noetherian schemes generates the targets derived
category of bounded complexes with coherent cohomology. There are two primary
consequences to our work. First, we show that classical generators in the
bounded derived category are preserved by derived pushforward along a proper
surjective morphism of Noetherian schemes. Secondly, the bounded derived
category coincides with the thick subcategory generated by perfect complexes
and structure sheaves of closed integral subschemes supported in the singular
locus of the ambient scheme.

###### Key words and phrases:

derived categories, thick subcategories, strong generation, classical
generation, devissage, modifications, alterations

###### 2020 Mathematics Subject Classification:

14A30 (primary), 14F08, 13D09, 32S45

##  1\. Introduction

This note presents a technical lemma that asserts, on an open set of a
Noetherian integral scheme, any pair of objects with full support in the
bounded derived category can locally finitely build each other in at most one
cone, cf. Lemma 3.1. An upshot to such a result is that it contributes a
powerful understanding towards the behavior of generation amongst functors on
derived categories arising in algebraic geometry and commutative algebra.

Recall a notion of generation for a triangulated category $\mathcal{T}$, which
was introduced in [BvdB03]. An object $G$ in $\mathcal{T}$ is called a
classical generator if the smallest thick subcategory containing $G$ coincides
with $\mathcal{T}$. In other words, every object in $\mathcal{T}$ can be
obtained from $G$ using a finite combination of shifts, cones, finite
coproducts, and retracts. If the number of cones required to finitely build an
object from $G$ is uniform, then $G$ is referred to as a strong generator. The
concept of strong generators has led to interesting invariants, as introduced
in [ABIM10, Rou08].

The primary focus is on the derived category of bounded complexes with
coherent cohomology on a Noetherian scheme $X$, which is denoted by
$D^{b}_{\operatorname{coh}}(X)$. There has been considerable progress made in
understanding instances where $D^{b}_{\operatorname{coh}}(X)$ admits the
aforementioned objects. These cases have been studied for quasi-excellent
separated schemes of finite Krull dimension [Aok21, Nee21], Noetherian
$J\textrm{-}2$ schemes [ELS20], Noetherian schemes admitting a separator
[Jat21], and affine Noetherian schemes [IT16, IT19, DLT23]. Furthermore, in
many concrete instances, there have been efforts to explicitly identify such
objects [Rou08, BIL+23, Ola23], including noncommutative techniques [BDL23].

An intermediate step in arguments for generic freeness à la dévissage asserts
that for any finitely generated module $M$ over a commutative Noetherian ring
$R$ there exists a finite descending chain of submodules $\cdots\subseteq
M_{1}\subseteq M_{0}=:M$ whose corresponding quotients $M_{i}/M_{i+1}$ are
isomorphic to objects of the form $R/\mathfrak{p}$ for some
$\mathfrak{p}\in\operatorname{Spec}(R)$. This is observed in the proof of
[Gro65, Lemme 6.9.2]. If interpreted in our context, this tells us that any
finitely generated $R$-module can be obtained from the collection of modules
of the form $R/\mathfrak{p}$ where $\mathfrak{p}\in\operatorname{Spec}(R)$
using kernels, cokernels, and extensions.

The main result of this work is to demonstrate that the essential image of the
derived pushforward along a proper surjective morphism of Noetherian schemes
generates bounded derived category of the target of the morphism.

######  Theorem 1.1.

If $\pi\colon Y\to S$ is a proper surjective morphism of Noetherian schemes,
then the thick subcategory generated by the essential image of
$\mathbb{R}\pi_{\ast}\colon D^{b}_{\operatorname{coh}}(Y)\to
D^{b}_{\operatorname{coh}}(X)$ coincides with $D^{b}_{\operatorname{coh}}(X)$.

Theorem 1.1 significantly extends prior work of the second author in [Lan23].
Along the way to proving Theorem 1.1, we establish a lemma which provides a
triangulated analogue of dévissage inspired arguments for generation in the
bounded derived category, see Lemma 3.2. This lemma tells us the smallest
thick subcategory generated by the structure sheaves of closed integral
subschemes coincides with $D^{b}_{\operatorname{coh}}(X)$.

In other words, objects in $D^{b}_{\operatorname{coh}}(X)$ can be finitely
built by structure sheaves of integral closed subschemes of $X$ using a finite
combination of cones, shifts, finite coproducts, and retracts. Subsequently,
we refine this to show that a similar statement holds where the closed
integral subschemes can be chosen to be those supported in the singular locus
of the ambient scheme, cf. Proposition 3.3. This refinement is a globalization
of [Tak14, Corollary 4.3.1], which asserts a similar statement for the case of
affine schemes, see Remark 3.4. It is worthwhile to note that our arguments
are very different in flavor than that of [Tak14].

