# Stably semiorthogonally indecomposable varieties

Dmitrii Pirozhkov

###### Abstract.

A triangulated category is said to be indecomposable if it admits no
nontrivial semiorthogonal decompositions. We introduce a definition of a
noncommutatively stably semiorthogonally indecomposable (NSSI) variety. This
propery implies, among other things, that each smooth proper subvariety has
indecomposable derived category of coherent sheaves, and that if $Y$ is NSSI,
then for any variety $X$ all semiorthogonal decompositions of $X\times Y$ are
induced from decompositions of $X$. We prove that any variety whose Albanese
morphism is finite is NSSI, and that the total space of a fibration over NSSI
base with NSSI fibers is also NSSI. We apply this indecomposability to deduce
that there are no phantom subcategories in some varieties, including surfaces
$C\times\mathbb{P}^{1}$, where $C$ is any smooth proper curve of positive
genus.

##  1\. Introduction

The derived category of coherent sheaves on an algebraic variety is an
interesting but complicated invariant. It is a triangulated category, and
sometimes it can be built out of smaller triangulated categories using the
notion of a semiorthogonal decomposition. One basic question is to identify
which smooth proper varieties have indecomposable derived categories, i.e.,
admit no nontrivial semiorthogonal decompositions. Examples of such varieties
are Calabi–Yau varieties [Bri99], curves of positive genus [Oka11], or more
generally varieties with a globally generated canonical bundle [KO15].

We propose a stronger notion of indecomposability for derived categories of
algebraic varieties. Let $Y$ be an algebraic variety over a field $\Bbbk$. Our
definition is a constraint on possible semiorthogonal decompositions for
categories equipped with the action of the symmetric monoidal category
$\mathrm{Perf}(Y)$ of perfect complexes on $Y$. We work in the formalism of
stable $\infty$-categories as in [Lur], but we mostly follow the exposition
given in [Per18].

Roughly speaking, a $\mathrm{Perf}(Y)$-linear structure on a $\Bbbk$-linear
stable $\infty$-category $\mathfrak{D}$ consists of an _action functor_
$a_{\mathfrak{D}}\colon\mathfrak{D}\otimes_{\Bbbk}\mathrm{Perf}(Y)\to\mathfrak{D}$
which is associative up to an isomorphism, together with higher associativity
data. When there is no risk of confusion, for objects $D\in\mathfrak{D}$ and
$S\in\mathrm{Perf}(Y)$ we denote the result of the action
$a_{\mathfrak{D}}(D\otimes S)\in\mathfrak{D}$ by $D\cdot S$. For the duration
of this paper, the term _$\mathrm{Perf}(Y)$ -linear category_ means a
$\Bbbk$-linear stable $\infty$-category with the $\mathrm{Perf}(Y)$-linear
structure as above.

We need a couple of preliminary definitions.

######  Definition 1.

Let $Y$ be a scheme over the field $\Bbbk$. Let $\mathfrak{D}$ be a
$\mathrm{Perf}(Y)$-linear category, and let $\mathcal{A}\subset\mathfrak{D}$
be a strictly full triangulated $\Bbbk$-linear subcategory. For a subset
$\mathcal{S}\subset\mathrm{Perf}(Y)$ we say that _$\mathcal{A}$ is closed
under the action of $\mathcal{S}$_ if for any object $S\in\mathcal{S}$ and any
object $A\in\mathcal{A}$, the result of the action $A\cdot S$ in
$\mathfrak{D}$ is an object in the subcategory $\mathcal{A}$.

######  Definition 2 ([BK90]).

Let $T$ be a triangulated category. A strictly full triangulated subcategory
$\mathcal{A}\subset T$ is a _left-admissible subcategory_ if the inclusion
functor $\mathcal{A}\hookrightarrow T$ has the left adjoint functor.

Now we can state the main definition of this paper. For the notion of a linear
category proper over the base, see [Per18, Lem. 4.7].

######  Definition 3.

Let $Y$ be a scheme over the field $\Bbbk$. We say that $Y$ is
_noncommutatively stably semiorthogonally indecomposable_ , or NSSI for
brevity, if for arbitrary choices of

  1. (1)

$\mathfrak{D}$, a $\mathrm{Perf}(Y)$-linear category which is proper over $Y$
and has a classical generator;

  2. (2)

$\mathcal{A}$, a left admissible subcategory of $\mathfrak{D}$ which is linear
over $\Bbbk$;

the subcategory $\mathcal{A}$ is closed under the action of $\mathrm{Perf}(Y)$
on $\mathfrak{D}$ in the sense of Definition 1.

######  Remark.

The first version of this paper [P20v1] was based on the notion of a _stably
semiorthogonally indecomposable_ variety. That definition is weaker in the
following sense. For a proper morphism $f\colon\mathfrak{X}\to Y$ from a
smooth variety $\mathfrak{X}$, the category $\mathrm{Perf}(\mathfrak{X})$ has
a natural $\mathrm{Perf}(Y)$-linear structure, with the action $D\cdot
S:=D\otimes_{\mathrm{Perf}(\mathfrak{X})}f^{*}S$. If we restrict Definition 3
only to linear categories arising in this way, we obtain exactly the
definition from [P20v1]. The definition was replaced since it is too weak for
the proof of Theorem 1.2.

######  Remark.

The same definition may be given for an arbitrary symmetric monoidal stable
$\infty$-category, not necessarily of the form $(\mathrm{Perf}(Y),\otimes)$.
The author does not know whether this would be useful. Note that the variety
$Y$ is completely determined by the monoidal structure on $\mathrm{Perf}(Y)$
[Lur04], so the definition above is a property of the variety $Y$, not of
something weaker.

When $T$ is a bounded derived category of coherent sheaves on a smooth
projective variety, a subcategory $\mathcal{A}\subset T$ is admissible if and
only if there exists a semiorthogonal decomposition of $T$ which has
$\mathcal{A}$ as a component [BK90]. Using this relation, in Lemma 7 we
confirm that the condition in Definition 3 implies the indecomposability in
the usual sense, i.e., all semiorthogonal decompositions of
$D^{b}_{\\!\mathrm{coh}}(Y)$ are trivial. We also show in Example 10 that
stable indecomposability is a strictly stronger condition than
indecomposability. In particular, it implies that the derived category of
sheaves on any smooth proper subvariety is also indecomposable.

The main results of this paper are the following two theorems.

######  Theorem 1.1 ( = Theorem 4.1).

Let $Y$ be a smooth projective variety over an algebraically closed field
$\Bbbk$ whose Albanese morphism is finite. Then $Y$ is a NSSI variety.

######  Theorem 1.2 ( = Theorem 5.4).

Let $\pi\colon Y\to B$ be a flat proper morphism of smooth proper varieties
over an algebraically closed field $\Bbbk$. Assume that $B$ is NSSI and that
for any $\Bbbk$-point $b\in B$ the fiber $Y_{b}:=\pi^{-1}(b)$ is a smooth
proper NSSI variety. Then $Y$ is NSSI.

We use these results to deduce the non-existence of phantom subcategories in
some varieties. Recall that a nonzero admissible subcategory
$\mathcal{A}\subset T$ of a triangulated category $T$ is called a _phantom
subcategory_ if the class of any object of $\mathcal{A}$ in the Grothendieck
group $K_{0}(T)$ is zero. In general, such subcategories exist, but it is
expected that they should not appear in many simple cases.

######  Proposition 4 ( = Proposition 25).

Let $\Bbbk$ be an algebraically closed field of characteristic zero. Let $Y$
be a smooth projective variety over $\Bbbk$ which is NSSI.

  1. (1)

Let $X$ be either the projective line $\mathbb{P}^{1}$ or a del Pezzo surface.
Then there are no phantom subcategories in the derived category
$D^{b}_{\\!\mathrm{coh}}(X\times Y)$.

  2. (2)

Let $\pi\colon\mathfrak{X}\to Y$ be an étale-locally trivial fibration with
fiber $\mathbb{P}^{1}$ or $\mathbb{P}^{2}$. Then there are no phantom
subcategories in the derived category $D^{b}_{\\!\mathrm{coh}}(\mathfrak{X})$.

Structure of the paper. In Section 2 we show some basic properties of
Definition 3. In Section 3 we prove a general rigidity statement for
admissible subcategories. In Section 4 we show that finite Albanese morphism
implies the NSSI property, thus proving Theorem 1.1. In Section 5 we prove
Theorem 1.2. Finally, in Section 6 we prove Proposition 4.

Notation and conventions. All triangulated categories and varieties in this
paper are over a field $\Bbbk$. Tensor products, pullbacks, and pushforwards
are assumed to be derived. For a subcategory $\mathcal{A}$ and an object $F$
in some category, the symbol $F\otimes\mathcal{A}$ denotes the set of all
tensor products $F\otimes A$ for objects $A\in\mathcal{A}$.

Acknowledgements. I thank Alexander Perry for asking a question that grew into
the current paper. I thank Alexander Kuznetsov and Shinnosuke Okawa for
helpful suggestions. The work on this paper was partially supported by
National Research University Higher School of Economics. This project has
received funding from the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme (Project HyperK — grant
agreement 854361).

