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This article is cited in 1 scientific paper (total in 1 paper) Producing new semi-orthogonal decompositions in arithmetic geometry M. V. Bondarkoab a St. Petersburg University, St. Petersburg, Russia b St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, St. Petersburg, Russia
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     § 1. IntroductionThis paper is devoted to constructing new semi-orthogonal decompositions (see Definitions 2.3, 1, and 2.9, II.2, below) of certain triangulated categories out of given ones; the relevance of this matter is discussed in Remark 1.4, 2. We prove some general existence statements and apply them to various triangulated subcategories of $D(\operatorname{Qcoh}(X))$ (see Definition 1.1, 7); we always assume $X$ to be a scheme that is proper over the spectrum of a Noetherian ring $R$. These applications rely on the main results of [21] and [22], along with a new theorem (Theorem 1.5). We reformulate some results of the paper in terms of admissible subcategories and concentrate them in an important theorem (Theorem 1.3). All these statements can easily be deduced from their decomposition versions (see Theorem 4.12) by rather straightforward applications of Proposition 2.5; see Remarks 4.22 and 2.4 below. In this paper all the subcategories we consider are assumed to be strictly full, and all coproducts are small. Let us introduce some definitions and notation. Definition 1.1. Let $\mathcal{D}$ be a triangulated category; assume that $\mathcal{T}$, $\mathcal{T}'$ and some $\mathcal{T}_i$ are (strictly full) triangulated subcategories of $\mathcal{D}$. 1. We say that $\mathcal{T}$ is left (right) admissible in $\mathcal{D}$ if the embedding $\mathcal{T}\to \mathcal{D}$ admits a left (right, respectively) adjoint; $\mathcal{T}$ is said to be admissible in $\mathcal{D}$ if it is both left and right admissible in it. 2. We write $\mathcal{T}\cap{\mathcal{T}'}$ for the subcategory of $\mathcal{D}$ whose object class equals ${\operatorname{Obj} \mathcal{T}\cap \operatorname{Obj}\mathcal{T}'}$. Moreover, we write $(\mathcal{T}_i)\cap{\mathcal{T}'}$ for the family $(\mathcal{T}_i\cap{\mathcal{T}'})$. 3. Given an additive category $C$ and $M,N\in\operatorname{Obj} C$, we write $C(M,N)$ for the group of morphisms from $M$ to $N$ in $C$. Moreover, for $D,E\subset \operatorname{Obj} \mathcal{D}$ we write $D\perp E$ if $\mathcal{D}(X,Y)=\{0\}$ for all $X\in D$ and $Y\in E$. 4. We write $\mathcal{T}^\perp_{\mathcal{T}'}$ for the subcategory of $\mathcal{T}'$ whose object class is
$$
\begin{equation*}
\{Y\in \operatorname{Obj} \mathcal{T}'\colon \{X\}\perp \{Y\}\ \forall\, X\in \operatorname{Obj} \mathcal{T}\}.
\end{equation*}
\notag
$$
Dually, we set the object class of the subcategory ${}^{\perp}_{\mathcal{T}'}\mathcal{T}$ to be
$$
\begin{equation*}
\{Y\in \operatorname{Obj} \mathcal{T}'\colon \{Y\}\perp \{X\}\ \forall\, X\in \operatorname{Obj} \mathcal{T}\}.
\end{equation*}
\notag
$$
Moreover, we write $(\mathcal{T}_i)^{\perp}_{\mathcal{T}'}$ for the family $(\mathcal{T}_i{}^{\perp}_{\mathcal{T}'})$. 5. Assume that $\mathcal{D}$ is closed with respect to (small) coproducts. Then we write $\mathcal{T}^{\coprod}$ for the smallest (strict) triangulated subcategory of $\mathcal{D}$ that is closed with respect to $\mathcal{D}$-coproducts and contains $\mathcal{T}$. Moreover, we write $\mathcal{T}^{\coprod}_{\mathcal{T}'}$ ($(\mathcal{T}_i)^{\coprod}$ and $({\mathcal{T}_i})^{\coprod}_{\mathcal{T}'}$) for the category $\mathcal{T}^{\coprod}\cap {\mathcal{T}'}$ (for the families $(\mathcal{T}_i^{\coprod})$ and $(\mathcal{T}_i^{\coprod}\cap {\mathcal{T}'})$, respectively). 6. Throughout this paper $R$ is a commutative unital ring. We set $R\text{-}\mathrm{mod}\subset R\text{-}\mathrm{Mod}$ to be the subcategory of finitely generated $R$-modules; $S=\operatorname{Spec} R$. 7. Fix a scheme $X$ proper over $S$. We set
$$
\begin{equation*}
\begin{aligned} \, \mathcal{D}_{\mathrm p} &= D_{\mathrm{perf}}(X)\subset \mathcal{D}^{\mathrm{b}}= D^{\mathrm{b}}_{\mathrm{coh}}(\operatorname{Qcoh}(X)) \subset \mathcal{D}^-= D^-_{\mathrm{coh}}(\operatorname{Qcoh}(X)) \subset \mathcal{D}^{\mathrm{u}} \\ &= D_{\mathrm{coh}}(\operatorname{Qcoh}(X)) \subset \mathcal{D}_{\mathrm{Q}}=D(\operatorname{Qcoh}(X)); \end{aligned}
\end{equation*}
\notag
$$
here $D_{\mathrm{perf}}(X)\subset \mathcal{D}_{\mathrm{Q}}$ is the subcategory of perfect complexes on $X$ (see [30], Definition 20.49.1 (tag 08CM)), and a complex $N$ (in $\mathcal{D}_{\mathrm{Q}}$) belongs to $\mathcal{D}^{\mathrm{u}}$ whenever all of its cohomology sheaves $H^i(N)$ are coherent; it also belongs to $\mathcal{D}^{\mathrm{b}}$ ($\mathcal{D}^-$) if we also have $H^i(N)=0$ for $i\gg 0$ and $i\ll 0$ (for $i\gg 0$ only, respectively). Moreover, we consider $\mathcal{D}^+= D^+_{\mathrm{coh}}(\operatorname{Qcoh}(X))\subset \mathcal{D}^{\mathrm{u}}$ that is defined similarly. We discuss these categories in Remark 1.4, 3, below; see also Remark 3.1. 8. We say that $X$ is projective over $S=\operatorname{Spec} R$ if $X$ is a closed subscheme of the projectivization $Y$ of a vector bundle $\mathcal{E}$ over $S$. 9. An additive functor $\mathcal{D}\to \mathfrak{E}$, where $\mathfrak{E}$ is an abelian category, is said to be homological if it converts distinguished triangles into long exact sequences. Remark 1.2. Clearly, all the subcategories of $\mathcal{D}$ that we describe in Definition 1.1, 2–5, are triangulated; recall the assumption of strictness. Theorem 1.3. Assume that $R$ is a Noetherian ring and $X$ is a proper scheme over $S=\operatorname{Spec}R$. I. Let $\mathcal{X}$ be a left (right) admissible subcategory of $\mathcal{D}_{\mathrm{p}}$ (see Definition 1.1, 7).
II. Assume that $\mathcal{W}$ is a left (right) admissible subcategory of $\mathcal{D}^{\mathrm{b}}$ and that either $X$ is regular of finite Krull dimension or that regular alterations exist for all integral closed subschemes of $X$.1
Remark 1.4. 1. The second and third statements in Theorem 1.3, II.1, generalize and extend Theorem A.1 of [13] broadly. 2. Recall that admissible subcategories and semi-orthogonal decompositions of some derived categories of (quasi)coherent sheaves are important for noncommutative geometry. 3. Recall that the obvious exact functors $D^-(\operatorname{coh}(X))\to D^-_{\mathrm{coh}}(\operatorname{Qcoh}(X))$ and $D^{\mathrm{b}}(\operatorname{coh}(X))\to D^{\mathrm{b}}_{\mathrm{coh}}(\operatorname{Qcoh}(X))$ are equivalences of categories; see [30], Lemma 36.11.1 (tag 0FDA). On the other hand a similar functor $D(\operatorname{coh}(X))\to D_{\mathrm{coh}}(\operatorname{Qcoh}(X))$ is not necessarily an equivalence: see § 3 in [24]. However, it is an equivalence if $X$ is regular and of finite Krull dimension: see Corollary 5.12 in [24]. 4. Note also that Theorem 1.3 becomes slightly more difficult to prove if $D_{\mathrm{perf}}(X)\neq D^{\mathrm{b}}_{\mathrm{coh}}(\operatorname{Qcoh}(X))$, that is, if $X$ is singular;2 cf. Corollary 4.18 below. This case is slightly less popular than the regular one. Yet some nontrivial semi-orthogonal decompositions of $D^{\mathrm{b}}(X)$ for possibly singular $X$ were provided by Theorem 6.7 and Corollary 6.10 of [2]. Moreover, semi-orthogonal decompositions in the case when $X$ is a singular surface were discussed in detail in [13]. 5. This paper was inspired by a certain duality between weight and $t$-structure statements that were studied by this author in several papers starting from [7], § 4.4. This matter was described in § 4.3 of the arXiv version [9] of this text (note however that some definitions there are slightly different form the ones used here). Note also that that version contains more remarks (of various sorts) than the current paper. In the new manuscript [8] a number of new statements on orthogonal weight and $t$-structures that generalize some central results of the current paper are established. 6. Other recent ingredients are the description of some of our categories as certain categories of functors from $\mathcal{D}_{\mathrm{p}}$ and $\mathcal{D}^{\mathrm{b}}$. These are essentially provided by (Neeman’s) Theorem 1.7 below in combination with the following theorem, which is both important for the current paper and interesting in itself. In the following statement we write $\operatorname{Fun}_R((\mathcal{D}_{\mathrm{p}})^{\mathrm{op}}, R\text{-}\mathrm{Mod})$ for the category of additive $R$-linear functors $((\mathcal{D}_{\mathrm{p}})^{\mathrm{op}})\to R\text{-}\mathrm{Mod}$. We write $\mathcal{Y}^{\mathrm{u}}\colon\mathcal{D}^{\mathrm{u}}\to \operatorname{Fun}_R((\mathcal{D}_{\mathrm{p}})^{\mathrm{op}},R\text{-}\mathrm{Mod})$ for the corresponding Yoneda map, that is, for $N\in \mathcal{D}^{\mathrm{u}}$ we set
$$
\begin{equation*}
\mathcal{Y}^{\mathrm u}(N)=\mathcal{D}^{\mathrm u}(\,\cdot\,,N)\big{|}_{\mathcal{D}_{\mathrm p}}.