Our main result yields valuable consequences towards studying the behavior of
classical generators under the derived pushforward. Specifically, we
demonstrate that classical generators in the bounded derived category are
preserved by derived pushforward along a proper surjective morphism of
Noetherian schemes, cf. Corollary 3.6. This allows for one to explicitly
identify strong generators in the bounded derived category of a variety over a
field through the use of alterations, cf. Example 3.8.

######  Notation 1.2.

Let $X$ be a Noetherian scheme.

  1. (1)

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

  2. (2)

$D_{\operatorname{Qcoh}}(X)$ is the strictly full subcategory of objects in
$D(X)$ whose cohomology sheaves are quasi-coherent $\mathcal{O}_{X}$-modules

  3. (3)

$D^{b}_{\operatorname{coh}}(X)$ is the strictly full subcategory of $D(X)$
whose objects are bounded with coherent cohomology sheaves

  4. (4)

$\operatorname{perf}X$ is the strictly full subcategory of $D(X)$ consisting
of perfect complexes

  5. (5)

$D^{b}_{\operatorname{coh}}(R)$ denotes, by abuse of notation,
$D^{b}_{\operatorname{coh}}(\operatorname{Spec}(R))$ for a commutative
Noetherian ring $R$.

######  Acknowledgements.

Souvik Dey 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.

##  2\. Background

First, we will briefly cover basic notions on generation in triangulated
categories. For a further treatment on necessary background, the reader is
referred to [BvdB03, Rou08]. Let $\mathcal{T}$ be a triangulated category with
shift functor $[1]\colon\mathcal{T}\to\mathcal{T}$.

######  Definition 2.1.

If $\mathcal{S}$ is a subcategory of $\mathcal{T}$, then we have the following
constructions:

  1. (1)

$\operatorname{add}(\mathcal{S})$ is the smallest strictly full subcategory of
$\mathcal{T}$ containing $\mathcal{S}$ that is closed under shifts, finite
coproducts, and retracts

  2. (2)

$\mathcal{S}$ is said to be thick if it is closed under retracts and is a
strictly full triangulated subcategory of $\mathcal{T}$

  3. (3)

$\langle\mathcal{S}\rangle$ is the smallest thick subcategory in $\mathcal{T}$
containing $\mathcal{S}$

  4. (4)

$\langle\mathcal{S}\rangle_{0}$ is the full subcategory consisting of all
objects isomorphic to the zero object

  5. (5)

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

  6. (6)

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

If $\mathcal{S}$ consists of a single object $G$, then we write $\langle
G\rangle$ for $\langle\mathcal{S}\rangle$.

######  Remark 2.2.

In the notation of Definition 2.1, we have an exhaustive filtration for the
smallest thick subcategory containing $\mathcal{S}$ in $\mathcal{T}$:

$\langle\mathcal{S}\rangle_{0}\subseteq\langle\mathcal{S}\rangle_{1}\subseteq\cdots\subseteq\bigcup_{n=0}^{\infty}\langle\mathcal{S}\rangle_{n}=\langle\mathcal{S}\rangle.$

######  Definition 2.3.

Given objects $E,G$ of $\mathcal{T}$, we have the following terminology:

  1. (1)

$E$ is said to be finitely built by $G$ if $E\in\langle G\rangle$

  2. (2)

$G$ is said to be a classical generator if $\langle G\rangle=\mathcal{T}$

  3. (3)

$G$ is said to be a strong generator if there exists $n\geq 0$ such that
$\langle G\rangle_{n}=\mathcal{T}$.

######  Example 2.4.

The following list, though far from comprehensive, are familiar instances in
both geometric and algebraic contexts where strong generators can be
explicitly identified:

  1. (1)

$R\oplus k$ is a strong generator for $D^{b}_{\operatorname{coh}}(R)$ if
$(R,\mathfrak{m},k)$ is a local Noetherian ring with an isolated singularity,
cf. [KMVdB11, Proposition A.2]

  2. (2)

$\mathcal{O}_{X}$ is a strong generator for $D^{b}_{\operatorname{coh}}(X)$ if
$X$ is a quasi-affine Noetherian regular scheme of finite Krull dimension, cf.
[Ola23, Corollary 5]

  3. (3)

$\operatorname{perf}X$ admits a classical generator for $X$ a Noetherian
scheme, cf. [BvdB03, Theorem 3.1.1]

  4. (4)

$\bigoplus^{\dim X}_{i=0}\mathcal{L}^{\otimes i}$ is a strong generator for
$D^{b}_{\operatorname{coh}}(X)$ if $\mathcal{L}$ is an ample line bundle on a
smooth quasi-projective variety $X$ over a field, cf. [Rou08, Proposition 7.9]

  5. (5)

$F_{\ast}^{e}(\bigoplus^{\dim X}_{i=0}\mathcal{L}^{\otimes i})$ is a strong
generator for $D^{b}_{\operatorname{coh}}(X)$ if $\mathcal{L}$ is an ample
line bundle on a singular quasi-projective variety $X$ over a perfect field of
positive characteristic where $F\colon X\to X$ is the Frobenius morphism, cf.
[BIL+23, Corollary 3.9].