##  2\. Preliminaries

We recall the notion and basic properties of semiorthogonal decompositions
from [BK90]. For a triangulated category $T$ a _semiorthogonal decomposition_
$T=\langle\mathcal{A},\mathcal{B}\rangle$ is a pair of strictly full
triangulated subcategories $\mathcal{A}$ and $\mathcal{B}$ such that for any
$A\in\mathcal{A}$ and any $B\in\mathcal{B}$ the graded $\mathrm{Hom}$-space
$\mathrm{Hom}^{\bullet}_{T}(B,A)$ vanishes, and such that for any object $E\in
T$ there exists a distinguished triangle

$B\to E\to A$

where $A\in\mathcal{A}$ and $B\in\mathcal{B}$. We call this the _projection
triangle_ of $E$. It is unique up to a unique isomorphism.

When $T=D^{b}_{\\!\mathrm{coh}}(Y)$ is a bounded derived category of coherent
sheaves on a smooth proper algebraic variety $Y$ over a field $\Bbbk$, in any
semiorthogonal decomposition both components $\mathcal{A}$ and $\mathcal{B}$
are _admissible subcategories_ of $D^{b}_{\\!\mathrm{coh}}(Y)$, i.e., strictly
full triangulated subcategories such that the inclusion functor into
$D^{b}_{\\!\mathrm{coh}}(Y)$ has both right and left adjoint functors. The
adjoint functors are called the right and the left projection functors,
respectively. Moreover, if $\mathcal{A}\subset D^{b}_{\\!\mathrm{coh}}(Y)$ is
an admissible subcategory, then the category

${}^{\perp}\mathcal{A}:=\\{B\in D^{b}_{\\!\mathrm{coh}}(Y)\,\,|\,\,\forall
A\in\mathcal{A}\,\,\mathrm{Hom}^{\bullet}(B,A)=0\\}$

is admissible, and there is a semiorthogonal decomposition
$D^{b}_{\\!\mathrm{coh}}(Y)=\langle\mathcal{A},{}^{\perp}\mathcal{A}\rangle$.

The derived category $D^{b}_{\\!\mathrm{coh}}(Y)$ is said to be
_indecomposable_ if for any semiorthogonal decomposition
$D^{b}_{\\!\mathrm{coh}}(Y)=\langle\mathcal{A},\mathcal{B}\rangle$ either
$\mathcal{A}=0$, or $\mathcal{B}=0$. Equivalently, the only admissible
subcategories of $D^{b}_{\\!\mathrm{coh}}(Y)$ are zero and the
$D^{b}_{\\!\mathrm{coh}}(Y)$.

######  Definition 5.

Let $\mathcal{T}$ be a triangulated category. An object $G\in\mathcal{T}$ is
called a _classical generator_ of $\mathcal{T}$ if the smallest triangulated
subcategory of $\mathcal{T}$ containing $G$ and closed under taking direct
summands is the whole $\mathcal{T}$.

######  Lemma 6.

Let $f\colon Y^{\prime}\to Y$ be a finite morphism of schemes. If
$G\in\mathrm{Perf}(Y)$ is a classical generator of $\mathrm{Perf}(Y)$, then
its pullback $f^{*}G$ is a classical generator of $\mathrm{Perf}(Y^{\prime})$.

###### Proof.

By [BB03, Th. 2.1.2] a perfect complex $f^{*}G$ is a classical generator for
$\mathrm{Perf}(Y^{\prime})$ if and only if for any nonzero $\mathcal{F}\in
D_{\mathrm{QCoh}}(Y^{\prime})$ the space
$\mathrm{Hom}_{Y^{\prime}}(f^{*}G,\mathcal{F})$ is nonzero. By the adjunction
there is an isomorphism

(6.1)
$\mathrm{Hom}_{Y^{\prime}}(f^{*}G,\mathcal{F})\cong\mathrm{Hom}_{Y}(G,f_{*}\mathcal{F}).$

Since $f$ is a finite morphism, the pushforward $f_{*}\mathcal{F}$ of a
nonzero object is nonzero. Then applying [BB03, Th. 2.1.2] on the variety $Y$
we see that the right-hand space in (6.1) is nonzero, and hence $f^{*}G$ is a
classical generator. ∎

Now we start working with the notion of NSSI varieties, introduced in
Definition 3.

######  Lemma 7.

Let $Y$ be a smooth proper variety. If $Y$ is a NSSI variety, then
$D^{b}_{\\!\mathrm{coh}}(Y)$ is an indecomposable category.

###### Proof.

Note that for a smooth variety $\mathrm{Perf}(Y)=D^{b}_{\\!\mathrm{coh}}(Y)$.
Let $D^{b}_{\\!\mathrm{coh}}(Y)=\langle\mathcal{A},\mathcal{B}\rangle$ be a
semiorthogonal decomposition. Consider $D^{b}_{\\!\mathrm{coh}}(Y)$ as a
$\mathrm{Perf}(Y)$-linear category, with the action given by tensor product.
Then from Definition 3 we deduce that both $\mathcal{A}$ and $\mathcal{B}$ are
closed under tensor products by arbitrary objects of $\mathrm{Perf}(Y)$. In
particular, for a dualizing line bundle $\omega_{Y}$ we have
$\mathcal{A}\otimes\omega_{Y}\subset\mathcal{A}$. By [KO15, Cor. 2.13] this is
only possible if either $\mathcal{A}=0$ or $\mathcal{B}=0$. ∎

######  Lemma 8.

Let $f\colon Y^{\prime}\to Y$ be a finite morphism between smooth projective
varieties. If $Y$ is a NSSI variety, then $Y^{\prime}$ is also NSSI.

###### Proof.

Since $Y$ is a smooth projective variety, there is a classical generator
$G\in\mathrm{Perf}(Y)$. The morphism $f$ is finite, so by Lemma 6 the pullback
object $f^{*}G\in\mathrm{Perf}(Y^{\prime})$ is a classical generator in
$\mathrm{Perf}(Y^{\prime})$.

Let $\mathfrak{D}$ be a $\mathrm{Perf}(Y^{\prime})$-linear category. By
definition we need to check that any admissible subcategory
$\mathcal{A}\subset\mathfrak{D}$ is closed under the action of
$\mathrm{Perf}(Y^{\prime})$. Since $\mathfrak{D}$ can also be considered as a
$\mathrm{Perf}(Y)$-linear category, by assumption that $Y$ is NSSI we know
that the subcategory $\mathcal{A}$ is closed under the action of the subset
$\operatorname{im}\,(f^{*}\colon\mathrm{Perf}(Y)\to\mathrm{Perf}(Y^{\prime}))$.
In particular, $\mathcal{A}$ is closed under the action of the classical
generator $f^{*}G\in\mathrm{Perf}(Y^{\prime})$.

Let $A\in\mathcal{A}$ be an object. Consider the subcategory
$\mathcal{C}\subset\mathrm{Perf}(Y^{\prime})$ consisting of those objects
$C\in\mathrm{Perf}(Y^{\prime})$ such that the result of the action $A\cdot C$
belongs to $\mathcal{A}$. It is easy to see that $\mathcal{C}$ is a
triangulated subcategory closed under taking direct summands. Since it
contains the classical generator $f^{*}G$, it coincides with the whole
$\mathrm{Perf}(Y^{\prime})$. This holds for an arbitrary object
$A\in\mathcal{A}$, hence $\mathcal{A}$ is closed under the action of
$\mathrm{Perf}(Y^{\prime})$ on $\mathfrak{D}$. This shows that $Y^{\prime}$ is
a NSSI variety. ∎

######  Corollary 9.

Let $Y$ be a smooth and proper variety which is NSSI. Then any smooth proper
subvariety $Y^{\prime}\subset Y$ has an indecomposable derived category.

###### Proof.

The inclusion $Y^{\prime}\subset Y$ is a finite morphism. By Lemma 8 this
implies that $Y^{\prime}$ is NSSI, and hence by Lemma 7 the derived category
$D^{b}_{\\!\mathrm{coh}}(Y^{\prime})$ is indecomposable. ∎

######  Example 10.

Let $Y$ be a K3 surface containing a smooth rational curve $C\subset Y$. Then
the derived category $D^{b}_{\\!\mathrm{coh}}(Y)$ is indecomposable, but $Y$
is not NSSI. Indecomposability follows from [KO15, Th. 1.2] since the
canonical bundle of $Y$ is globally generated. However, $Y$ is not NSSI by
Corollary 9: the curve $C\simeq\mathbb{P}^{1}$ in $Y$ admits a nontrivial
semiorthogonal decomposition
$D^{b}_{\\!\mathrm{coh}}(\mathbb{P}^{1})=\langle\mathcal{O},\mathcal{O}(1)\rangle$.

##  3\. Topological rigidity of admissible subcategories

Let $Y$ be a smooth and proper variety over the field $\Bbbk$. Let
$\mathcal{A}\subset D^{b}_{\\!\mathrm{coh}}(Y)$ be a left-admissible
subcategory. Kawatani and Okawa proved in [KO15, Th. 1.4] that $\mathcal{A}$
is closed under tensor products by line bundles from $\mathrm{Pic}^{0}(Y)$. In
this section we generalize their result to left-admissible subcategories for
sufficiently nice $\mathrm{Perf}(Y)$-linear categories (Theorem 3.2). In order
to do this, we first establish a general topological rigidity property for
admissible subcategories for the same class of $\mathrm{Perf}(Y)$-linear
categories. The ideas for proofs are more or less the same as in [KO15], but
translated to a more general setting.