\end{equation*}
\notag
$$
Theorem 1.5. Assume that $X$ is a proper scheme over $S=\operatorname{Spec} R$, where $R$ is a Noetherian ring. 1. Then the functor $\mathcal{Y}^{\mathrm{u}}$ is full. 2. Suppose that $X$ is projective over $S$ (in the sense specified in Definition 1.1, 8). Then, given an object $N$ of $\mathcal{D}_{\mathrm{Q}}$, we have $\mathcal{D}_{\mathrm{Q}}(M,N)\in R\text{-}\mathrm{mod}$ for all $M\in \operatorname{Obj} \mathcal{D}_{\mathrm{p}}$ if and only if $N\in \operatorname{Obj} \mathcal{D}^{\mathrm{u}}$ (see Definition 1.1, 7 and 6). Moreover, an object $N$ of $\mathcal{D}^{\mathrm{u}}$ belongs to $\mathcal{D}^-$ ($\mathcal{D}^+$, $\mathcal{D}^{\mathrm{b}}$) if and only if for each $M\in \operatorname{Obj} \mathcal{D}_{\mathrm{p}}$ we have $\{M[i]\}\perp \{N\}$ if $i\ll 0$ (respectively $i\gg 0$ in both cases). Remark 1.6. Theorem 1.5 extends substantially Corollary 0.5 of [21]; see Theorem 1.7, 1, below. Correspondingly, it is perhaps sufficient to assume that $X$ is proper over $\operatorname{Spec} R$ in this theorem (as well as in Corollary 1.10 below; see Remark 1.11, 2). Recall also that Corollary 0.5 of [21] was inspired by the question of the existence of certain adjoint functors; see Remark 0.7 ibidem. We prove a nice general result of this sort in Corollary 4.5 below and combine it with Theorem 1.5, 1, in Remark 4.8. These statements are perhaps more practical than the corresponding result of Corollary 0.4 in [21]. Now let us recall some results of Neeman’s that are really important for the current paper. This will help the reader to understand the context of Theorem 1.5 better. We assume that $R$ is a commutative Noetherian ring. Theorem 1.7. Assume that $X$ is a proper scheme over $S=\operatorname{Spec} R$. 1. Let $\mathcal{Y}^-\colon \mathcal{D}^-\to \operatorname{Fun}_R((\mathcal{D}_{\mathrm{p}})^{\mathrm{op}},R\text{-}\mathrm{Mod})$ denote the corresponding Yoneda map, that is, for $N\in \mathcal{D}^-$ let $\mathcal{Y}^-(N)$ be the restriction of $\mathcal{D}^-(\,\cdot\,,N)$ to $\mathcal{D}_{\mathrm{p}}$. Then the functor $\mathcal{Y}^-\colon \mathcal{D}^-\to \operatorname{Fun}_R((\mathcal{D}_{\mathrm{p}})^{\mathrm{op}},R\text{-}\mathrm{Mod})$ is full and a homological functor $H\colon (\mathcal{D}_{\mathrm{p}})^{\mathrm{op}}\to R\text{-}\mathrm{Mod}$ belongs to the essential image of $\mathcal{Y}^-$ if and only if it satisfies the following conditions: for any $M\in \operatorname{Obj} \mathcal{D}_{\mathrm{p}}$ the module $H(M)$ is finitely generated over $R$ and there exists $c_M>0$ such that $H(M[i])=0$ whenever $i>c_M$. Moreover, the restriction $\mathcal{Y}^{\mathrm{b}}$ of $\mathcal{Y}^-$ to $\mathcal{D}^{\mathrm{b}}$ is faithful and its essential image consists of those homological functors that satisfy the following conditions: for any $M\in \operatorname{Obj} \mathcal{D}_{\mathrm{p}}$ the module $H(M)$ is finitely generated and there exists $c_M>0$ such that $H(M[i])=0$ whenever $|i|>c_M$. 2. Assume in addition that either $X$ is regular of finite Krull dimension or that regular alterations exist for all integral closed subschemes of $X$. Then a functor $F\colon \mathcal{D}^{\mathrm{b}}\to R\text{-}\mathrm{Mod}$ is corepresented by an object of $\mathcal{D}_{\mathrm{p}}$ if and only if $\bigoplus_{i\in \mathbb{Z}}F(N[i])$ is a finitely generated $R$-module for each $N\in\operatorname{Obj} \mathcal{D}^{\mathrm{b}}$. Proof. 1. This is essentially Corollary 0.5 of [21]; note here that we replace the corresponding derived categories mentioned there by their equivalent modifications described in Definition 1.1, 7 (see Remark 1.4, 3).
2. In the case when regular alterations exist for all integral closed subschemes of $X$ this statement immediately follows from Theorem 0.2 of [22] (yet one has to take Remark 1.4, 3, into account again). If $X$ is regular of finite Krull dimension, then this statement follows easily from assertion 1: see Corollary 4.18 for the proof. The theorem is proved. It is worth noting that both parts of this theorem are corollaries of general and rather abstract statements which rely on the theory of approximability developed by Neeman; see § 3.2 below. Now let us pass to one of the main results of [19], which is closely related to the (new) result of Corollary 1.10. We need some well-known definitions. Definition 1.8. Let $\mathcal{D}$ be a triangulated category closed with respect to coproducts. 1. An object $M$ of $\mathcal{D}$ is said to be compact (in $\mathcal{D}$) if the functor $\mathcal{D}(M,\,\cdot\,)\colon \mathcal{D}\to \operatorname{Ab}$ respects coproducts. 2. We say that a (triangulated) subcategory $\mathcal{T}$ of $\mathcal{D}$ compactly generates $\mathcal{D}$ whenever $\mathcal{T}$ is essentially small, its objects are compact in $\mathcal{D}$ and $\mathcal{D}=\mathcal{T}^{\coprod}$ (see Definition 1.1, 5). 3. We say that a triangulated category $\mathcal{T}$ is countable if the class of isomorphism classes of objects in $\mathcal{T}$ and all $\mathcal{T}$-morphism sets are countable. Proposition 1.9. Assume that $\mathcal{T}$ is countable and compactly generates $\mathcal{D}$. Then all homological functors $\mathcal{T}^{\mathrm{op}}\to \operatorname{Ab}$ are represented by objects of $\mathcal{D}$. This statement immediately follows from Proposition 4.11 and Theorem 5.1 of [19] (see the sentence following the formulation of that theorem). We also prove the following statement; we do not apply it in this paper. Corollary 1.10. Assume that $X$ is proper over $S=\operatorname{Spec} R$ (where $R$ is a Noetherian ring), $R$ is either countable or self-injective, that is, injective as a module over itself, and $H\colon (\mathcal{D}_{\mathrm{p}})^{\mathrm{op}}\to R\text{-}\mathrm{mod}$ is a homological functor. 1. Then $H$ is represented by an object of $\mathcal{D}_{\mathrm{Q}}$. 2. Assume in addition that $X$ is projective over $S$. Then $H$ is represented by an object of $\mathcal{D}^{\mathrm{u}}$. Moreover, if for each $M\in \operatorname{Obj} \mathcal{D}_{\mathrm{p}}$ there exists $c_M>0$ such that $H(M[i])=0$ whenever $i<-c_M$, then $H$ is represented by an object of $\mathcal{D}^+$. Remark 1.11. 1. The assumption that $R$ is countable comes from Proposition 1.9. On the other hand our proof of the self-injective version of part 1 originates from the proof of Theorem A.1 in [6]. 2. Note that the $\mathcal{D}^-$ and $\mathcal{D}^{\mathrm{b}}$-versions of Corollary 1.10, 2, are provided by Corollary 0.5 of [21] (see Theorem 1.7, 1); moreover, in that corollary only the properness of $X$ over $S$ is assumed. 