######  Remark 2.5.

([ELS20, $\S 4.2$]) If $\mathcal{K}$ is a thick subcategory of $\mathcal{T}$
and $\mathcal{T}/\mathcal{K}$ denotes the Verdier quotient of $\mathcal{T}$ by
$\mathcal{K}$, then the sequence of natural functors

$\mathcal{K}\xrightarrow{i}\mathcal{T}\xrightarrow{q}\mathcal{T}/\mathcal{K}$

is called a Verdier localization. Note that $q\circ i=0$, $i$ is the natural
inclusion, and $q$ is the natural quotient functor.

######  Lemma 2.6.

Suppose $\mathcal{K}$ is a thick subcategory of $\mathcal{T}$. If
$\mathcal{S}$ is a subcategory of $\mathcal{T}$ such that
$\langle\mathcal{S}\rangle=\mathcal{T}/\mathcal{K}$, then
$\langle\mathcal{K}\cup\mathcal{S}\rangle=\mathcal{T}$.

###### Proof sketch.

If $E$ is an object of $\mathcal{T}/\mathcal{K}$, then there exists an $n\geq
0$ such that $E\in\langle\mathcal{S}\rangle_{n}$ in $\mathcal{T}/\mathcal{K}$.
Recall maps in $\mathcal{T}/\mathcal{K}$ corresponds to a roof of a pair of
maps in $\mathcal{T}$ whose cones belong $\mathcal{K}$. An induction argument
on $n$ will tell us $E$ belongs to $\langle\mathcal{K}\cup\mathcal{S}\rangle$
in $\mathcal{T}$. ∎

######  Definition 2.7.

Let $X$ be a Noetherian scheme, $E$ an object of
$D^{b}_{\operatorname{coh}}(X)$, and $i\colon Z\to X$ a closed immersion.

  1. (1)

$\operatorname{Supp}(E):=\bigcup^{\infty}_{j=-\infty}\operatorname{Supp}(\mathcal{H}^{j}(E))$
is called the support of $E$

  2. (2)

$E$ is supported on $Z$ whenever $\operatorname{Supp}(E)$ is contained in $Z$

  3. (3)

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

######  Remark 2.8.

Let $X$ be a Noetherian scheme, $i\colon U\to X$ an open immersion, and
$Z:=X\setminus U$ the associated closed subscheme with reduced induced
structure. There exists a Verdier localization $i^{\ast}\colon
D^{b}_{\operatorname{coh}}(X)\to D^{b}_{\operatorname{coh}}(U)$ with kernel
$D^{b}_{\operatorname{coh},Z}(X)$, cf. [ELS20, Theorem 4.4].

######  Remark 2.9.

Let $X$ be a Noetherian scheme. The Verdier quotient of
$D^{b}_{\operatorname{coh}}(X)$ by $\operatorname{perf}X$ is called the
singularity category of $X$, which we denote by $D_{\operatorname{sg}}(X)$.
This was introduced by [Buc21] in the case of Gorenstein local rings, and
[Orl04] for the case of varieties over a field.

##  3\. Results

Next up, we prove our main results and to start, we need a establish our
technical lemma that is akin to generic freeness of coherent
$\mathcal{O}_{X}$-modules on a Noetherian integral scheme.

######  Lemma 3.1.

Let $X$ be a Noetherian integral scheme. If $E$ and $G$ are objects of
$D^{b}_{\operatorname{coh}}(X)$ with full support, then there exists an open
immersion $i\colon U\to X$ such that in the category
$D^{b}_{\operatorname{coh}}(U)$ one has $i^{\ast}E\in\langle
i^{\ast}G\rangle_{1}$ and $i^{\ast}G\in\langle i^{\ast}E\rangle_{1}$.