We start by some well-known properties of classical generators and their
implications for admissible subcategories.

######  Lemma 11.

Let $\mathfrak{D}$ be a category, and let $\mathcal{A}\subset\mathfrak{D}$ be
a left-admissible subcategory. If an object $E\in\mathfrak{D}$ is a classical
generator of the subcategory ${}^{\perp}\mathcal{A}$, then for any object
$F\in\mathfrak{D}$ the vanishing $\mathrm{RHom}_{\mathfrak{D}}(E,F)=0$ holds
if and only if $F\in\mathcal{A}$.

###### Proof.

Since $\mathcal{A}$ is a left-admissible subcategory, it induces a
semiorthogonal decomposition
$\mathfrak{D}=\langle\mathcal{A},\mathcal{B}\rangle$, where $\mathcal{B}$ is
the orthogonal subcategory ${}^{\perp}\mathcal{A}$, and $\mathcal{B}$ is a
right-admissible subcategory [Per18, Lem. 3.6]. Define the functor
$\mathcal{B}_{R}\colon\mathfrak{D}\to\mathfrak{D}$ as the composition

$\mathfrak{D}\to\mathcal{B}\hookrightarrow\mathfrak{D}$

of the embedding functor $\mathcal{B}\hookrightarrow\mathfrak{D}$ with its
right adjoint, and the functor $\mathcal{A}_{L}$ similarly.

Consider the projection triangle:

$\mathcal{B}_{R}(F)\to F\to\mathcal{A}_{L}(F).$

An application of the functor $\mathrm{RHom}_{\mathfrak{D}}(E,-)$ to it
results in a distinguised triangle of graded vector spaces:

$\mathrm{RHom}(E,\mathcal{B}_{R}(F))\to\mathrm{RHom}(E,F)\to\mathrm{RHom}(E,\mathcal{A}_{L}(F)).$

Here the rightmost object $\mathrm{RHom}(E,\mathcal{A}_{L}(F))$ is zero since
$E\in\mathcal{B}={}^{\perp}\mathcal{A}$. Thus $\mathrm{RHom}(E,F)=0$ if and
only if $\mathrm{RHom}(E,\mathcal{B}_{R}(F))=0$. But the object
$\mathcal{B}_{R}(F)$ lies in $\mathcal{B}$, and $E$ is a classical generator
of that subcategory. Since any classical generator is a generator [BB03,
(2.1)], the vanishing of $\mathrm{RHom}(E,\mathcal{B}_{R}(F))$ implies that
$\mathcal{B}_{R}(F)$ is zero, i.e., that the object $F$ lies in $A$, as
claimed. ∎

######  Lemma 12.

Let $\mathfrak{D}$ be a category which admits a classical generator. Let
$\mathcal{A}\subset\mathfrak{D}$ be a left-admissible or a right-admissible
subcategory. Then $\mathcal{A}$ has a classical generator.

###### Proof.

Let $G\in\mathfrak{D}$ be a classical generator. For any essentially
surjective functor $\Phi\mathfrak{D}\to\mathfrak{D}^{\prime}$ between
triangulated categories it is easy to see that $\Phi(G)$ is a classical
generator in $\mathfrak{D}^{\prime}$. By assumption the inclusion functor
$\mathcal{A}\hookrightarrow\mathfrak{D}$ has either a left or a right adjoint
functor, which is a projection and hence surjective on objects. Thus
$\mathcal{A}$ has a classical generator. ∎

######  Lemma 13.

Let $\mathfrak{D}$ be a category which admits a classical generator. Let
$\mathcal{A}\subset\mathfrak{D}$ be a left-admissible subcategory. Then there
exists an object $E\in\mathfrak{D}$ such that an object $F\in\mathfrak{D}$
lies in $\mathcal{A}$ if and only if $\mathrm{RHom}_{\mathfrak{D}}(E,F)=0$.

###### Proof.

By [Per18, Lem. 3.6] the subcategory
${}^{\perp}\mathcal{A}\subset\mathfrak{D}$ is right-admissible. Then by Lemma
12 it has a classical generator. So we conclude by Lemma 11. ∎

Informally speaking, the following theorem shows that in sufficiently nice
situations, given a family of objects in some $\Bbbk$-linear category
$\mathfrak{D}$, the condition that an object lies in a fixed left-admissible
subcategory $\mathcal{A}\subset\mathfrak{D}$ is open. To make this rigorous,
we need to put some restrictions on the category and the family, as seen
below.

######  Theorem 3.1.

Let $\mathfrak{D}$ be a proper category over a field $\Bbbk$ which has a
classical generator. Let $U$ be a quasi-compact quasi-separated scheme, and
let $u_{0}\in U$ be a $\Bbbk$-point. Let $\widetilde{\mathfrak{D}}$ be the
base change $\mathfrak{D}\boxtimes\mathrm{Perf}(U)$ of $\mathfrak{D}$ from
$\Bbbk$ to $U$. Let $F\in\widetilde{\mathfrak{D}}$ be an object.

Let $\mathcal{A}\subset\mathfrak{D}$ be a left-admissible subcategory. Assume
that the base change $F_{0}\in\mathfrak{D}$ of the object $F$ along the
inclusion $\\{u_{0}\\}\hookrightarrow U$ lies in the subcategory
$\mathcal{A}$. Then there exists a Zariski-open neighborhood
$U^{\prime}\subset U$ of the point $u_{0}$ such that the base change of $F$ to
$\mathfrak{D}\boxtimes\mathrm{Perf}(U^{\prime})$ lies in the subcategory
$\mathcal{A}\boxtimes\mathrm{Perf}(U^{\prime})$. In particular, for any
$\Bbbk$-point $u\in U^{\prime}$ the restriction $F|_{\\{u\\}}\in\mathfrak{D}$
lies in $\mathcal{A}$.

###### Proof.

Consider the base change diagram:

${{\mathfrak{D}}}$${{\widetilde{\mathfrak{D}}:=\mathfrak{D}\boxtimes\mathrm{Perf}(U)}}$${{\mathfrak{D}}}$${{\mathrm{Spec}\,\Bbbk}}$${{U}}$${{\mathrm{Spec}\,\Bbbk}}$$\scriptstyle{u_{0}}$

Since $\mathfrak{D}$ has a classical generator, by Lemma 13 there exists an
object $E\in\mathfrak{D}$ such that $\mathcal{A}=E^{\perp}$. Since $U$ is
quasi-compact and quasi-separated, there exists a classical generator
$G_{U}\in\mathrm{Perf}(U)$ [BB03, Th. 3.1.1]. Consider the object $E\boxtimes
G_{U}\in\widetilde{\mathfrak{D}}$. Since $\widetilde{\mathfrak{D}}$ is a
$U$-linear category which is proper over $U$ [Per18, Lem 4.10] the internal
$\mathrm{Hom}$-object $\mathcal{H}om(E\boxtimes G_{U},F)$ exists and lies in
$\mathrm{Perf}(U)$ [Per18, Lem. 4.7]. Denote it by $\mathcal{H}$.

By the base change property of internal $\mathrm{Hom}$-objects [Per18, Lem.
2.10] the pullback of $\mathcal{H}$ to the point $\\{u_{0}\\}$ is a graded
vector space isomorphic to
$\mathrm{RHom}_{\mathfrak{D}}(E\otimes(G_{U}|_{\\{u_{0}\\}}),F_{0})$. By
assumption $F_{0}\in\mathcal{A}$, and hence by the choice of $E$ this graded
space is zero. Thus the point $u_{0}\in U$ does not lie in the set-theoretic
support of the perfect complex $\mathcal{H}$ on $U$. Then there exists a
Zariski-open neighborhood $U^{\prime}$ of $u_{0}$ such that the restriction of
$\mathcal{H}$ to $U^{\prime}$ is zero.

The restriction of the object $E\boxtimes G_{U}$ to the open subset
$U^{\prime}\subset U$ is isomorphic to $E\boxtimes(G_{U}|_{U^{\prime}})$. Note
that the object $E\in\mathfrak{D}$ is a classical generator of the subcategory
${}^{\perp}\mathcal{A}$ by construction in Lemma 13. Since
$G_{U}\in\mathrm{Perf}(U)$ is a classical generator, its restriction to an
open set $U^{\prime}\subset U$ is a classical generator for
$\mathrm{Perf}(U^{\prime})$. Thus by [Per18, Lem. 2.7] the object
$E\boxtimes(G_{U}|_{U^{\prime}})$ is a classical generator for the left-
admissible subcategory
${}^{\perp}\mathcal{A}\boxtimes\mathrm{Perf}(U^{\prime})$. By Lemma 11 the
vanishing of
$\mathcal{H}|_{U^{\prime}}\cong\mathcal{H}om(E|_{U^{\prime}},F|_{U^{\prime}})$
implies that $F|_{U^{\prime}}$ lies in
$\mathcal{A}\boxtimes\mathrm{Perf}(U^{\prime})$, as claimed. ∎

######  Theorem 3.2.

Let $Y$ be a smooth and proper scheme. Let $\mathfrak{D}$ be an $Y$-linear
category which is proper over $Y$ and has a classical generator. Let
$\mathcal{A}\subset\mathfrak{D}$ be a left-admissible subcategory. Then
$\mathcal{A}$ is closed under the action of $\mathrm{Pic}^{0}(Y)$.