3. An object of $\mathcal{D}$ that represents $H$ is unique up to an isomorphism according to Lemma 3.5, 1, below. Let us now describe the contents of the paper. Some further information of this type can be found at the beginnings of sections. In § 2 we give some basic definitions related (mostly) to triangulated categories and semi-orthogonal decompositions, and prove some simple (and mostly not new) statements. In § 3 we establish Theorem 1.5, Corollary 1.10 and a few other statements related to Neeman’s results. In § 4 we prove an abstract result (Theorem 4.4) on the existence of certain semi-orthogonal decompositions. We use it to prove Theorem 1.3 (see Theorem 4.12 and Remark 4.22), to deduce an easy result (Corollary 4.5) on the existence of adjoint functors (see Remark 4.8), and to study certain support subcategories of $\mathcal{D}_{\mathrm{Q}}$ (see Proposition 4.14). AcknowledgementThe author is deeply grateful to the referee for several important comments on the text. § 2. PreliminariesIn this section we discuss some simple notions related to triangulated categories and semi-orthogonal decompositions. In § 2.1 we recall some definitions and statements related to triangulated categories. In § 2.2 we define and study semi-orthogonal decompositions of length $2$. In § 2.3 we recall some basics on semi-orthogonal decompositions of arbitrary length. We need them in §§ 3.1 and 4.4 only. In § 2.4 we recall the definition of triangulated countable homotopy colimits along with a few of their simple properties. We need them only in § 3.2. 2.1. Some basic definitions and statements$\bullet$ For categories $C'$ and $C$ we write $C'\subset C$ if $C'$ is a subcategory of $C$; recall that we only consider strictly full subcategories in this paper. $\bullet$ The symbol $\mathcal{T}$ below always denotes some triangulated category. The symbols $\mathcal{D}$, $\mathcal{A}$ and $\mathcal{B}$ (possibly, endowed with indices) are also used for triangulated categories only. $\bullet$ For any $A,B,C \in \operatorname{Obj} \mathcal{T}$ we say that $C$ is an extension of $B$ by $A$ if there exists a distinguished triangle $\bullet$ Given a distinguished triangle $X\xrightarrow{f}Y\to Z$, we write $Z=\operatorname{Cone}(f)$; recall that $Z$ is determined by $f$ up to a non-canonical isomorphism. Recall that we only consider small coproducts in this paper. Lemma 2.1. Let $\mathcal{A}$ and $\mathcal{B}$ be (strictly full) triangulated subcategories of $\mathcal{T}$. Take $\mathcal{C}$ to be the class of those $M\in \operatorname{Obj} \mathcal{T}$ for which there exists a distinguished triangle $B\to M\to A\to B[1]$ with $B\in \mathcal{B}$ and $A\in \mathcal{A}$. 1. If $\mathcal{B}\perp \mathcal{A}$, then $\mathcal{C}$ gives a triangulated subcategory of $\mathcal{T}$ too. 2. If $\mathcal{T}$ is closed with respect to (small) coproducts, and $\mathcal{B}$ and $\mathcal{A}$ are closed with respect to $\mathcal{T}$-coproducts, then $\mathcal{C}$ is closed with respect to $\mathcal{T}$-coproducts too. These statements follow easily from Proposition 2.1.1, (1,2), of [10] (as concerns assertion 1, note that $\mathcal{C}[1]=\mathcal{C}$); assertion 2 also follows from Remark 1.2.2 of [20]. The following statements are simple and well known. Lemma 2.2. Assume that $\mathcal{T}$ is closed with respect to coproducts and $\mathcal{B}$ is a triangulated subcategory of $\mathcal{T}$. 1. Then $\mathcal{B}^{\perp}_{\mathcal{T}}=(\mathcal{B}^{\coprod})^{\perp}_{\mathcal{T}}$. 2. If $\mathcal{B}$ is essentially small and consists of compact objects (see Definition 1.8, 1) and $\mathcal{B}^{\perp}_{\mathcal{T}}=\{0\}$, then $\mathcal{B}$ compactly generates $\mathcal{T}$. Proof. Assertion 1 is very easy. It suffices to note that for any object $N$ of $\mathcal{T}$ the class ${}^\perp_{\mathcal{T}} \{N\}$ is closed with respect to $\mathcal{T}$-coproducts since for any family $\{M_i\}$ of objects of $\mathcal{T}$.
Assertion 2 follows from Proposition 8.4.1 of [20]. The lemma is proved. 2.2. Semi-orthogonal decompositions of length 2Let us give some other central definitions. Definition 2.3. Assume that $\mathcal{T}\subset \mathcal{D}$. 1. Let $\mathcal{B}$ and $\mathcal{A}$ be (strictly full triangulated) subcategories of $\mathcal{T}$. We say that the pair $D=(\mathcal{A},\mathcal{B})$ is a semi-orthogonal decomposition of $\mathcal{T}$ (or gives a decomposition of $\mathcal{T}$) if $\operatorname{Obj} \mathcal{B}\perp \operatorname{Obj} \mathcal{A}$ and for any $M\in \operatorname{Obj} \mathcal{T}$ there exists a distinguished triangle with $B\in \mathcal{B}$ and $A\in \mathcal{A}$.We call (2.1) a $D$-decomposition triangle (for $M$). 2. If $D_{\mathcal{D}}=(\mathcal{A}_{\mathcal{D}},\mathcal{B}_{\mathcal{D}})$ is a semi-orthogonal decomposition of $\mathcal{D}$ and
$$
\begin{equation*}
D_{\mathcal{D}}\cap\mathcal{T}=(\mathcal{A}_{\mathcal{D}}\cap \mathcal{T},\mathcal{B}_{\mathcal{D}}\cap \mathcal{T})
\end{equation*}
\notag
$$
(see Definition 1.1, 2) is a semi-orthogonal decomposition of $\mathcal{T}$, then we say that $D_{\mathcal{D}}$ restricts to $\mathcal{T}$, $D_{\mathcal{D}}\cap\mathcal{T}$ is the corresponding restriction and $D_{\mathcal{D}}$ is an extension of $D_{\mathcal{D}}\cap\mathcal{T}$ to $\mathcal{D}$. 3. If $\mathcal{T},\mathcal{T}'\subset \mathcal{D}$ and $D=(\mathcal{A},\mathcal{B})$ is a semi-orthogonal decomposition of $\mathcal{T}$, then we write $D^{\perp}_{\mathcal{T}'}$ (${}^{\perp}_{\mathcal{T}'}D$) for the pair $(\mathcal{A}^{\perp}_{\mathcal{T}'},\mathcal{B}^{\perp}_{\mathcal{T}'})$ ($({}^{\perp}_{\mathcal{T}'}\mathcal{A},{}^{\perp}_{\mathcal{T}'}\mathcal{B})$, respectively); see Definition 1.1, 4. Similarly, if the category $\mathcal{D}\supset \mathcal{T}$ is closed with respect to small coproducts and $\mathcal{T}'\subset \mathcal{D}$, then $D^{\coprod}$ ($D^{\coprod}_{\mathcal{T}'}$) denotes the pair $(\mathcal{A}^{\coprod},\mathcal{B}^{\coprod})$ ($(\mathcal{A}^{\coprod}_{\mathcal{T}'}, \mathcal{B}^{\coprod}_{\mathcal{T}'})$, respectively); see Definition 1.1, 5. Remark 2.4. Semi-orthogonal decompositions described in Definition 2.3, 1, are essentially the length $2$ ones in the sense of Definition 2.9, II.2, below. We defer decompositions of arbitrary length and their properties to § 2.3. The reason for this is that these more general decompositions do not help us in proving any new properties of (left or right) admissible subcategories. On the other hand semi-orthogonal decompositions of length $2$ are important for our proofs even though Theorem 4.12 below contains just a little more information than the corresponding statements in Theorem 1.3 (cf. Remark 4.22). Proposition 2.5. 1. Assume that $D=(\mathcal{A},\mathcal{B})$ is a semi-orthogonal decomposition of $\mathcal{T}$. Then $\mathcal{B}^\perp_{\mathcal{T}}= \mathcal{A}$ and $\mathcal{B}={}^\perp_{\mathcal{T}} \mathcal{A}$, and there exists an exact right adjoint $R_D$ to the embedding $\mathcal{B}\to \mathcal{T}$ and an exact left adjoint $L_D$ to the embedding $\mathcal{A}\to\mathcal{T}$. It is convenient to assume that $L_D$ and $R_D$ send $\mathcal{T}$ into $\mathcal{T}$ itself. Moreover, the triangle (2.1) is functorially determined by $M$, and the arrows $B\to M\to A$ in it come from the transformations corresponding to the aforementioned adjunctions. 2. The correspondence $D\mapsto \mathcal{A}$ ($D\mapsto \mathcal{B}$) induces a bijection between the class of semi-orthogonal decompositions of $\mathcal{T}$ and that of left (right, respectively) admissible subcategories of $\mathcal{T}$; see Definition 1.1, 1. 