###### Proof.

Consider a nonempty open affine $U$ in $X$. The set $S$ of points $p$ in $U$
such that $E_{p}\in\langle G_{p}\rangle_{1}$ is Zariski open, cf. [Let21,
Proposition 3.5]. It is evident that $S$ contains the generic point of $X$,
and so $S$ is nonempty. By the same reasoning, the set $T$ of points $p$ in
$X$ such that $G_{p}\in\langle E_{p}\rangle_{1}$ is Zariski open and nonempty.
If both $S$ and $T$ contain the generic point $\eta$ of $X$, then there exists
a nonempty open affine $W$ in $S\cap T$, i.e. consider a component in an open
affine covering of $S\cap T$. Let $j\colon W\to X$ be the associated open
immersion. For each $p\in W$, one has $E_{p}\in\langle G_{p}\rangle_{1}$ and
$G_{p}\in\langle E_{p}\rangle_{1}$. Therefore, it follows from a local-to-
global principle on generation for affine schemes that $j^{\ast}E\in\langle
j^{\ast}G\rangle_{1}$ and $j^{\ast}G\in\langle j^{\ast}E\rangle_{1}$, cf.
[Let21, Corollary 3.4]. ∎

######  Lemma 3.2.

(Derived dévissage) Let $X$ be a Noetherian scheme. For any perfect complex
$P$ on $X$ with full support, one has that $D_{\operatorname{coh}}^{b}(X)$
coincides with the thick subcategory generated by $P$ and objects of the form
$i_{\ast}\mathbb{L}i^{\ast}P$ where $i\colon Z\to X$ is a closed immersion
from an integral scheme $Z$.

###### Proof.

If $X=\emptyset$, there is nothing to check, so without loss of generality,
$X$ is nonempty. We argue by Noetherian induction, and so, assume the claim
holds for any properly contained closed subscheme $Z$ of $X$. Let $E$ be an
object of $D^{b}_{\operatorname{coh}}(X)$. If $E$ has support properly
contained in $X$, then the induction hypothesis tells us that $E$ belongs to
$\mathcal{S}$, and so we may reduce to the case where $E$ has full support.
Moreover, as objects in $D^{b}_{\operatorname{coh}}(X)$ are finitely built by
their cohomology sheaves, we can further impose that $E$ is a coherent
$\mathcal{O}_{X}$-module.

Suppose the closed subset $\operatorname{Supp}(E)$ has irreducible components
$Z_{1},\ldots,Z_{n}$. There exists a short exact sequence of coherent
$\mathcal{O}_{X}$-modules:

$0\to E_{1}\to E\to E_{1}^{\prime}\to 0$

where $\operatorname{Supp}(E_{1})\subseteq Z_{1}$ and
$\operatorname{Supp}(E_{1}^{\prime})\subseteq\cup^{n}_{i=2}Z_{i}$, cf. [Sta23,
Tag 01YD]. By applying this observation to $E_{1}^{\prime}$, then an induction
argument on the minimal number of irreducible components allows us reduce to
the case where $X$ is an integral scheme. This means $E$ is a coherent
$\mathcal{O}_{X}$-module with full support on the integral scheme. There
exists a dense open immersion $j\colon U\to X$ such that $j^{\ast}E$ and
$j^{\ast}P$ finitely build one another in at most one cone in
$D^{b}_{\operatorname{coh}}(U)$, see Lemma 3.1. If $U=X$, then we are done, so
assume $U$ is a properly contained subset of $X$. Note that $j^{\ast}E$ is a
direct summand of an object of the form
$\oplus_{n\in\mathbb{Z}}j^{\ast}P^{\oplus r_{n}}[n]$, so we have an object
$j^{\ast}E$ and isomorphism $\phi\colon j^{\ast}A\oplus
j^{\ast}E\to\oplus_{n\in\mathbb{Z}}j^{\ast}P^{\oplus r_{n}}[n]$ in
$D^{b}_{\operatorname{coh}}(U)$. There exists a roof in
$D^{b}_{\operatorname{coh}}(X)$:

${B}$${{\bigoplus_{n\in\mathbb{Z}}P^{\oplus r_{n}}[n]}}$${{A\oplus
E}.}$$\scriptstyle{f}$$\scriptstyle{g}$

Observe that the cones of $f,g$ are supported on the properly contained closed
subset $Z$ of $X$ (see Remark 2.8), and so the objects
$\operatorname{cone}(f),\operatorname{cone}(g)$ are in $\mathcal{S}$ by our
inductive hypothesis. Therefore, $B$ belongs to $\mathcal{S}$, and hence, so
does $A\oplus E$. ∎

Lemma 3.2 tells us that the smallest thick subcategory generated by objects of
the form $i_{\ast}\colon Z\to X$ with $i$ a closed immersion from an integral
scheme coincides with that of $D^{b}_{\operatorname{coh}}(Z)$. However, the
following result refines this to the class of closed subschemes supported in
the singular locus of the ambient scheme. Moreover, this is a globalization of
[Tak14, Corollary 4.3.1].