###### Proof.

Consider the base change of the $Y$-linear category $\mathfrak{D}$ to the
product $Y\times\mathrm{Pic}^{0}(Y)$:

${{\mathcal{A}\boxtimes\mathrm{Perf}(\mathrm{Pic}^{0}(Y))}}$${{\mathcal{A}}}$${{\widetilde{\mathfrak{D}}:=\mathfrak{D}\boxtimes\mathrm{Perf}(\mathrm{Pic}^{0}(Y))}}$${{\mathfrak{D}}}$${{Y}}$${{Y\times\mathrm{Pic}^{0}(Y)}}$${{Y}}$${{\\{\mathcal{O}_{Y}\\}}}$${{\mathrm{Pic}^{0}(Y)}}$${{\\{*\\}}}$${\lrcorner}$${\lrcorner}$

Let $A\in\mathcal{A}$ be an arbitrary object, and let $\mathcal{P}$ be a
Poincaré line bundle on $Y\times\mathrm{Pic}^{0}(Y)$. Since
$\widetilde{\mathfrak{D}}$ is linear over $Y\times\mathrm{Pic}^{0}(Y)$, we can
consider an object
$\widetilde{A}:=\mathcal{P}\cdot(A\boxtimes\mathcal{O}_{\mathrm{Pic}^{0}(Y)})$
in $\widetilde{\mathfrak{D}}$. The base change of $\widetilde{A}$ to the point
corresponding to a trivial line bundle
$\\{\mathcal{O}_{Y}\\}\in\mathrm{Pic}^{0}(Y)$ is isomorphic to
$A\in\mathcal{A}$.

Note that $\mathfrak{D}$ is proper over $Y$, and $Y$ is proper over the base
field, so we can apply Theorem 3.1 to the object $\widetilde{A}$. It shows
that there exists an open subset $U\subset\mathrm{Pic}^{0}(Y)$ containing
$\mathcal{O}_{Y}$ such that for any line bundle $L\in U$ the restriction of
the object $\widetilde{A}$ to the point $\\{L\\}\in\mathrm{Pic}^{0}(Y)$, that
is, the result of the action $L\cdot A\in\mathfrak{D}$, is an object in the
subcategory $\mathcal{A}\subset\mathfrak{D}$.

By Lemma 12 we can choose some classical generator
$G_{\mathcal{A}}\in\mathcal{A}$. Let $U\subset\mathrm{Pic}^{0}(Y)$ be the set
of line bundles $L$ such that $L\cdot G_{\mathcal{A}}$ lies in $\mathcal{A}$.
It is an open set by the argument above. We claim that $U$ is closed under
multiplication in $\mathrm{Pic}^{0}(Y)$. Indeed, let $L_{1},L_{2}\in U$ be two
line bundles. We know that $L_{1}\cdot G_{\mathcal{A}}$ lies in
$\mathcal{A}=\langle G_{\mathcal{A}}\rangle$. Using the action of $L_{2}$ we
see that

$L_{2}\cdot(L_{1}\cdot G_{\mathcal{A}})\in\langle L_{2}\cdot
G_{\mathcal{A}}\rangle.$

But we also know that $L_{2}\cdot G_{\mathcal{A}}$ lies in $\mathcal{A}$. Thus
$L_{2}\cdot(L_{1}\cdot G_{\mathcal{A}})$ lies in $\mathcal{A}$, and by the
definition of the linear category this object is isomorphic to $(L_{2}\otimes
L_{1})\cdot G_{\mathcal{A}}$. So we see that if $L_{1}$ and $L_{2}$ lie in
$U$, then their tensor product also lies in $U$.

Since $Y$ is smooth and proper, the group scheme $\mathrm{Pic}^{0}(Y)$ is an
abelian variety, in particular irreducible. Then any open subset which is
closed under multiplication is necessarily the whole $\mathrm{Pic}^{0}(Y)$. We
conclude that $U=\mathrm{Pic}^{0}(Y)$, i.e., for any line bundle
$L\in\mathrm{Pic}^{0}(Y)$ the result of the action $L\cdot G_{\mathcal{A}}$
lies in $\mathcal{A}$. Let $A\in\mathcal{A}$ be an arbitrary object. Since
$G_{\mathcal{A}}$ is a classical generator of $\mathcal{A}$, we have
$A\in\langle G_{\mathcal{A}}\rangle$, which implies that $L\cdot A\in\langle
L\cdot G_{\mathcal{A}}\rangle\subset\mathcal{A}$. Thus the subcategory
$\mathcal{A}$ is closed under the action of $\mathrm{Pic}^{0}(Y)$. ∎

##  4\. Finite Albanese morphism implies NSSI

In this section we prove Theorem 1.1. In the proof we use the notion of a
spanning set.

######  Definition 14.

Let $T$ be a triangulated category. A set of objects $T_{0}\subset T$ is a
_spanning set_ of $T$ if any object $t\in T$ such that
$\mathrm{RHom}(t,T_{0})=0$ or $\mathrm{RHom}(T_{0},t)=0$ is the zero object.

######  Lemma 15.

Let $A$ be an abelian variety over an algebraically closed field $\Bbbk$. Then
the set of objects $\mathrm{Pic}^{0}(A)\subset D^{b}_{\\!\mathrm{coh}}(A)$
forms a spanning set.

###### Proof.

Let $\hat{A}$ be the dual abelian variety. Under the Fourier–Mukai equivalence
[Muk81] of categories
$D^{b}_{\\!\mathrm{coh}}(A)\xrightarrow{\sim}D^{b}_{\\!\mathrm{coh}}(\hat{A})$
the set of line bundles of degree zero on $A$ corresponds to the set of
skyscraper sheaves on $\hat{A}$, and skyscraper sheaves always form a spanning
set for varieties over algebraically closed fields (see, for example, [Huy06,
Prop. 3.17]). ∎

######  Lemma 16.

Let $Y$ be a smooth projective variety over an algebraically closed field
$\Bbbk$. The following are equivalent:

  1. (1)

The Albanese map $\mathrm{alb}_{Y}\colon Y\to\operatorname{Alb}(Y)$ is a
finite morphism.

  2. (2)

The set of objects $\mathrm{Pic}^{0}(Y)\subset D^{b}_{\\!\mathrm{coh}}(Y)$
forms a spanning set.

###### Proof.

Assume that the Albanese map of $Y$ is not finite. Since $Y$ is proper, this
implies that at least one of the fibers of $\mathrm{alb}_{Y}$ is a positive-
dimensional proper scheme. Using Chow’s lemma we may find a non-constant
morphism $f\colon C\to Y$ from a smooth projective curve $C$ whose image is
contained in a fiber of $\mathrm{alb}_{Y}$, i.e., the composition
$C\xrightarrow{f}Y\to\operatorname{Alb}(Y)$ is a constant map. Pick some line
bundle $M$ on $C$ of degree $g-1$, where $g$ is a genus of $C$, such that $M$
does not have nonzero global sections. Then $\mathrm{RHom}_{C}(f^{*}L,M)=0$
for any line bundle
$L\in\mathrm{Pic}^{0}(Y)\cong\mathrm{Pic}^{0}(\operatorname{Alb}(Y))$ since
the pullback $f^{*}L$ is a trivial line bundle on $C$. By adjunction this
implies that $\mathrm{RHom}_{Y}(L,f_{*}M)=0$. Since $f\colon C\to Y$ is a
nonconstant map from a smooth connected curve, it is a finite map, hence
$f_{*}M$ is a nonzero object semiorthogonal to all line bundles from
$\mathrm{Pic}^{0}(Y)$, so $\mathrm{Pic}^{0}(Y)$ does not form a spanning set.