3. $D=(\mathcal{A},\mathcal{B})$ is a semi-orthogonal decomposition of $\mathcal{T}$ if and only if $D^{\mathrm{op}}=(\mathcal{B}^{\mathrm{op}},\mathcal{A}^{\mathrm{op}})$ is a semi-orthogonal decomposition of the category $\mathcal{T}^{\mathrm{op}}$. Assertions 1 and 2 are well known. They follow easily from Lemma 3.1 of [4]. Note however that one should look at the proof of that lemma, where the corresponding morphisms in (2.1) are calculated, and that the functoriality of (2.1) was stated in Lemma 2.3 of [16]. Assertion 3 is obvious. We also need the following properties of our notions. Proposition 2.6. Assume that $\mathcal{T},\mathcal{T}'\subset \mathcal{D}$, where $\mathcal{D}$ is an $R$-linear category (see Definition 1.1, 6), and let $D$ ($D'$) be a semi-orthogonal decomposition of $\mathcal{T}$ ($\mathcal{T}'$, respectively). If $D'=D^{\perp}_{\mathcal{T}'}$ (see Definition 2.3, 3), then the following bi-functors $\mathcal{T}^{\mathrm{op}}\times \mathcal{T}'\to R\text{-}\mathrm{Mod}$ are canonically isomorphic:3 and Proof. Let $T \in \mathcal{T}$ and $T' \in \mathcal{T}'$. Then there are distinguished triangles
$$
\begin{equation}
R_D(T)\to T\to L_D (T) \quad\text{and}\quad R_{D'}(T')\to T'\to L_{D'} (T')
\end{equation}
\tag{2.2}
$$
which are functorial in $T$ and $T'$, respectively: see Proposition 2.5, 1. Moreover, this proposition (in combination with Definition 2.3, 1) implies that $L_D(T)[i]\perp L_{D'}(T')$ and $R_D(T)[i]\perp R_{D'}(T')$ for any $i\in \mathbb{Z}$. Therefore, applying $\mathcal{D}(L_D(T),\,\cdot\,)$ to the second triangle in (2.2) we obtain an isomorphism
$$
\begin{equation*}
\mathcal{D}(L_D(T),R_{D'}(T'))\to\mathcal{D}(L_D(T),T'),
\end{equation*}
\notag
$$
which is functorial in $T$ and $T'$. Similarly, applying the functor $\mathcal{D}(\,\cdot\,,R_{D'}(T'))$ to the first triangle in (2.2) we obtain an isomorphism which is functorial in $T$ and $T'$. Taking the composition of these two isomorphisms we obtain an isomorphism $\mathcal{D}(L_D(\,\cdot\,),\,\cdot\,)\cong \mathcal{D}(\,\cdot\,,R_{D'}(\,\cdot\,))$ of bi-functors.
Analogously, to obtain the second isomorphism in question one takes the composition of the isomorphisms $\mathcal{D}(R_D(T),T')\to \mathcal{D}(R_D(T),L_{D'}(T'))$ and $\mathcal{D}(R_D(T), L_{D'}(T'))\to \mathcal{D}(T,L_{D'}(T'))$ that are constructed similarly to the isomorphisms above. The proposition is proved. Let us now introduce an order on semi-orthogonal decompositions. Definition 2.7. We write $D_1\leqslant_{R} D_2$ if the $D_i=(\mathcal{A}_i,\mathcal{B}_i)$, $i=1,2$, are semi-orthogonal decompositions of $\mathcal{T}$ and $\mathcal{B}_1\subset \mathcal{B}_2$. Proposition 2.8. Assume that $D_i=(\mathcal{A}_i,\mathcal{B}_i)$ for $i=1,2$ are semi-orthogonal decompositions of $\mathcal{T}$. I. Then the following results hold.
II. Assume that $\mathcal{T},\mathcal{T}'\subset \mathcal{D}$ and $D'_i=(\mathcal{A}'_i,\mathcal{B}'_i)$, $i=1,2$, are decompositions of $\mathcal{T}'$.
Proof. I. This is obvious from our definitions along with Proposition 2.5, 1 and 3.
II.1. Obviously, $\mathcal{B}_1'\subset \mathcal{B}_2'$ and $\mathcal{A}'_1\subset \mathcal{A}'_2$. Applying assertion I we conclude that $D'_1=D'_2$ indeed. 2. If $D_1\leqslant_{R} D_2$, then, clearly, $\mathcal{A}_1'\subset \mathcal{A}_2'$. Thus $D'_2\leqslant_{R} D'_1$ according to assertion I.1. Next we must prove the following: if $\mathcal{T}\subset \mathcal{T}'$ and $\mathcal{B}_2'\subset \mathcal{B}'_1$, then $D_1\leqslant_{R} D_2$. By assertion I.1 the latter condition is fulfilled if (and only if) $\mathcal{A}_2\subset \mathcal{A}_1$. Now, $\mathcal{B}_i'=(\mathcal{B}_i)^{\perp}_{\mathcal{D}}\cap \mathcal{T}'$ and $\mathcal{A}_i=(\mathcal{B}_i)^{\perp}_{\mathcal{D}}\cap \mathcal{T}$ (for $i=1,2$; see Proposition 2.5, 1, again); hence $\mathcal{A}_i=\mathcal{B}_i'\cap \mathcal{T}$. Thus, $\mathcal{A}_2\subset \mathcal{A}_1$ indeed. The proposition is proved. 2.3. Semi-orthogonal decompositions of arbitrary lengthNow we pass to semi-orthogonal decompositions of arbitrary length. These are closely related to certain chains of subcategories. Until the end of this section we assume that $n$ is a positive integer. Definition 2.9. I. Assume that $C=(C_{<i})$, $1\leqslant i\leqslant n$, is a family of subcategories of $\mathcal{T}$. 1. We say that $C$ is a length $n$ left admissible chain (cf. Definition 4.1 of [5]) in $\mathcal{T}$ if all the $C_{<i}$ are left admissible in $\mathcal{T}$ and $C_{<i}\subset C_{<j}$ whenever $0< i<j\leqslant n$. 2. For $0\leqslant i\leqslant n$ we set $C_{\geqslant i}={}^{\perp}_{\mathcal{T}}(C_{<i})$ and $\mathcal{D}ec(C)_i=C_{<i+1}{} \cap C_{\geqslant i}$; here we set $C_{<0}=\{0\}$ and $C_{<n+1}=\mathcal{T}$. Correspondingly, we set
$$
\begin{equation*}
\mathcal{D}ec(C)=(\mathcal{D}ec(C)_i,\ 0\leqslant i\leqslant n).
\end{equation*}
\notag
$$
II. Assume that $D=(D_i)$, $0\leqslant i\leqslant n$, is a system of (strictly full) triangulated subcategories of $\mathcal{T}$. 1. Then for any $j$, $0\leqslant j\leqslant n+1$, we write $D_{< j}$ ($D_{\geqslant j}$) for the smallest (strictly full) triangulated subcategory of $\mathcal{T}$ that contains $D_i$ for all $i< j$ ($i\geqslant j$, respectively).4 We set $D_{<}$ to be the family $(D_{< j})$, $1\leqslant i\leqslant n$. 2. We say that the family $D$ gives a length $n+1$ semi-orthogonal decomposition of $\mathcal{T}$ (or just a decomposition of $\mathcal{T}$) if $D_j\perp D_i$ whenever $0\leqslant i<j\leqslant n$, and $D_{< n+1}=\mathcal{T}$. Now let us relate left admissible chains to semi-orthogonal decompositions. Proposition 2.10. Assume that $D=(D_i)$ is a length $n+1$ semi-orthogonal decomposition of $\mathcal{T}$ and $C$ is a length $n$ left admissible chain in $\mathcal{T}$. 1. Then for any $j$, $1\leqslant j\leqslant n$, the pair $(D_{<j},D_j)$ gives a semi-orthogonal decomposition of $D_{<j+1}$ in the sense of Definition 2.3, 1, and $(D_{< j}, D_{\geqslant j})$ is a decomposition of $\mathcal{T}$ (in the same sense). 2. The maps $D\mapsto D_{<}$ and $C\mapsto \mathcal{D}ec(C)$ induce mutually inverse bijections between the class of length $n+1$ semi-orthogonal decompositions of $\mathcal{T}$ and the class of length $n$ left admissible chains in $\mathcal{T}$. Moreover, for any $j$, $0\leqslant j\leqslant n$, we have $\mathcal{D}ec(C)_{\geqslant j}=C_{\geqslant j}$. Proof. 1. Clearly, $D_j\perp D_{<j}$ (see the easy result of Lemma 2.1 of [16]). The existence of decompositions of the type (2.1) for all objects of $D_{<j+1}$ follows easily from Lemma 2.1, 1. Indeed, the class $G$ of those $M\in \operatorname{Obj} D_{<j+1}$ for which there exists a distinguished triangle $B\to M\to A\to B[1]$ with $B\in D_j$ and $A\in D_{<j}$ gives a triangulated subcategory of $\mathcal{T}$, which clearly contains the $D_i$ for $0\leqslant i\leqslant j$; hence $G=\operatorname{Obj} D_{<j+1}$ indeed.