######  Proposition 3.3.

Let $X$ be a Noetherian scheme. For any classical generator $P$ of
$\operatorname{perf}X$, one has that $D_{\operatorname{coh}}^{b}(X)$ coincides
with the thick subcategory generated by $P$ and objects of the form
$i_{\ast}\mathbb{L}i^{\ast}P$ where $i\colon Z\to X$ is a closed immersion
from an integral scheme $Z$ supported in the singular locus of $X$.

###### Proof.

By Lemma 3.2, we know that $D^{b}_{\operatorname{coh}}(X)$ coincides with the
thick subcategory generated by $P$ and objects of the form
$i_{\ast}\mathbb{L}i^{\ast}P$ where $i\colon Z\to X$ is a closed immersion
from an integral scheme $Z$. Denote this collection of objects by
$\mathcal{S}$. As $D_{\operatorname{sg}}(X)$ is a Verdier quotient of
$D_{\operatorname{coh}}^{b}(X)$, the thick subcategory generated by the images
of the objects in $\mathcal{S}$ coincides with $D_{\operatorname{sg}}(X)$.
However, we can refine the collection of images of objects in $\mathcal{S}$ to
those which have nonzero image in $D_{\operatorname{sg}}(X)$.

From this collection of objects, let us check which objects of $\mathcal{S}$
have zero image in $D_{\operatorname{sg}}(X)$. Clearly, $P$ being a perfect
complex on $X$ has a zero image in $D_{\operatorname{sg}}(X)$.

We look at those closed immersions $i\colon Z\to X$ with $Z$ integral and
$i_{\ast}\mathbb{L}i^{\ast}P$ has zero image in $D_{\operatorname{sg}}(X)$.
This is equivalent to $i_{\ast}\mathbb{L}i^{\ast}P$ being an object of
$\operatorname{perf}X$. Since $P$ is a classical generator for
$\operatorname{perf}X$ and $i$ is a finite morphism, we know that
$\mathbb{L}i^{\ast}P$ is a classical generator for $\operatorname{perf}Z$, and
so $\mathcal{O}_{Z}$ is finitely build by $\mathbb{L}i^{\ast}P$, cf. [Sta23,
Tag 0BQT]. Hence, $i_{\ast}\mathcal{O}_{Z}$ belongs to $\operatorname{perf}X$.
If $\eta$ is the generic point of $Z$, then $\mathcal{O}_{X,\eta}$ finitely
builds $\kappa(\eta)$, and so $\eta$ is in the regular locus of $X$.

Our work tells us $i_{\ast}\mathbb{L}i^{\ast}P$ being a perfect complex
ensures $Z$ is not supported in the singular locus. Consequently,
$D_{\operatorname{sg}}(X)$ coincides with the thick subcategory generated by
the objects of the form $i_{\ast}\mathbb{L}i^{\ast}P$ where $i\colon Z\to X$
is a closed immersion from an integral scheme whose generic point does not
belong to the regular locus of $X$. Since the singular locus of $X$ is
specialization closed, we see that any such closed subscheme in the collection
is supported in the singular locus of $X$. Thus, the desired claim follows
from Lemma 2.6. ∎

######  Remark 3.4.

Let $X$ be a Noetherian scheme. Recall that Lemma 3.2 tells us
$D_{\operatorname{coh}}^{b}(X)$ coincides with the thick subcategory generated
by $\mathcal{O}_{X}$ and objects of the form $i_{\ast}\mathcal{O}_{Z}$ where
$i\colon Z\to X$ is a closed immersion from an integral scheme $Z$. Suppose
$P$ is a classical generator for $\operatorname{perf}X$. The proof of
Proposition 3.3 can be adapted to show $D_{\operatorname{coh}}^{b}(X)$
coincides with the thick subcategory generated by $P$ and objects of the form
$i_{\ast}\mathcal{O}_{Z}$ where $i\colon Z\to X$ is a closed immersion from an
integral scheme $Z$ supported in the singular locus of $X$.