Conversely, assume that $\mathrm{alb}_{Y}$ is a finite morphism. Let $F\in
D^{b}_{\\!\mathrm{coh}}(Y)$ be a nonzero object. Then the pushforward
$\mathrm{alb}_{Y*}(F)$ is a nonzero object in
$D^{b}_{\\!\mathrm{coh}}(\operatorname{Alb}(Y))$. Since
$\mathrm{Pic}^{0}(\operatorname{Alb}(Y))$ forms a spanning set for
$D^{b}_{\\!\mathrm{coh}}(\operatorname{Alb}(Y))$ by Lemma 15, for any nonzero
$F\in D^{b}_{\\!\mathrm{coh}}(Y)$ there exists some line bundle
$L\in\mathrm{Pic}^{0}(\operatorname{Alb}(Y))\cong\mathrm{Pic}^{0}(Y)$ such
that

$0\neq\mathrm{RHom}_{\operatorname{Alb}(Y)}(L,\mathrm{alb}_{Y*}(F))\cong\mathrm{RHom}_{Y}(\mathrm{alb}_{Y}^{*}(L),F).$

This is one half of the definition of the spanning set in
$D^{b}_{\\!\mathrm{coh}}(Y)$. It implies the full definition, i.e., the
nonvanishing of $\mathrm{RHom}$’s from $F$ to line bundles in
$\mathrm{Pic}^{0}(Y)$, by Serre duality (see, for example, [Huy06, Ex. 1.48]).
∎

######  Lemma 17.

Let $Y$ be a smooth and proper variety, and let
$\mathcal{S}\subset\mathrm{Perf}(Y)$ be a spanning set. Let $\mathfrak{D}$ be
a proper $Y$-linear subcategory which has a classical generator. Let
$\mathcal{A}\subset\mathfrak{D}$ be an admissible subcategory. Assume that it
is closed under the action of all objects in
$\mathcal{S}\subset\mathrm{Perf}(Y)$. Then $\mathcal{A}$ is closed under the
action of $\mathrm{Perf}(Y)$.

###### Proof.

By Lemma 13 there exists an object $E\in\mathfrak{D}$ such that
$\mathcal{A}=E^{\perp}$. Let $A\in\mathcal{A}$ be an arbitrary object. By
[Per18, (2.3.1)] there exists an internal $\mathrm{Hom}$-object
$\mathcal{H}om(E,A)$, which is a perfect complex on $Y$ since $\mathfrak{D}$
is proper over $Y$ [Per18, Lem. 4.7]. By definition this object represents the
functor

(17.1) $\mathrm{RHom}_{\mathfrak{D}}((-)\cdot
E,A)\colon\mathrm{Perf}(Y)\to\mathrm{Perf}(\Bbbk).$

By assumption, for any object $S\in\mathcal{S}$ the functor (17.1) vanishes on
the dual object $S^{\scriptstyle\vee}$.

Let $\mathcal{S}^{\scriptstyle\vee}$ denote the collection of duals to objects
in $\mathcal{S}$. Thus we see that the representing object
$\mathcal{H}om(E,A)\in\mathrm{Perf}(Y)$ is right-orthogonal to any object in
$\mathcal{S}^{\scriptstyle\vee}$. It is easy to see that
$\mathcal{S}^{\scriptstyle\vee}$ is a spanning set in $\mathrm{Perf}(Y)$,
hence $\mathcal{H}om(E,A)=0$. Since the functor (17.1) is represented by the
zero object, we conclude that the orbit of the object $A$ under the action of
$\mathrm{Perf}(Y)$ lies in the subcategory $E^{\perp}=\mathcal{A}$. The choice
of an object $A\in\mathcal{A}$ was arbitrary, so the category $\mathcal{A}$ is
closed under the action of $\mathrm{Perf}(Y)$. ∎

Using the lemmas above we can now prove Theorem 1.1.

######  Theorem 4.1 ( = Theorem 1.1).

Let $Y$ be a smooth projective variety over an algebraically closed field
$\Bbbk$ whose Albanese morphism is finite. Then $Y$ is a NSSI variety.

###### Proof.

Let $\mathfrak{D}$ be a $Y$-linear proper subcategory which has a classical
generator. Let $\mathcal{A}\subset\mathfrak{D}$ be a $\Bbbk$-linear admissible
subcategory. According to Definition 3 we need to check that $\mathcal{A}$ is
closed under the action of $\mathrm{Perf}(Y)$ on $\mathfrak{D}$.

We know by Theorem 3.2 that the subcategory $\mathcal{A}\subset\mathfrak{D}$
is closed under the action of the subset
$\mathrm{Pic}^{0}(Y)\subset\mathrm{Perf}(Y)$. Since $Y$ has finite Albanese
morphism, by Lemma 16 the set $\mathrm{Pic}^{0}(Y)$ is a spanning set for the
category $\mathrm{Perf}(Y)$. But then Lemma 17 shows that closedness under the
action of $\mathrm{Perf}(Y)$ follows from the closedness under the action of
$\mathrm{Pic}^{0}(Y)$, and the proof is finished. ∎

##  5\. Families of NSSI varieties

In this section we show that the converse to Theorem 1.1 does not hold. In
Theorem 5.4 we give a new way to construct NSSI varieties, and the result of
this procedure does not necessarily have finite Albanese morphism. Namely, we
show that the total space of a smooth proper family of NSSI varieties over a
NSSI base is itself a NSSI variety. This can be applied, for example, to the
Albanese map of a bielliptic surface: all fibers are elliptic curves, and the
base of the fibration is an elliptic curve. This example is discussed in more
detail in Corollary 22.

The proof of Theorem 5.4 uses some technical properties of linear categories.
We discuss them in the first two subsections. The last subsection contains the
proof of Theorem 5.4.

###  5.1. Linearity for subcategories closed under the action

Let $Y$ be a scheme over the field $\Bbbk$. Let $\mathfrak{D}$ be a
$\mathrm{Perf}(Y)$-linear category. Assume that
$\mathcal{A}\subset\mathfrak{D}$ is a strictly full subcategory that is closed
under the action of $\mathrm{Perf}(Y)$ on $\mathfrak{D}$ in the sense of
Definition 1. Then it would be reasonable to expect that $\mathcal{A}$ has the
structure of a $\mathrm{Perf}(Y)$-linear category on its own. To define it, we
would need to construct some higher associativity data on $\mathcal{A}$. The
next proposition shows that this can be done in a canonical way if the
subcategory is left-admissible (or right-admissible, with an analogous proof).
The result essentially follows from the definition of an action of a symmetric
monoidal $\infty$-category given in [Lur]. Since we were not able to find this
statement in the literature, we give a sketch of the proof.

######  Proposition 18.

Let $Y$ be a scheme over the field $\Bbbk$. Let $\mathfrak{D}$ be a
$\mathrm{Perf}(Y)$-linear category, and let $\mathcal{A}\subset\mathfrak{D}$
be a left-admissible $\Bbbk$-linear subcategory. Assume that $\mathcal{A}$ is
closed under the action of $\mathrm{Perf}(Y)$ on $\mathfrak{D}$. Then the
subcategory $\mathcal{A}$ is canonically equipped with the structure of a
$\mathrm{Perf}(Y)$-linear category, and the inclusion functor
$\mathcal{A}\hookrightarrow\mathfrak{D}$ has a canonical lift to a
$\mathrm{Perf}(Y)$-linear functor between $\mathrm{Perf}(Y)$-linear
categories.

###### Sketch of the proof.

The structure of a $\mathrm{Perf}(Y)$-action on the stable $\infty$-category
$\mathcal{A}$ consists of an action functor
$a_{\mathcal{A}}\colon\mathcal{A}\otimes_{\Bbbk}\mathrm{Perf}(Y)\to\mathcal{A}$
together with higher associativity data. As a first step, we will define the
action functor on $\mathcal{A}$ and check that it agrees with the action
functor on $\mathfrak{D}$.

Since $\mathcal{A}$ is a left-admissible subcategory, the inclusion functor
$\iota_{\mathcal{A}}\colon\mathcal{A}\hookrightarrow\mathfrak{D}$ has a left
adjoint $\Phi\colon\mathfrak{D}\to\mathcal{A}$. We use the action functor
$a_{\mathfrak{D}}\colon\mathfrak{D}\otimes\mathrm{Perf}(Y)\to\mathfrak{D}$ on
$\mathfrak{D}$ to define a functor $a_{\mathcal{A}}$ on $\mathcal{A}$ as a
composition:

$a_{\mathcal{A}}\colon\mathcal{A}\otimes_{\Bbbk}\mathrm{Perf}(Y)\xrightarrow{\iota_{\mathcal{A}}\otimes\mathrm{id}}\mathfrak{D}\otimes_{k}\mathrm{Perf}(Y)\xrightarrow{a_{\mathfrak{D}}}\mathfrak{D}\xrightarrow{\Phi}\mathcal{A}.$

By definition the functor $a_{\mathcal{A}}$ fits into the following square of
functors with a natural transformation from the upper path to the lower path
given by the composition of the morphism
$a_{\mathfrak{D}}\circ(\iota_{\mathcal{A}}\otimes\mathrm{id})$ with the counit
natural transformation
$\eta\colon\iota_{\mathcal{A}}\circ\Phi\Rightarrow\mathrm{id}_{\mathfrak{D}}$:

(18.1)
${{\mathcal{A}\otimes_{\Bbbk}\mathrm{Perf}(Y)}}$${\mathcal{A}}$${{\mathfrak{D}\otimes_{\Bbbk}\mathrm{Perf}(Y)}}$${{\mathfrak{D}}}$$\scriptstyle{\iota_{\mathcal{A}}}$$\scriptstyle{\iota_{\mathcal{A}}\otimes\mathrm{id}}$$\scriptstyle{a_{\mathcal{A}}}$$\scriptstyle{a_{\mathfrak{D}}}$

Since $\mathcal{A}$ is closed under the action of $\mathrm{Perf}(Y)$, the
image of the functor
$a_{\mathfrak{D}}\circ(\iota_{\mathcal{A}}\otimes\mathrm{id})$ lies in the
subcategory $\mathcal{A}$. The counit natural transformation $\eta$ between
endofunctors of $\mathfrak{D}$ is an isomorphism on each object of
$\mathcal{A}\subset\mathfrak{D}$ by definition. Thus the natural
transformation in the square (18.1) is an isomorphism of functors. This
property is one of the ingredients for the lift of the inclusion functor
$\iota_{\mathcal{A}}\colon\mathcal{A}\hookrightarrow\mathfrak{D}$ to a
$\mathrm{Perf}(Y)$-linear functor.