The proof of the fact that $(D_{< j}, D_{\geqslant j})$ is a decomposition of $\mathcal{T}$ is similar. It is easily seen that $D_{\geqslant j}\perp D_{< j}$, and the class $G'$ of those $M\in \operatorname{Obj} \mathcal{T}$ for which there exists a distinguished triangle $B\to M\to A\to B[1]$ with $B\in D_{\geqslant j}$ and $A\in D_{<j}$ forms a triangulated subcategory of $\mathcal{T}$, which contains the $D_i$ for $0\leqslant i\leqslant n$. 2. Applying assertion 1 along with Proposition 2.5, 1, we obtain that $D_{<i}$ is left admissible in $D_{< i+1}$ for $1\leqslant i\leqslant n$. Since $D_{<n+1}=\mathcal{T}$, we obtain that all the $D_{<i}$ are left admissible in $\mathcal{T}$. Thus, $D_<$ is a (length $n$) left admissible chain in $\mathcal{T}$ indeed. Conversely, all the $\mathcal{D}ec(C)_i$ are clearly triangulated subcategories of $\mathcal{T}$ and $\mathcal{D}ec(C)_j\perp \mathcal{D}ec(C)_i$ whenever $0\leqslant i<j\leqslant n$. Next, simple induction, in combination with assertion 1, yields easily that $(\mathcal{D}ec(C))_{< i+1}=C_{<i+1}$ for any $i$, $0\leqslant i \leqslant n$. In particular, $(\mathcal{D}ec(C))_{< n+1}=\mathcal{T}$. Consequently, $\mathcal{D}ec(C)$ is a length $n+1$ semi-orthogonal decomposition of $\mathcal{T}$ and $(\mathcal{D}ec(C))_<=C$. Lastly, take $D'=\mathcal{D}ec(D_<)$. Clearly, $D'_i$ contains $D_i$ for any $i$, $0\leqslant i\leqslant n$. Applying assertion 1 along with downward induction on $i$ we easily deduce that $D'_i=D_i$; see Proposition 2.8, I. The proposition is proved. Remark 2.11. Our proposition is closely related to (the categorical dual of) Proposition 4.4 of [5]. 2.4. A reminder on countable homotopy colimitsNow we recall a few basic properties of triangulated homotopy colimits (as introduced in [3]). We apply them in § 3.2 below. We assume that $\mathcal{D}$ is a triangulated category closed with respect to countable coproducts. Definition 2.12. For a sequence of objects $Y_i$ of $\mathcal{D}$ for $i\geqslant 0$ and maps $f_i\colon {Y_{i}\!\to\! Y_{i+1}}$ we set where $f$ is defined on $Y_i$ as the composition $Y_i\xrightarrow{f_i}Y_{i+1}{\hookrightarrow}\coprod Y_i$. We call $\operatorname{\underrightarrow{\mathrm{hocolim}}} Y_i$ a homotopy colimit of $Y_i$. Remark 2.13. 1. Note that these homotopy colimits are not really canonical since the choice of a cone is not canonical. They are only defined up to noncanonical isomorphisms. 2. Our definition gives a canonical morphism $\coprod Y_i\to \operatorname{\underrightarrow{\mathrm{hocolim}}} Y_i$; correspondingly, we also have canonical morphisms $Y_i\to\operatorname{\underrightarrow{\mathrm{hocolim}}} Y_i$. Lemma 2.14. Assume that $Y=\operatorname{\underrightarrow{\mathrm{hocolim}}} Y_i$ (in $\mathcal{D}$). Then the following hold. 1. A homotopy colimit of $Y_{i_j}$ is isomorphic to $Y$ for each subsequence $Y_{i_j}$ of $Y_i$. In particular, any (finite) number of first terms in $(Y_i)$ can be discarded. 2. Denote by $c_i$ the canonical morphisms $Y_i\to Y$ mentioned in Remark 2.13, 2. Let $H\colon \mathcal{D}\to R\text{-}\mathrm{Mod}$, where $R$ is a ring, be a homological functor (see Definition 1.1, 9); assume that $H$ respects countable coproducts. Then the morphisms $H(c_i)$ yield an isomorphism $\varinjlim H(Y_i)\to H(Y)$. Assertion 1 is Lemma 1.7.1 in [20]. Assertion 2 is a particular case of Lemma 13.33.8 (tag 0CRK) in [30] (cf. also Lemma 2.8 in [18]). § 3. On Theorem 1.5 and approximabilityIn § 3.1 we prove Theorem 1.5, 2. In § 3.2 we prove an abstract result (Theorem 3.8) closely related to [21] (cf. Theorem 1.7, 1). We use it to prove Theorem 1.5, 1; next we prove Corollary 1.10. 3.1. The proof of Theorem 1.5, 2We recall some statements from [30] that allow us to apply the results there to various categories of quasicoherent sheaves. Remark 3.1. The ‘main’ derived categories in [30] are the derived categories of $\mathcal{O}_X$-modules. However, $D(\mathcal{O}_X)$ contains a full triangulated subcategory $D_{\operatorname{Qcoh}}(\mathcal{O}_X)$ consisting of those complexes whose cohomology is quasi-coherent. Now, if $X$ is Noetherian, then the obvious functor $\mathcal{D}_{\mathrm{Q}}=D({\operatorname{Qcoh}}(X))\to D_{\operatorname{Qcoh}}(\mathcal{O}_X)$ is an equivalence: see [30], Proposition 36.8.3 (tag 09T4). It clearly follows that $D_{\operatorname{coh}}({\operatorname{Qcoh}}(X))\cong D_{\operatorname{coh}} (\mathcal{O}_X)$ (cf. Definition 1.1, 7, or § 13.17 (tag 06UP) in [30] for categorical notation of this sort). Moreover, the direct and inverse image functors (that is, $f_*\colon D(\mkern-1mu\mathcal{O}_X\mkern-1.5mu) \!\leftrightarrows\! D(\mkern-1mu\mathcal{O}_Y\mkern-1.5mu) : \! f^* $ for a quasi-separated and quasi-compact morphism $f\colon X\to Y$ of schemes) and tensor products respect the subcategories $D_{\operatorname{Qcoh}}(\mathcal{O})$ of $D(\mathcal{O})$; see [30], Lemmas 36.3.8, 36.4.1 and 36.3.9 (tags 08DW, 08D5 and 08DX). These observations allow us to apply results of [30] to the categories $D(\operatorname{Qcoh}(\,\cdot\,))$ and their subcategories mentioned in Definition 1.1, 7, instead of $D_{\operatorname{Qcoh}}(\mathcal{O})\subset D(\mathcal{O})$ and the corresponding triangulated subcategories that are defined in terms of the cohomology of complexes of sheaves of modules (similarly to Definition 1.1, 7). Now we prove an implication in Theorem 1.5, 2, in a slightly more general case. Lemma 3.2. Assume that $X$ is a proper scheme over $S\,{=}\,\operatorname{Spec} R$ and ${M\,{\in}\, \operatorname{Obj} \mathcal{D}_{\mathrm{p}}}$. Then the following hold. 1. For any $N\in \operatorname{Obj} \mathcal{D}^{\mathrm{u}}$ we have $\mathcal{D}_{\mathrm{Q}}(M,N)\in R\text{-}\mathrm{mod}$. 2. $\mathcal{D}_{\mathrm{Q}}(M[i],N)=\{0\}$ whenever $N$ is an object of $\mathcal{D}^-$ ($\mathcal{D}^+$) and $i$ is sufficiently small (large, respectively). Proof. 1. The spectral sequence argument in the proof of Lemma 36.18.2 (tag 09M4) in [30], Ch. 36, reduces the statement to the case when $N$ is just a coherent sheaf. In this case the statement easily follows from Lemma 36.11.7 (tag 0D0D) in [30], Ch. 36 (if one also applies Remark 3.1 to compare the corresponding categories).