######  Example 3.5.

If $X$ is a Noetherian scheme which has finitely many singular points, then
$D^{b}_{\operatorname{coh}}(X)$ admits a classical generator via Proposition
3.3. For example, suppose $X$ is a normal quasi-projective surface over a
field with an ample line bundle $\mathcal{L}$ and singular points
$p_{1},\ldots,p_{n}$. By [Orl09, Theorem 4], the object
$\mathcal{O}_{X}\oplus\mathcal{L}\oplus\mathcal{L}^{\otimes 2}$ is a classical
generator for $\operatorname{perf}X$. Let
$\pi_{j}\colon\operatorname{Spec}(\kappa(p_{j}))\to X$ denote the associated
closed immersion for each singular point with the reduced induced closed
subscheme structure. By Proposition 3.3, the object

$\mathcal{O}_{X}\oplus\mathcal{L}\oplus\mathcal{L}^{\otimes
2}\oplus\pi_{1,\ast}\mathcal{O}_{\operatorname{Spec}(\kappa(p_{1}))}\oplus\cdots\oplus\pi_{n,\ast}\mathcal{O}_{\operatorname{Spec}(\kappa(p_{n}))}$

is a strong generator for $D^{b}_{\operatorname{coh}}(X)$.

###### Proof of Theorem 1.1.

If $S=\emptyset$, there is nothing to check, so without loss of generality,
$S$ is nonempty. Similar to Lemma 3.2, we argue via Noetherian induction, so
assume the claim holds for any properly contained closed subscheme $Z$ of $S$.
Let $\mathcal{T}$ be the thick subcategory generated by the essential image of
$\mathbb{R}\pi_{\ast}\colon D^{b}_{\operatorname{coh}}(Y)\to
D^{b}_{\operatorname{coh}}(S)$. By Lemma 3.2, it suffices to show both
$\mathcal{O}_{X}$ and $i_{\ast}\mathcal{O}_{Z}$ belong to $\mathcal{T}$ where
$i\colon Z\to S$ ranges over all closed immersion with $Z$ integral.

The induction hypothesis ensures $i_{\ast}\mathcal{O}_{Z}$ belongs to
$\mathcal{T}$ for each closed immersion $i\colon Z\to S$ with $Z$ integral and
properly contained in $S$. Indeed, let $i$ denote such a morphism. There
exists a fibered square:

${{Y\times_{S}Z}}$${Z}$${Y}$${S}$$\scriptstyle{\pi_{2}}$$\scriptstyle{i}$$\scriptstyle{\pi}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\lrcorner}$

Note that $\pi_{2}$ is a proper surjective morphism and $\pi_{1}$ is a closed
immersion as these properties are stable under base change. Since the claim
holds on all properly contained closed subschemes of $S$, the thick
subcategory generated by the essential image of $\mathbb{R}\pi_{2,\ast}\colon
D^{b}_{\operatorname{coh}}(Y\times_{S}Z)\to D^{b}_{\operatorname{coh}}(Z)$
coincides with $D^{b}_{\operatorname{coh}}(Z)$. Consequently, it follows that
$i_{\ast}\mathcal{O}_{Z}$ belongs to $\mathcal{T}$ as any object in the
essential image of $i_{\ast}\circ\mathbb{R}\pi_{2,\ast}\colon
D^{b}_{\operatorname{coh}}(Y\times_{S}Z)\to D^{b}_{\operatorname{coh}}(S)$
belongs to $\mathcal{T}$ via the commutativity of the diagram above.

We are left to verify that $\mathcal{O}_{X}$ is in $\mathcal{T}$. First,
consider the case $X$ is integral. However, one can argue in a similar fashion
for the proof of Lemma 3.2 to show the desired claim. There is an open
immersion $i\colon U\to S$ with $U$ dense such that
$i^{\ast}\mathbb{R}\pi_{\ast}\mathcal{O}_{Y}$ and $\mathcal{O}_{U}$ finitely
build one another in at most one cone in $D^{b}_{\operatorname{coh}}(U)$, see
Lemma 3.1. Note that $j^{\ast}\mathcal{O}_{X}=\mathcal{O}_{U}$. If $U=S$, then
we would be done, so consider the case where $U$ is a properly contained
subset of $S$. Since $\mathcal{O}_{U}$ is a direct summand of an object of the
form
$\oplus_{n\in\mathbb{Z}}j^{\ast}\mathbb{R}\pi_{\ast}\mathcal{O}_{Y}^{\oplus
r_{n}}[n]$, there is an object $j^{\ast}E$ and isomorphism $\phi\colon
j^{\ast}A\oplus\mathcal{O}_{U}\to\oplus_{n\in\mathbb{Z}}j^{\ast}\mathbb{R}\pi_{\ast}\mathcal{O}_{Y}^{\oplus
r_{n}}[n]$ in $D^{b}_{\operatorname{coh}}(U)$. This gives us a roof in
$D^{b}_{\operatorname{coh}}(S)$:

${B}$${{\bigoplus_{n\in\mathbb{Z}}\mathbb{R}\pi_{\ast}\mathcal{O}_{Y}^{\oplus
r_{n}}[n]}}$${{A\oplus\mathcal{O}_{X}}.}$$\scriptstyle{f}$$\scriptstyle{g}$

The cones of $f,g$ are supported on the properly contained closed subsets of
$S$. Hence, the objects $\operatorname{cone}(f)$ and $\operatorname{cone}(g)$
are in $\mathcal{T}$ by our inductive hypothesis. This tells us $B$ belongs to
$\mathcal{T}$, and thus, so does $A\oplus\mathcal{O}_{X}$.