We also need to provide associativity morphisms for the action functor
$a_{\mathcal{A}}$. Among other things, we need a natural isomorphism of
functors that makes the square below commute, where
$m\colon\mathrm{Perf}(Y)\otimes\mathrm{Perf}(Y)\to\mathrm{Perf}(Y)$ is the
tensor product of objects:

(18.2)
${{\mathcal{A}\otimes\mathrm{Perf}(Y)\otimes\mathrm{Perf}(Y)}}$${{\mathcal{A}\otimes\mathrm{Perf}(Y)}}$${{\mathcal{A}\otimes\mathrm{Perf}(Y)}}$${\mathcal{A}}$$\scriptstyle{\mathrm{id}\otimes
m}$$\scriptstyle{a_{\mathcal{A}}}$$\scriptstyle{a_{\mathcal{A}}\otimes\mathrm{id}}$$\scriptstyle{a_{\mathcal{A}}}$

Note that we already have an analogous natural isomorphism for the action on
the category $\mathfrak{D}$. Using the inclusion functor $\iota_{\mathcal{A}}$
and the projection functor $\Phi$ we can induce the natural transformation as
follows.

${{\mathcal{A}\otimes\mathrm{Perf}(Y)\otimes\mathrm{Perf}(Y)}}$${{\mathfrak{D}\otimes\mathrm{Perf}(Y)\otimes\mathrm{Perf}(Y)}}$${{\mathfrak{D}\otimes\mathrm{Perf}(Y)}}$${{\mathfrak{D}\otimes\mathrm{Perf}(Y)}}$${\mathfrak{D}}$${\mathcal{A}}$$\scriptstyle{\mathrm{id}\otimes
m}$$\scriptstyle{a_{\mathfrak{D}}}$$\scriptstyle{a_{\mathfrak{D}}\otimes\mathrm{id}}$$\scriptstyle{a_{\mathfrak{D}}}$$\scriptstyle{\iota_{\mathcal{A}}}$$\scriptstyle{\Phi}$

Using the fact that $\mathcal{A}$ is closed under the action of
$\mathrm{Perf}(Y)$ we can again check that this induced natural transformation
is an isomorphism of functors. However, this is not exactly the natural
transformation we need, since its source and target functors are not the same
as in (18.2). Still, using the inclusion functor of $\mathcal{A}$, its
adjoint, and the counit natural transformation, we can construct the required
natural isomorphism using vertical and horizontal compositions. Similar
arguments can be used to induce the full $\mathrm{Perf}(Y)$-linear structure
on $\mathcal{A}$ and the inclusion functor. ∎

###  5.2. Preliminaries about linear categories

For the proof of Theorem 1.2 we need some further properties of linear
categories. They are more or less formal consequences of definitions as given
in [Lur] or [Per18].

######  Definition 19.

Let $f\colon X\to Y$ be a morphism of varieties such that the pushforward
$f_{*}$ sends perfect complexes on $X$ to perfect complexes on $Y$. Let
$\mathfrak{D}$ be a $Y$-linear category, and let
$\widetilde{\mathfrak{D}}:=\mathfrak{D}\otimes_{\mathrm{Perf}(Y)}\mathrm{Perf}(X)$
be its base change along the morphism $f$. Then we define the pushforward and
pullback functors between $\mathfrak{D}$ and $\widetilde{\mathfrak{D}}$ as
follows:

$\displaystyle\widetilde{f_{*}}\colon\widetilde{\mathfrak{D}}\cong\mathfrak{D}\otimes_{\mathrm{Perf}(Y)}\mathrm{Perf}(F)\xrightarrow{\mathrm{id}\otimes
f_{*}}\mathfrak{D}\otimes_{\mathrm{Perf}(Y)}\mathrm{Perf}(Y)\cong\mathfrak{D};$
$\displaystyle\widetilde{f^{*}}\colon\mathfrak{D}\cong\mathfrak{D}\otimes_{\mathrm{Perf}(Y)}\mathrm{Perf}(Y)\xrightarrow{\mathrm{id}\otimes
f^{*}}\mathfrak{D}\otimes_{\mathrm{Perf}(Y)}\mathrm{Perf}(F)\cong\widetilde{\mathfrak{D}}.$

For the future use we record two properties of those functors.

######  Lemma 20.

Let $f\colon X\to Y$ be a morphism of varieties such that the pushforward
$f_{*}$ sends perfect complexes on $X$ to perfect complexes on $Y$. Let
$\mathfrak{D}$ be a $Y$-linear category, and let
$\widetilde{\mathfrak{D}}:=\mathfrak{D}\otimes_{\mathrm{Perf}(Y)}\mathrm{Perf}(X)$
be its base change along the morphism $f$.

  1. (1)

For any objects $D\in\mathfrak{D}$ and $E\in\mathrm{Perf}(X)$ the following
variant of the projection formula holds in $\mathfrak{D}$:

(20.1) $f_{*}(E)\cdot D\cong\widetilde{f_{*}}(E\cdot\widetilde{f^{*}}(D)),$

where on the left hand side we use the $\mathrm{Perf}(Y)$-action on
$\mathfrak{D}$ and on the right hand side we use the $\mathrm{Perf}(X)$-action
on $\widetilde{\mathfrak{D}}$.

  2. (2)

If $\mathcal{A}\subset\mathfrak{D}$ is a $Y$-linear left-admissible
subcategory and
$\widetilde{\mathcal{A}}:=\mathcal{A}\otimes_{\mathrm{Perf}(Y)}\mathrm{Perf}(X)\subset\widetilde{\mathfrak{D}}$
is its base change along the morphism $f$, then
$\widetilde{f^{*}}(\mathcal{A})\subset\widetilde{\mathcal{A}}$ and
$\widetilde{f_{*}}(\widetilde{\mathcal{A}})\subset\mathcal{A}$.

###### Proof.

Follows directly from Definition 19. ∎

######  Lemma 21.

Let $f\colon X\to Y$ be a morphism of varieties such that the pushforward
$f_{*}$ sends perfect complexes on $X$ to perfect complexes on $Y$. Suppose
that $f$ fits into a Cartesian square of schemes:

${{X}}$${{Y}}$${{X^{\prime}}}$${{Y^{\prime}}}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f}$$\scriptstyle{g}$${\lrcorner}$

such that the morphism $g$ is flat.

Let $\mathfrak{D}$ be a $Y$-linear category. It can be considered as a
category linear over $Y^{\prime}$ via the morphism $g\colon Y\to Y^{\prime}$.
Let $\widetilde{\mathfrak{D}}$ be the base change of $\mathfrak{D}/Y$ along
the morphism $f$, and let $\widetilde{\mathfrak{D}}^{\prime}$ be the base
change of $\mathfrak{D}/Y^{\prime}$ along the morphism $f^{\prime}$. Then

  1. (1)

There is a natural equivalence of $X^{\prime}$-linear categories
$\widetilde{\mathfrak{D}}\cong\widetilde{\mathfrak{D}}^{\prime}$;

  2. (2)

That equivalence identifies the functors $\widetilde{f_{*}}$ and
$\widetilde{f^{*}}$ with the functors $\widetilde{f^{\prime}_{*}}$ and
$\widetilde{f^{\prime*}}$, respectively.

###### Proof.

According to [Per18, Sec. 2.3] the base change $\widetilde{\mathfrak{D}}$ of
the category $\mathfrak{D}$ is obtained as a tensor product
$\mathfrak{D}\otimes_{\mathrm{Perf}(Y)}\mathrm{Perf}(X)$. Since the morphism
$g$ is flat, by [Per18, (2.1)] the derived fiber product
$Y\times^{L}_{Y^{\prime}}X^{\prime}$ is isomorphic to the usual geometric
fiber product $X$, which means, explicitly, that

$\mathrm{Perf}(X)\cong\mathrm{Perf}(Y)\otimes_{\mathrm{Perf}(Y^{\prime})}\mathrm{Perf}(X^{\prime}).$

Thus we get a natural equivalence of categories

(21.1)
$\displaystyle\widetilde{\mathfrak{D}}:=\mathfrak{D}\otimes_{\mathrm{Perf}(Y)}\mathrm{Perf}(X)\cong\mathfrak{D}\otimes_{\mathrm{Perf}(Y)}\left(\mathrm{Perf}(Y)\otimes_{\mathrm{Perf}(Y^{\prime})}\mathrm{Perf}(X^{\prime})\right)\cong$
$\displaystyle\cong\mathfrak{D}\otimes_{\mathrm{Perf}(Y^{\prime})}\mathrm{Perf}(X^{\prime})=:\widetilde{\mathfrak{D}}^{\prime}.$

It is easy to check that this equivalence is
$\mathrm{Perf}(X^{\prime})$-linear and that it respects the pushforward and
the pullback functors. ∎

###  5.3. Proof of the theorem

We are now ready to prove the main theorem of this section.