2. This is immediate from Remark 3.1 combined with Lemma 36.18.2 (tag 09M4) of [30], Ch. 36. The lemma is proved. Remark 3.3. 1. $N\in \operatorname{Obj} \mathcal{D}_{\mathrm{Q}}$ belongs to $\mathcal{D}^{\mathrm{b}}$ if and only if it belongs to both $\mathcal{D}^-$ and $\mathcal{D}^+$. Hence if $M\in \operatorname{Obj} \mathcal{D}_{\mathrm{p}}$ and $N\in \mathcal{D}^{\mathrm{b}}$, then $\mathcal{D}_{\mathrm{Q}}(M[i],N)=\{0\}$ whenever $i$ is either sufficiently small or sufficiently large. 2. Similarly, to verify the converse implication in Theorem 1.5, 2, it suffices to establish the criteria corresponding to the conditions $N\in \operatorname{Obj} \mathcal{D}^-$ and $N\in \operatorname{Obj} \mathcal{D}^+$; see below. Proof of Theorem 1.5, 2. We must prove that an object $N$ of $\mathcal{D}_{\mathrm{Q}}$ belongs to $\mathcal{D}^{\mathrm{u}}$ if $\mathcal{D}_{\mathrm{Q}}(M,N)\in R\text{-}\mathrm{mod}$ for all $M\in \operatorname{Obj} \mathcal{D}_{\mathrm{p}}$. Moreover, for $N\in \mathcal{D}^{\mathrm{u}}$ we have $N\in \operatorname{Obj} \mathcal{D}^-$ ($N\in \operatorname{Obj} \mathcal{D}^+$) if and only if $\{M[i]\}\perp \{N\}$ for $i\ll 0$ ($i\gg 0$, respectively) and (any fixed) $M\in \operatorname{Obj} \mathcal{D}_{\mathrm{p}}$. We denote the projection $X\to S$ by $p$.
We argue similarly to the proof of Theorem A.1 in [6]. Since $X$ is projective over $S$, it is a closed subscheme of the projectivization $Y$ of a vector bundle over $S$. Recall that an object $N$ of $\mathcal{D}_{\mathrm{Q}}$ belongs to $\mathcal{D}^{\mathrm{u}}$ if and only if all of its (quasi-coherent) cohomology sheaves $H^i(N)$ are coherent. Moreover, $N$ belongs to $\mathcal{D}^-$ ($\mathcal{D}^+$) whenever we also have $H^i(N)=0$ for $i\gg 0$ ($i\ll 0$, respectively). Let us reduce the statements in question to the case $X\!=\!Y$. For any ${M\mkern-1mu\!\in\! D_{\mathrm{perf}}(Y)}$ we have $\mathcal{D}_{\mathrm{Q}}(Li^*M,N)\cong D(\operatorname{Qcoh}(Y))(M, i_*N)$, where $i$ is the embedding $X\to Y$. Since $Li^*M\in\mathcal{D}_{\mathrm{p}}$, the functor represented by the object $i_*N$ fulfils the corresponding assumptions. Thus it remains to note that $N$ belongs to $\mathcal{D}^{\mathrm{u}}$ ($\mathcal{D}^+$, $\mathcal{D}^-$) if and only if $i_*N$ belongs to $D_{\mathrm{coh}}(\operatorname{Qcoh}(Y))$ ($D^+_{\mathrm{coh}}(\operatorname{Qcoh}(Y))$, $D^-_{\mathrm{coh}}(\operatorname{Qcoh}(Y))$, respectively); see [30], Lemmas 29.4.1 and 30.9.8 (tags 01QY and 087T) along with Remark 3.1. Now we assume that $X=Y$ and $X$ is of dimension $d\geqslant 0$ over $S$. We use Theorem 6.7 of [2]. It gives fully faithful functors
$$
\begin{equation*}
\Phi_j\colon D(R)\to \mathcal{D}_{\mathrm{Q}} \quad \text{and}\quad F\mapsto p^*F(j)=p^*F\otimes \mathcal{O}(j)
\end{equation*}
\notag
$$
for $j\in \mathbb{Z}$ (see [30], Definition 27.10.1 (tag 01MN)); here we identify $D(\operatorname{Qcoh}(S))$ with $D(R)$ (see Corollary 3.3.5 in [6]). Moreover, it gives us a length $d+1$ semi-orthogonal decomposition $D$ of $\mathcal{D}_{\mathrm{Q}}$ whose components $D_j$ are the essential images $\operatorname{Im} \Phi_j$ for $0\leqslant j\leqslant d$; see Definition 2.9, II.2 (or Definition 5.3 in [2]).
Let us prove by induction on $m$, $-1\leqslant m\leqslant d$, that $N$ belongs to $\mathcal{D}^{\mathrm{u}}$ (respectively, to $\mathcal{D}^+$ or $\mathcal{D}^-$) if we assume in addition that $N$ belongs to $D_{<m+1}$; recall that $D_{<m+1}$ is the smallest triangulated subcategory of $\mathcal{D}_{\mathrm{Q}}$ that contains $D_j$ for $0\leqslant j\leqslant m$, and $D_{<d+1}=\mathcal{D}_{\mathrm{Q}}$. Our statement is vacuous if $m=-1$. Suppose that the inductive assumption is fulfilled for $m=m_0-1$ (where $0\leqslant m_0\leqslant d$) and $N\in \operatorname{Obj} D_{< m_0+1}$. Now the subcategories $D_{< m_0}$ and $D_{m_0}$ give a semi-orthogonal decomposition of $ D_{< m_0+1}$; see Proposition 2.10, 1 (or Definition 2.2 in [16]). Hence there exists a distinguished triangle with $N'\in D_{m_0}$ and $N''\in \operatorname{Obj} D_{< m_0}$, and $N'\cong \Phi_{m_0}\circ \Phi_{m_0}^! (N)$; here $\Phi_{m_0}^!$ is the right adjoint to the functor $\Phi_{m_0}\colon D(R)\to D_{< m_0+1}$. Now, the cohomology of the complex $\Phi_{m_0}^! (N)$ is given by $\mathcal{D}_{\mathrm{Q}}(p^*R(m_0),N[i])\cong \mathcal{D}_{\mathrm{Q}}(\mathcal{O}_X(m_0),N[i])$ for $i\in \mathbb{Z}$ (here $R\in D(R)\cong D(\operatorname{Qcoh}(S))$ is the complex whose only nonzero term is $R$ in degree $0$). Since $\mathcal{O}_X(m_0)$ is a line bundle on $X$ (thus it is a perfect complex), $M=\Phi_{m_0}^! (N)$ belongs to $ D_{\mathrm{coh}}(\operatorname{Qcoh}(S))\subset D(R\text{-}\mathrm{Mod})$ (to $D^+_{\mathrm{coh}}(\operatorname{Qcoh}(S))$ or $D^-_{\mathrm{coh}}(\operatorname{Qcoh}(S))$, respectively). Thus, the object $\Phi_{m_0}(M)$ (equal to $N'$) belongs to $\mathcal{D}^{\mathrm{u}}$ (to $\mathcal{D}^+$, $\mathcal{D}^-$, respectively); note that the functor $\Phi_{m_0}$ sends $D_{\mathrm{coh}}(\operatorname{Qcoh}(S))$ ($D^+_{\mathrm{coh}}(\operatorname{Qcoh}(S))$ or $D^-_{\mathrm{coh}}(\operatorname{Qcoh}(S))$) into $\mathcal{D}^{\mathrm{u}}$ ($\mathcal{D}^+$ or $\mathcal{D}^-$, respectively); cf. Proposition 3.5 in [2]. Moreover, applying (3.1) to the functors corepresented by objects of $\mathcal{D}_{\mathrm{p}}$ we obtain that $\mathcal{D}_{\mathrm{Q}}(M,N'')$ belongs to $R\text{-}\mathrm{mod}$ for any $M\in \operatorname{Obj} \mathcal{D}_{\mathrm{p}}$ (and we also have $\mathcal{D}_{\mathrm{Q}}(M[-i],N'')=\{0\}$ for $i\ll 0$, for the $\mathcal{D}^+$-version of the argument, whereas $\mathcal{D}_{\mathrm{Q}}(M[-i],N'')=\{0\}$ for $i\gg 0$, for the $\mathcal{D}^-$-version). Applying the inductive assumption we deduce that $N''$ is an object of $ \mathcal{D}^{\mathrm{u}}$ ($\mathcal{D}^+$ or $\mathcal{D}^-$, respectively) too; hence the same is valid for $N$ itself.Theorem 1.5, 2 is proved. 3.2. Neeman’s approximability, Theorem 1.5, 1, and Corollary 1.10Let us prove some rather technical (and general) results that yield Theorem 1.5, 1. We need some definitions and notation used in [21]; our arguments are also closely related to [21]. We start from a particular case of [21], Definition 5.1 (see also Remark 5.6 ibidem). Definition 3.4. Assume that $\mathcal{T}$ is $R$-linear (recall that $R$ is a commutative unital ring) and compactly generated (see Definition 1.8, 2) by its triangulated subcategory $\mathcal{T}^{\mathrm{c}}$. 1. Let us denote the corresponding Yoneda map $\mathcal{T}\to\operatorname{Fun}_R((\mathcal{T}^{\mathrm{c}})^{\mathrm{op}}, R\text{-}\mathrm{Mod})$ by $\mathcal{Y}$, that is, for $N\in \mathcal{T}$ we set
$$
\begin{equation*}
\mathcal{Y}(N)=\mathcal{T}(\,\cdot\,,N)\big{|}_{\mathcal{T}^{\mathrm c}}.