Lastly, we need to check that $\mathcal{O}_{X}$ is in $\mathcal{T}$ when $X$
is not integral. Let $Z_{1},\ldots,Z_{n}$ be the irreducible components of
$X$. There exists a short exact sequence of coherent
$\mathcal{O}_{X}$-modules:

$0\to E_{1}\to\mathcal{O}_{X}\to E_{1}^{\prime}\to 0$

where $\operatorname{Supp}(E_{1})\subseteq Z_{1}$ and
$\operatorname{Supp}(E_{1}^{\prime})\subseteq\cup^{n}_{i=2}Z_{i}$, cf. [Sta23,
Tag 01YD]. Since $E_{1}$ and $E_{1}^{\prime}$ are supported on properly
contained closed subsets of $X$, our induction hypothesis tells us that
$E_{1}$ and $E_{1}^{\prime}$ belongs to $\mathcal{T}$. Thus, $\mathcal{O}_{X}$
is in $\mathcal{T}$ as desired. ∎

######  Corollary 3.6.

Suppose $\pi\colon Y\to X$ is a proper surjective morphism of Noetherian
schemes. If $G$ is a classical generator for $D^{b}_{\operatorname{coh}}(Y)$,
then $\mathbb{R}\pi_{\ast}G$ is a classical generator for
$D^{b}_{\operatorname{coh}}(X)$.

###### Proof.

By Theorem 1.1, the thick subcategory generated by the essential image of the
functor

$\mathbb{R}\pi_{\ast}\colon D^{b}_{\operatorname{coh}}(Y)\to
D^{b}_{\operatorname{coh}}(X)$

coincides with $D^{b}_{\operatorname{coh}}(X)$. Since $G$ is a classical
generator for $D^{b}_{\operatorname{coh}}(Y)$, $\mathbb{R}\pi_{\ast}G$
finitely builds any object in the essential image of the functor above. Hence,
any object of $D^{b}_{\operatorname{coh}}(X)$ is finitely built by
$\mathbb{R}\pi_{\ast}G$. ∎

The following is a slight alternative to the proof of [LO24, Lemma 3.14].

######  Corollary 3.7.

Suppose $\pi\colon Y\to X$ is a proper surjective morphism of Noetherian
schemes. If $G$ is a strong generator for $D^{b}_{\operatorname{coh}}(Y)$,
then $\mathbb{R}\pi_{\ast}G$ is a strong generator for
$D^{b}_{\operatorname{coh}}(X)$.

###### Proof.

By Corollary 3.6, $\mathbb{R}\pi_{\ast}G$ is a classical generator for
$D^{b}_{\operatorname{coh}}(X)$. There exists an $n\geq 0$ such that
$\mathcal{O}_{X}\in\langle\mathbb{R}\pi_{\ast}G\rangle_{n}$. Given
$P\in\operatorname{perf}X$, one has from the projection formula that
$P\in\langle\mathbb{R}\pi_{\ast}(G\overset{\mathbb{L}}{\otimes}\mathbb{L}\pi^{\ast}P)\rangle_{n}$.
If $\langle G\rangle_{k}=D^{b}_{\operatorname{coh}}(Y)$, then
$P\in\langle\mathbb{R}\pi_{\ast}G\rangle_{nk}$. The desired claim follows from
[LO24, Theorem 1.1]. ∎

######  Example 3.8.

Let $X$ be a variety over a field. There exists a proper surjective morphism
$\pi\colon\widetilde{X}\to X$ of varieties over $k$ such that
$\dim\widetilde{X}=\dim X$ and $\widetilde{X}$ is a regular quasi-projective
variety, cf. [dJ96, Theorem 4.1]. Let $\mathcal{L}$ be an ample line bundle on
$\widetilde{X}$. By [Rou08, Proposition 7.31], we know that
$G:=\bigoplus^{\dim X}_{i=-\dim X}\mathcal{L}^{\otimes i}$ is a strong
generator for $D^{b}_{\operatorname{coh}}(\widetilde{X})$, and so
$\mathbb{R}\pi_{\ast}G$ is a strong generator for
$D^{b}_{\operatorname{coh}}(X)$ via Corollary 3.7.