######  Theorem 5.4 ( = Theorem 1.2).

Let $\pi\colon Y\to B$ be a flat proper morphism of smooth proper varieties
over an algebraically closed field $\Bbbk$. Assume that $B$ is NSSI and that
for any $\Bbbk$-point $b\in B$ the fiber $Y_{b}:=\pi^{-1}(b)$ is a smooth
proper NSSI variety. Then $Y$ is NSSI.

###### Proof.

Let $\mathfrak{D}$ be a $Y$-linear category which is proper over $Y$ and has a
classical generator. Let $\mathcal{A}\subset\mathfrak{D}$ be a left-admissible
subcategory. Via the morphism $\pi\colon Y\to B$ we may consider
$\mathfrak{D}$ as a $B$-linear subcategory. Since $\pi$ is a proper flat
morphism, the category $\mathfrak{D}$ is proper over $B$ [Per18, Lem. 4.9]. By
assumption $B$ is NSSI, which implies that the left-admissible subcategory
$\mathcal{A}\subset\mathfrak{D}$ is closed under the action of
$\mathrm{Perf}(B)$. By Proposition 18 this implies that $\mathcal{A}$ can be
considered as a $B$-linear category. In particular, it is valid to consider
the base change of $\mathcal{A}$ along an arbitrary map to $B$.

Pick some point $b\in B$ and let $j\colon F\hookrightarrow Y$ be the fiber
over that point. Consider the base change diagram:

(21.2)
${{\widetilde{\mathcal{A}}\subset\widetilde{\mathfrak{D}}}}$${{\mathcal{A}\subset\mathfrak{D}}}$${{F}}$${{Y}}$${{\\{b\\}}}$${{B}}$

Here $\widetilde{\mathcal{A}}$ and $\widetilde{\mathfrak{D}}$ are defined as
base changes of $B$-linear categories $\mathcal{A}$ and $\mathfrak{D}$,
respectively, along the inclusion $\\{b\\}\hookrightarrow B$. By [Per18, Lem.
3.17] the subcategory $\widetilde{\mathcal{A}}$ is left-admissible in
$\widetilde{\mathfrak{D}}$. Since the category $\mathfrak{D}$ has a classical
generator by assumption, and the category $\widetilde{\mathfrak{D}}$ is
obtained from $\mathfrak{D}$ by the base change to a point
$\\{b\\}\simeq\mathrm{Spec}\,\Bbbk$, we conclude by [Per18, Lem. 2.7] that
$\widetilde{\mathfrak{D}}$ has a classical generator as well.

Moreover, since $\mathfrak{D}$ is equipped with a $Y$-linear structure, the
category $\widetilde{\mathfrak{D}}$ has the induced structure of a $F$-linear
category as the base change of $\mathfrak{D}$ along the morphism $j\colon
F\hookrightarrow Y$, as shown in Lemma 21. Thus over $F$ we have an $F$-linear
category $\widetilde{\mathfrak{D}}$ which is proper over $F$ and has a
classical generator, and a left-admissible $\Bbbk$-linear subcategory
$\widetilde{\mathcal{A}}\subset\widetilde{\mathfrak{D}}$. By assumption the
fiber $F$ is NSSI, so we conclude that $\widetilde{\mathcal{A}}$ is closed
under the action of $\mathrm{Perf}(F)$ on $\widetilde{\mathfrak{D}}$.

Choose arbitrary objects $E\in\mathrm{Perf}(F)$ and $A\in\mathcal{A}$. Recall
from Definition 19 that the morphism $j$ induces the pullback functor
$\widetilde{j^{*}}\colon\mathfrak{D}\to\widetilde{\mathfrak{D}}$. By Lemma 21
the same functor is also induced by the inclusion $\\{b\\}\hookrightarrow B$,
and since $\mathcal{A}$ is a $B$-linear category, the pullback $j^{*}$
restricts to a functor between the subcategories $\mathcal{A}$ and
$\widetilde{\mathcal{A}}$. Then the object $\widetilde{j^{*}}(A)$ lies in
$\widetilde{\mathcal{A}}$. Since $\widetilde{A}$ is closed under the action of
$\mathrm{Perf}(F)$, we know that the result of the action
$\widetilde{j^{*}}(A)\cdot E$ lies in the subcategory
$\widetilde{\mathcal{A}}\subset\widetilde{\mathfrak{D}}$. Using the
pushforward functor
$\widetilde{j_{*}}\colon\widetilde{\mathcal{A}}\to\mathcal{A}$ and the
projection formula from Lemma 20 we get an isomorphism

$\widetilde{j_{*}}(\widetilde{j^{*}}(A)\cdot E)\simeq A\cdot j_{*}(E).$

Since the left-hand object lies in the subcategory $\mathcal{A}$ and the
object $A$ is arbitrary, we see that the subcategory
$\mathcal{A}\subset\mathfrak{D}$ is closed under the action of the subset
$\operatorname{im}\,(j_{*}\colon\mathrm{Perf}(F)\to\mathrm{Perf}(Y))$.

In particular, this subset contains all skyscraper sheaves at the points of
$F=\pi^{-1}(b)\subset Y$. Since the choice of the point $b\in B$ was
arbitrary, this holds for all fibers, and hence $\mathcal{A}$ is closed under
the action of the set of all skyscrapers in $Y$. Since the base field $\Bbbk$
is assumed to be algebraically closed, by [Huy06, Prop. 3.17] the skyscraper
sheaves form a spanning set in $\mathrm{Perf}(Y)$, so by Lemma 17 we conclude
that $\mathcal{A}$ is closed under the action of $\mathrm{Perf}(Y)$. Therefore
$Y$ is a NSSI variety. ∎

######  Remark.

In the argument above, even if the category $\mathfrak{D}$ is of geometric
origin, i.e., for some morphism $f\colon\mathfrak{X}\to Y$ of varieties there
is an equivalence $\mathfrak{D}\simeq\mathrm{Perf}(\mathfrak{X})$ of
$Y$-linear categories, we cannot guarantee the same for the base change
$\widetilde{\mathfrak{D}}$ unless the morphism $f$ is flat, but this is very
restrictive. Thus in order to freely use base change with nice properties such
as ones from Lemma 20 we had to replace the notion of stably semiorthogonally
indecomposable varieties from the first version of this paper [P20v1] with the
noncommutative generalization given in Definition 3.

As an explicit application of the theorem, we show that bielliptic surfaces
are NSSI varieties, even though their Albanese morphism is not finite.

######  Corollary 22.

Any bielliptic surface over an algebraically closed field $\Bbbk$ is a NSSI
variety.

###### Proof.

Let $Y$ be a bielliptic surface. Recall from [Bea96, (VI)] that the Albanese
morphism of $Y$ is a fibration $f\colon Y\to E$ for some elliptic curve $E$,
and each fiber is isomorphic to some fixed elliptic curve $F$. Since elliptic
curves are NSSI by Theorem 1.1, we can apply Theorem 1.2 to the Albanese
morphism of $Y$ to conclude that $Y$ is a NSSI variety. ∎

##  6\. Products and fibrations

In this section we discuss some applications of the NSSI property for
varieties. In Lemma 24 we show that NSSI varieties are stably indecomposable
in the following naive sense: if $Y$ is a NSSI variety, then for an arbitrary
smooth projective variety $X$, any admissible subcategory in
$D^{b}_{\\!\mathrm{coh}}(X\times Y)$ is induced by an admissible subcategory
in $D^{b}_{\\!\mathrm{coh}}(X)$. Note that when applied to the case
$X=\\{\mathrm{pt}\\}$ this result recovers Lemma 7.

After that, in Proposition 25 we use the NSSI property to deduce the non-
existence of phantom subcategories in some varieties, such as total spaces of
$\mathbb{P}^{1}$\- and $\mathbb{P}^{2}$-families over NSSI varieties. For
example, this shows that there are no phantom subcategories in the surface
$\mathbb{P}^{1}\times E$, where $E$ is an elliptic curve.

We start with an observation about linear admissible subcategories. Given a
morphism $\pi\colon\mathfrak{X}\to U$, a $U$-linear admissible subcategory in
$D^{b}_{\\!\mathrm{coh}}(\mathfrak{X})$ can be thought of as a family of
admissible subcategories in the fibers of the morphism $\pi$. The moduli
spaces of such fiberwise admissible subcategories have been studied in the
paper [BOR20]. When applied to a special situation where $\pi$ is an étale-
locally trivial fibration, their results imply the following.

######  Lemma 23.

Let $\pi\colon\mathfrak{X}\to U$ be a smooth and proper morphism which is an
étale-locally trivial fibration with fiber $X$. Assume that $U$ is a connected
excellent scheme over $\mathbb{Q}$ and that $\pi$ admits a relative ample
invertible sheaf. Then for any point $u\in U$ the base change map

$\left\\{\parbox{99.58464pt}{ \begin{center} $U$-linear admissible \\\
subcategories \\\ $\mathcal{A}\subset D^{b}_{\\!\mathrm{coh}}(\mathfrak{X})$
\end{center} }\right\\}\xrightarrow{\text{restriction to
$\mathfrak{X}_{u}\simeq X$}}\left\\{\parbox{99.58464pt}{ \begin{center}
admissible \\\ subcategories \\\ $\mathcal{A}_{X}\subset
D^{b}_{\\!\mathrm{coh}}(X)$ \end{center} }\right\\}$

is an injection.