\end{equation*}
\notag
$$
2. We call an (infinite) chain of morphisms $E_0\to E_1\to \dotsb$ an approximating system for an object $F$ of $\mathcal{T}$ whenever 3. If an approximating system for $F$ exists, then we say that $F$ is $\mathcal{T}^{\mathrm{c}}$-approximable. We write $\mathcal{T}_{\mathrm{a}}$ for the full subcategory of $\mathcal{T}^{\mathrm{c}}$-approximable objects of $\mathcal{T}$. Note that this subcategory is obviously strict. 4. Let $\mathcal{T}'$ be a subcategory of $\mathcal{T}$ such that the restriction of $\mathcal{Y}$ to $\mathcal{T}'$ is full. Then we say that this restriction is conservative whenever for any $F,F'\in \operatorname{Obj} \mathcal{T}'$ such that $\mathcal{Y}(F)\cong \mathcal{Y}(F')$ we have $F\cong F'$.5 Lemma 3.5. Assume that $\mathcal{T}$ is ($R$-linear and) compactly generated by a triangulated subcategory $\mathcal{T}^{\mathrm{c}}$, and let $F\in \operatorname{Obj} \mathcal{T}$. 1. If $F$ is $\mathcal{T}^{\mathrm{c}}$-approximable then for each $G\in \operatorname{Obj} \mathcal{T}$ any $\operatorname{Fun}_R((\mathcal{T}^{\mathrm{c}})^{\mathrm{op}}, R\text{-}\mathrm{Mod})$-morphism $\varphi\colon \mathcal{Y}(F)\to \mathcal{Y}(G)$ equals $\mathcal{Y}(h)$ for some $h\in \mathcal{T}(F,G)$. Consequently, the restriction of the functor $\mathcal{Y}$ to the subcategory $\mathcal{T}_{\mathrm{a}}\subset \mathcal{T}$ is full. Moreover, this restriction is conservative (see Definition 3.4, 4). 2. $E_0\to E_1\to\dotsb$ (for some $E_i\in \mathcal{T}^{\mathrm{c}}$) is an approximating system for $F$ if and only if $F\cong \operatorname{\underrightarrow{\mathrm{hocolim}}} E_i$ (see Definition 2.12). Proof. 1. The existence of $h$ such that $\varphi=\mathcal{Y}(h)$ is ensured by Lemma 5.8 in [21]; see Remark 5.7 of [21]. We obtain the fullness of the restriction of $\mathcal{Y}$ to $\mathcal{T}_{\mathrm{a}}$ as a particular case of this statement.
Next, fullness implies that any $\mathcal{T}_{\mathrm{a}}$-isomorphism $\mathcal{Y}(F)\to \mathcal{Y}(F')$ is of the form $\mathcal{Y}(h')$ for some $h'\in \mathcal{T}(F,F')$. Since the category $\mathcal{T}^{\mathrm{c}}$ is stable with respect to shifts, $\mathcal{Y}(h'[i])$ is an isomorphism for any $i\in \mathbb{Z}$. Since $\mathcal{Y}$ is a homological functor, it follows that $\mathcal{Y}(\operatorname{Cone}(h'))=0$. Hence Lemma 2.2, 1, implies that $\operatorname{Cone}(h')=0$, and we obtain that $h'$ is an isomorphism indeed. 2. Since the functor $\mathcal{Y}$ respects coproducts, it sends $\operatorname{\underrightarrow{\mathrm{hocolim}}} E_i$ into $\varinjlim \mathcal{Y}(E_i)$: see Lemma 2.14, 2. Thus, $(E_i)$ is an approximating system for $F$ if and only if $\mathcal{Y}(F)\cong \mathcal{Y}(\operatorname{\underrightarrow{\mathrm{hocolim}}} E_i)$. Lastly, the previous assertion says that the latter condition is equivalent to $F\cong \operatorname{\underrightarrow{\mathrm{hocolim}}} E_i$. The lemma is proved. We also recall the following definition (this is Definition 1.3.1 of [1]). Definition 3.6. A pair $\mathcal{T}^{\geqslant 0},\mathcal{T}^{\leqslant 0}\subset \mathcal{T}$ of full subcategories gives a $t$-structure on $\mathcal{T}$ whenever the following properties are fulfilled: Moreover, for $n\in \mathbb{Z}$ one considers the following categories: $\mathcal{T}^{\geqslant n}=\mathcal{T}^{\geqslant 0}[-n]$ and $\mathcal{T}^{\leqslant n}=\mathcal{T}^{\leqslant 0}[-n]$. We need the following property of $t$-structures. Lemma 3.7. For any $n\in \mathbb{Z}$ there exists a right adjoint to the embedding ${\mathcal{T}^{\geqslant n}\,{\to}\, \mathcal{T}}$. Moreover, for any $X\in \mathcal{T}$ we write $X^{\leqslant n}\to X$ for the corresponding counit morphism; in this notation, $\operatorname{Cone}(X^{\leqslant n}\to X)\in \mathcal{T}^{\geqslant n+1}$. The lemma follows immediately from Proposition 1.3.3 in [1]. Theorem 3.8. Assume that $\mathcal{T}$ is ($R$-linear and) compactly generated by its triangulated subcategory $\mathcal{T}^{\mathrm{c}}$; $F\in \mathcal{T}$. I. Suppose that there exists a chain of $\mathcal{T}$-morphisms $F_0'\to F'_1\to \dotsb$, and the $F'_i$ are equipped with compatible morphisms $t_i$ into $F$ that yield an isomorphism $\varinjlim \mathcal{Y}(F'_i)\cong \mathcal{Y}(F)$ (cf. Definition 3.4, 2). Moreover, assume that there exist morphisms $c_i\colon E_i'\to F_i'$ such that $E'_i\in \operatorname{Obj} \mathcal{T}^{\mathrm{c}}$ and for each $T\in \operatorname{Obj} \mathcal{T}^{\mathrm{c}}$ there exists $N_T\geqslant 0$ such that $\{T\}\perp\{\operatorname{Cone}(c_i)\}$ for all $i>N_T$. Then one can choose a subsequence $E_i$ of $E'_i$, along with some connecting morphisms between them, that give an approximating system for $F$. II. Assume that $\mathcal{T}$ is generated by a single object $G\in \operatorname{Obj} \mathcal{T}^{\mathrm{c}}$, that is,
$$
\begin{equation*}
\{G[i],\ i\in\mathbb{Z}\}^{\perp}_{\mathcal{T}}=\{0\}.
\end{equation*}
\notag
$$
Then $F$ is $\mathcal{T}^{\mathrm{c}}$-approximable whenever one of the following assumptions is fulfilled. 1. There exist $E_i'\in \operatorname{Obj} \mathcal{T}^{\mathrm{c}}$, $F_i'\in \operatorname{Obj} \mathcal{T}$, $c_i\colon E_i'\to F_i'$, a chain of $\mathcal{T}$-morphisms $F_0'\to F'_1\to \dotsb$, and compatible $t_i\colon F_i'\to F$ such that
$$
\begin{equation*}
\{G[s],\ -j\leqslant s\leqslant j\}\perp \{\operatorname{Cone}(c_j),\operatorname{Cone}(t_j)\}
\end{equation*}
\notag
$$
for all $j\geqslant 0$. 2. There exist a $t$-structure on $\mathcal{T}$ and $N\!\in\! \mathbb{Z}$ such that $G\!\in\! \mathcal{T}^{\leqslant N}$ and ${\{G\}\!\perp\! \mathcal{T}^{\leqslant -N}}$, and for each $i\geqslant 0$ there exists a morphism $c'_i\colon \mathcal{E}_i\to F^{\leqslant i}$ (see Lemma 3.7 for this notation) such that $\mathcal{E}_i\in \operatorname{Obj}\mathcal{T}^{\mathrm{c}}$ and $\operatorname{Cone}(c'_i)\in \mathcal{T}^{\leqslant -i}$. Proof. I. Let us prove that for each $j\geqslant 0$ there exists $l>j$, along with a morphism $E'_j\to E'_l$, such that the square is commutative. Now, we have an exact sequence hence we can take any $l>\max(N_{{E'_j}},j)$.
Next, we apply this argument repeatedly, starting from $n_0=0$, to obtain an infinite commutative diagram We set $E_i=E'_{n_i}$.Lastly, for any object $T$ of $\mathcal{T}^{\mathrm{c}}$ we have
$$
\begin{equation*}
\varinjlim_{i\geqslant 0} \mathcal{T}(T,E_i) \cong \varinjlim_{i>N_{T\oplus T[1]}} \mathcal{T}(T,E_i)\cong \varinjlim_{i>N_{T\oplus T[1]}} \mathcal{T}(T,F'_{n_i})\cong \mathcal{T}(T,F).