## References

  * [ABIM10] Luchezar L. Avramov, Ragnar-Olaf Buchweitz, Srikanth B. Iyengar, and Claudia Miller.  Homology of perfect complexes.  Adv. Math., 223(5):1731–1781, 2010.
  * [Aok21] Ko Aoki.  Quasiexcellence implies strong generation.  J. Reine Angew. Math., 780:133–138, 2021.
  * [BDL23] Anirban Bhaduri, Souvik Dey, and Pat Lank.  A note on generation and descent for derived categories of noncommutative schemes.  arXiv preprint arXiv:2312.02840, 2023.
  * [BIL+23] Matthew R. Ballard, Srikanth B. Iyengar, Pat Lank, Alapan Mukhopadhyay, and Josh Pollitz.  High frobenius pushforwards generate the bounded derived category.  arXiv preprint arXiv:2303.18085, 2023.
  * [Buc21] Ragnar-Olaf Buchweitz.  Maximal Cohen-Macaulay modules and Tate cohomology. With appendices by Luchezar L. Avramov, Benjamin Briggs, Srikanth B. Iyengar and Janina C. Letz, volume 262 of Math. Surv. Monogr. Providence, RI: American Mathematical Society (AMS), 2021.
  * [BvdB03] A. Bondal and M. van den Bergh.  Generators and representability of functors in commutative and noncommutative geometry.  Mosc. Math. J., 3(1):1–36, 2003.
  * [dJ96] A. J. de Jong.  Smoothness, semi-stability and alterations.  Publ. Math., Inst. Hautes Étud. Sci., 83:51–93, 1996.
  * [DLT23] Souvik Dey, Pat Lank, and Ryo Takahashi.  Strong generation and (co)ghost index for module categories.  arXiv preprint arXiv:2307.13675, 2023.
  * [ELS20] Alexey Elagin, Valery A. Lunts, and Olaf M. Schnürer.  Smoothness of derived categories of algebras.  Mosc. Math. J., 20(2):277–309, 2020.
  * [Gro65] A. Grothendieck.  Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas. (Séconde partie).  Publ. Math., Inst. Hautes Étud. Sci., 24:1–231, 1965.
  * [IT16] Srikanth B. Iyengar and Ryo Takahashi.  Annihilation of cohomology and strong generation of module categories.  Int. Math. Res. Not., 2016(2):499–535, 2016.
  * [IT19] Srikanth B. Iyengar and Ryo Takahashi.  Openness of the regular locus and generators for module categories.  Acta Math. Vietnam., 44(1):207–212, 2019.
  * [Jat21] V. B. Jatoba.  Strong generators in $D_{\mathrm{perf}}(X)$ for schemes with a separator.  Proc. Am. Math. Soc., 149(5):1957–1971, 2021.
  * [KMVdB11] Bernhard Keller, Daniel Murfet, and Michel Van den Bergh.  On two examples by Iyama and Yoshino.  Compos. Math., 147(2):591–612, 2011.
  * [Lan23] Pat Lank.  Descent conditions for generation in derived categories.  arXiv preprint arXiv:2308.08080, 2023.
  * [Let21] Janina C. Letz.  Local to global principles for generation time over commutative Noetherian rings.  Homology Homotopy Appl., 23(2):165–182, 2021.
  * [LO24] Pat Lank and Noah Olander.  Approximation by perfect complexes detects rouquier dimension.  arXiv preprint arXiv:2401.10146, 2024.
  * [Nee21] Amnon Neeman.  Strong generators in $\mathbf{D}^{\mathrm{perf}}(X)$ and $\mathbf{D}^{b}_{\mathrm{coh}}(X)$.  Ann. Math. (2), 193(3):689–732, 2021.
  * [Ola23] Noah Olander.  Ample line bundles and generation time.  J. Reine Angew. Math., 800:299–304, 2023.
  * [Orl04] D. O. Orlov.  Triangulated categories of singularities and D-branes in Landau-Ginzburg models.  In Algebraic geometry. Methods, relations, and applications. Collected papers. Dedicated to the memory of Andrei Nikolaevich Tyurin., pages 227–248. Moscow: Maik Nauka/Interperiodica, 2004.
  * [Orl09] Dmitri Orlov.  Remarks on generators and dimensions of triangulated categories.  Mosc. Math. J., 9(1):143–149, 2009.
  * [Rou08] Raphaël Rouquier.  Dimensions of triangulated categories.  J. $K$-Theory, 1(2):193–256, 2008.
  * [Sta23] The Stacks Project Authors.  Stacks Project.  https://stacks.math.columbia.edu, 2023.
  * [Tak14] Ryo Takahashi.  Reconstruction from Koszul homology and applications to module and derived categories.  Pac. J. Math., 268(1):231–248, 2014.