###### Proof.

By [BOR20, Th. A] there exists an algebraic space $\mathrm{SOD}_{\pi}\to U$
which is étale over $U$ such that the set of global sections
$\mathrm{SOD}_{\pi}(U)$ is the set of $U$-linear admissible subcategories of
$D^{b}_{\\!\mathrm{coh}}(\mathfrak{X})$, and the fiber of $\mathrm{SOD}_{\pi}$
over the point $u\in U$ is the set of admissible subcategories in
$D^{b}_{\\!\mathrm{coh}}(X)$. Since $\pi$ is assumed to be étale-locally
trivial, the étale morphism $\mathrm{SOD}_{\pi}\to U$ is by construction also
étale-locally trivial over $U$, in particular separated [Stacks, Tag 02KU]. A
global section of a separated étale morphism over a connected scheme is
uniquely determined by its value over a single point $u$, and this is exactly
the claim of the lemma. ∎

######  Lemma 24.

Let $Y$ be a smooth projective variety which is NSSI. Let $X$ be any smooth
projective variety, and let $\mathcal{A}\subset
D^{b}_{\\!\mathrm{coh}}(X\times Y)$ be an admissible subcategory. Then there
exists an admissible subcategory $\mathcal{A}_{X}\subset
D^{b}_{\\!\mathrm{coh}}(X)$ such that $\mathcal{A}=\mathcal{A}_{X}\boxtimes
D^{b}_{\\!\mathrm{coh}}(Y)$.

###### Proof.

Consider the projection morphism

$\pi\colon X\times Y\to Y.$

The category $D^{b}_{\\!\mathrm{coh}}(X\times Y)$ thus may be viewed as a
$Y$-linear category, with action given by the tensor product with the
pullbacks along $\pi$. Since $Y$ is assumed to be a NSSI variety, the
subcategory $\mathcal{A}$ is closed under the action of $\mathrm{Perf}(Y)$,
and hence by Proposition 18 it is naturally a $Y$-linear subcategory as well.
Pick a point $y\in Y$. By Lemma 23 the subcategory $\mathcal{A}$ is uniquely
determined by its restriction to the fiber $\pi^{-1}(y)$, which is an
admissible subcategory $\mathcal{A}_{X}\subset D^{b}_{\\!\mathrm{coh}}(X)$.
Since the base change $\mathcal{A}_{X}\boxtimes D^{b}_{\\!\mathrm{coh}}(Y)$ is
a $Y$-linear admissible subcategory of $D^{b}_{\\!\mathrm{coh}}(X\times Y)$
with the same restriction to the fiber, we conclude that
$\mathcal{A}=\mathcal{A}_{X}\boxtimes D^{b}_{\\!\mathrm{coh}}(Y)$. ∎

This work started with the following application in mind. It provides many new
examples of varieties which have non-trivial semiorthogonal decompositions,
but do not contain any phantom subcategories.

######  Proposition 25 ( = Proposition 4).

Let $\Bbbk$ be an algebraically closed field of characteristic zero. Let $Y$
be a smooth projective variety over $\Bbbk$ which is NSSI.

  1. (1)

Let $X$ be either the projective line $\mathbb{P}^{1}$ or a del Pezzo surface.
Then there are no phantom subcategories in the derived category
$D^{b}_{\\!\mathrm{coh}}(X\times Y)$.

  2. (2)

Let $\pi\colon\mathfrak{X}\to Y$ be an étale-locally trivial fibration with
fiber $\mathbb{P}^{1}$ or $\mathbb{P}^{2}$. Then there are no phantom
subcategories in the derived category $D^{b}_{\\!\mathrm{coh}}(\mathfrak{X})$.

###### Proof.

We deal with part (1) first. Let $\mathcal{A}\subset
D^{b}_{\\!\mathrm{coh}}(X\times Y)$ be a admissible subcategory. Since $Y$ is
assumed to be NSSI, by Lemma 24 there exists an admissible subcategory
$\mathcal{A}_{X}\subset D^{b}_{\\!\mathrm{coh}}(X)$ such that
$\mathcal{A}=\mathcal{A}_{X}\boxtimes D^{b}_{\\!\mathrm{coh}}(Y)$. In
particular, $\mathcal{A}$ contains pullbacks of all objects in
$\mathcal{A}_{X}$ along the projection $p\colon X\times Y\to X$. Since the
pullback map $p^{*}\colon K_{0}(X)\to K_{0}(X\times Y)$ is an injection,
$\mathcal{A}$ is a phantom in $D^{b}_{\\!\mathrm{coh}}(X\times Y)$ if and only
if $\mathcal{A}_{X}$ is in $D^{b}_{\\!\mathrm{coh}}(X)$.

By assumption $X$ is either $\mathbb{P}^{1}$, or a del Pezzo surface. For
$\mathbb{P}^{1}$ it is easy to show that any nontrivial admissible subcategory
in $D^{b}_{\\!\mathrm{coh}}(\mathbb{P}^{1})$ is generated by a single line
bundle $\mathcal{O}_{\mathbb{P}^{1}}(n)$, and in particular there are no
phantom subcategories. If $X$ is a del Pezzo surface, then under our
assumptions on the base field $D^{b}_{\\!\mathrm{coh}}(X)$ has no phantom
subcategories by [Pir20, Th. 6.35]. So in both cases there are no phantoms in
$D^{b}_{\\!\mathrm{coh}}(X)$. Thus $\mathcal{A}$ is not a phantom subcategory.

Consider now part (2). Let $\mathcal{A}\subset
D^{b}_{\\!\mathrm{coh}}(\mathfrak{X})$ be an admissible subcategory. By
Theorem 1.1 and Proposition 18 it is a $Y$-linear subcategory. Note that since
the fibers of the morphism $\pi$ are projective spaces, the relative
anticanonical bundle provides a relatively ample line bundle. For any étale-
locally trivial fibration admitting a relatively ample line bundle Lemma 23
shows that $\mathcal{A}$ is uniquely determined by its restriction
$\mathcal{A}_{X}$ to some fiber $j\colon X\subset\mathfrak{X}$. However, since
the base change procedure in Lemma 23 involves not only the restriction of
objects from $\mathcal{A}$ to the fiber, but also the closure of the resulting
subcategory with respect to direct summands, it is in principle possible that
a phantom subcategory in the total space may restrict to a non-phantom
subcategory in the fiber.

Another compatibility condition, proved in [Kuz11, Th. 5.6], is that for any
object $A\in\mathcal{A}_{X}$ the pushforward $j_{*}A\in
D^{b}_{\\!\mathrm{coh}}(\mathfrak{X})$ lies in $\mathcal{A}$. For general
étale-locally trivial fibrations the induced pushforward map $j_{*}\colon
K_{0}(X)\to K_{0}(\mathfrak{X})$ does not have to be an injection. However,
for the cases considered in (2), it is, as we show below.

Assume that $\pi\colon\mathfrak{X}\to Y$ is an étale-locally trivial fibration
whose fiber is a projective space $\mathbb{P}^{n}$. By Blanchard–Deligne
theorem the Leray–Serre spectral sequence computing the cohomology of the
total space $\mathfrak{X}$ degenerates at $E_{2}$, and the answer is given in
terms of the local systems formed by the cohomology of the fibers of the map
$\pi$. Since all fibers are projective spaces, the cohomology is generated by
a unique primitive ample class, hence the action of the (étale) fundamental
group of $Y$ on the cohomology $H^{*}(\mathbb{P}^{n})$ can only be trivial.
Thus $H^{*}(\mathfrak{X})$ is abstractly isomorphic to the tensor product
$H^{*}(Y)\otimes H^{*}(\mathbb{P}^{n})$. In particular, the pushforward
$j_{*}\colon K_{0}(\mathbb{P}^{n})\to K_{0}(\mathfrak{X})$ is an injection.

To sum up, any admissible subcategory $\mathcal{A}\subset
D^{b}_{\\!\mathrm{coh}}(\mathfrak{X})$ is uniquely determined by the
admissible subcategory $\mathcal{A}_{X}\subset D^{b}_{\\!\mathrm{coh}}(X)$ on
some fiber. If $X$ is $\mathbb{P}^{1}$ or $\mathbb{P}^{2}$, then as explained
in part (1), there are no phantom subcategories in
$D^{b}_{\\!\mathrm{coh}}(X)$. In particular, $\mathcal{A}_{X}$ contains some
object $A\in\mathcal{A}_{X}$ with nonzero class in $K_{0}(X)$. The object
$j_{*}A$ lies in $\mathcal{A}$, and as argued above, when $X$ is a projective
space, the class of the pushforward in $K_{0}(\mathfrak{X})$ is nonzero, hence
$\mathcal{A}$ is not a phantom subcategory. ∎

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