\end{equation*}
\notag
$$
Hence the $E_i$ form an approximating system for $F$.
II.1. According to assertion I, it suffices to verify that $\varinjlim \mathcal{Y}(F'_i)\cong \mathcal{Y}(F)$ and for any object $T$ of $\mathcal{T}^{\mathrm{c}}$ there exists $N_T\geqslant 0$ such that $\{T\}\perp\{\operatorname{Cone}(c_i),\ i>N_T\}$. To verify the first statement we must check that for any object $T$ of $\mathcal{T}^{\mathrm{c}}$ there exists $N'_T\geqslant 0$ such that $\mathcal{T}(T,t_i)$ is an isomorphism for all $i>N'_T$. The latter condition is clearly valid whenever $\{T, T[1]\}\perp \{\operatorname{Cone}(t_i),\ i>N'_T\}$. Now we fix $T\!\in\! \mathcal{T}^{\mathrm{c}}$. Recall that there exists $N\!>\!0$ such that $T$ is a direct summand of some $T'$ that belongs to the extension closure of the set ${\{G[i]\colon -N\leqslant i\leqslant N\}}$, that is, to the smallest class $EC$ of objects of $\mathcal{T}$ that contains $\{G[i]\colon -N\leqslant i\leqslant N\} \cup\{0\}$ and also all extensions of (pairs of) its elements; see Example 0.13 and Remark 0.15 in [21] or Proposition 4.4.1 in [20]. Since $\{G[s],\ -j\leqslant s\leqslant j\}\perp \{\operatorname{Cone}(c_j),\operatorname{Cone}(t_j)\}$, it follows that one can take $N_T=N'_T=N+1$. 2. We apply the previous assertion to $c_i=c'_{i+N}$ and $t_i$ being the counit morphisms $F^{\leqslant N+i}\to F$ (see Lemma 3.7); correspondingly, we set $F'_i=F^{\leqslant N+i}$ and $E'_i=\mathcal{E}_{i+N}$. So we must prove that $\{G[i],\ -j\leqslant i\leqslant j\}\perp \{\operatorname{Cone}(c'_{j+N}),\operatorname{Cone}(t_j)\}$ for all $j\geqslant 0$. By our assumptions $\{G[i],\ -j\leqslant i\leqslant j\}\subset \mathcal{T}^{\leqslant N+j}\cap {}^{\perp}_{\mathcal{T}} \mathcal{T}^{\leqslant -N-j}$ for each $j\geqslant 0$: see (axiom (ii) in) Definition 3.6; also recall that $\operatorname{Cone}(c_j)=\operatorname{Cone}(c'_{j+N}) \in \mathcal{T}^{\leqslant -j-N}$. According to Lemma 3.7, we have $\operatorname{Cone}(t_j)\in \mathcal{T}^{\geqslant j+N+2}$. Thus the statement in question holds. The theorem is proved. Corollary 3.9. Assume that $X$ is proper over $S=\operatorname{Spec} R$. 1. Then $\mathcal{D}_{\mathrm{Q}}$ is generated by an object $G$ of $\mathcal{D}_{\mathrm{p}}$, and $\mathcal{D}_{\mathrm{p}}$ is the subcategory of compact objects of $\mathcal{D}_{\mathrm{Q}}$. 2. Set $\mathcal{T}=\mathcal{D}_{\mathrm{Q}}$ and $\mathcal{T}^{\mathrm{c}}=\mathcal{D}_{\mathrm{p}}$. Then $\mathcal{D}^{\mathrm{u}}\subset \mathcal{T}_{\mathrm{a}}$. 3. The restriction $\mathcal{Y}^{\mathrm{u}}\colon \mathcal{D}^{\mathrm{u}}\to \operatorname{Fun}_R((\mathcal{D}_{\mathrm{p}})^{\mathrm{op}}, R\text{-}\mathrm{Mod})$ of the corresponding functor $\mathcal{Y}$ to $\mathcal{D}^{\mathrm{u}}$ (note that for $N\in \mathcal{D}_{\mathrm{Q}}$ we set $\mathcal{Y}(N)$ to be the restriction of $\mathcal{D}_{\mathrm{Q}}(\,\cdot\,,N)$ to $\mathcal{D}_{\mathrm{p}}$) is full. Moreover, if $\mathcal{Y}(N')\cong \mathcal{Y}(N)$ for $N\in \operatorname{Obj} \mathcal{D}^{\mathrm{u}}$ and $N'\in \operatorname{Obj} \mathcal{D}_{\mathrm{Q}}$, then $N'\cong N$ (cf. Definition 3.4, 4). Proof. 1. This is Theorem 3.1.1 of [6].
2. We take the standard $t$-structure on $\mathcal{D}_{\mathrm{Q}}$ (in the terminology of [21]; cf. Example 1.3.2, (i), in [1]), and choose a compact generator $G$ of $\mathcal{D}_{\mathrm{Q}}$ provided by the previous assertion. Then there exists $N\in \mathbb{Z}$ such that $G\in \mathcal{T}^{\leqslant N}$ and $\{G\}\perp \mathcal{T}^{\leqslant -N}$; see Example 3.4 in [21]. According to Theorem 3.8, II.2, it remains to verify that for any $F\in \operatorname{Obj} \mathcal{D}^{\mathrm{u}}$ and $i\geqslant 0$ there exists a choice of $c'_i\colon \mathcal{E}_i\to F^{\leqslant i}$ such that $\mathcal{E}_i\in \operatorname{Obj}\mathcal{T}^{\mathrm{c}}$ and ${\operatorname{Cone}(c'_i)\in \mathcal{T}^{\leqslant -i}}$. Next, recall that $F^{\leqslant i}$ belongs to $\mathcal{D}^-$ for each $i\in \mathbb{Z}$; cf. Example 1.3.2, (i) and (ii), in [1]. According to Example 3.4 in [21], in this case $\mathcal{D}^-$ equals the corresponding subcategory $\mathcal{T}^-_c$ of $\mathcal{T}$; see Definition 0.16 ibidem. Hence the existence of $c'_i$ that satisfies our assumptions follows from Lemma 7.5, (ii), of [21]; see Remarks 7.4 and 7.6 there. 3. This is a straightforward combination of the previous assertion with Lemma 3.5. The corollary is proved. Remark 3.10. 1. Corollary 3.9, 2, 3, proves Theorem 1.5, 1. 2. To prove Corollary 3.9, 3, it suffices to apply the fullness statement in Lemma 3.5, 1, along with Lemma 3.5, 2. To prove Corollary 1.10 we need a simple lemma. Lemma 3.11. Let $\mathfrak{B}$ be an $R$-linear additive category. 1. Then the category $\operatorname{Fun}_R(\mathfrak{B},R\text{-}\mathrm{Mod})$ (see Theorem 1.5 for the notation) is equivalent to $\operatorname{Fun}_{\mathbb{Z}}(\mathfrak{B},\operatorname{Ab})$. 2. A functor $\mathfrak{B}^{\mathrm{op}}\to R\text{-}\mathrm{Mod}$ is representable in $\mathfrak{B}$ whenever its composition with the forgetful functor $R\text{-}\mathrm{Mod}\to \operatorname{Ab}$ is representable, provided that we consider $\mathfrak{B}$ as a (mere) additive category. Proof. 1. Any additive functor $F\colon \mathfrak{B}\to \operatorname{Ab}$ becomes naturally an $R$-linear one if we define multiplication by $r\in R$ on $F(B)$ for $B\in \operatorname{Obj} \mathfrak{B}$ by means of the endomorphism $F(r\operatorname{id}_{B})$. It remains to note that this construction provides a unique way to lift $F$ to a $R$-linear functor into $R\text{-}\mathrm{Mod}$.
2. This is immediate from assertion 1. The lemma is proved. Proof of Corollary 1.10. 1. First assume that $R$ is countable.
Let us verify that for any $Y$ that is of finite type over $S=\operatorname{Spec} R$ the category $D^{\mathrm{b}}(\operatorname{coh}(Y))$ is countable in the sense of Definition 1.8, 3. This is Lemma 57.17.3 (tag 0G0W) in [30], but no proof is given there. So we recall that $D^{\mathrm{b}}(\operatorname{coh}(Y))$ is a Verdier quotient of $K^{\mathrm{b}}(\operatorname{coh}(Y))$. Hence it suffices to verify that $K^{\mathrm{b}}(\operatorname{coh}(Y))$ is countable. Indeed, the set of isomorphism classes of $D^{\mathrm{b}}(\operatorname{coh}(Y))$ is clearly a quotient of the set of isomorphism classes of $K^{\mathrm{b}}(\operatorname{coh}(Y))$, and the countability of morphism sets in Verdier quotients of countable triangulated categories follows immediately from the well-known description of these morphism sets given by Proposition 2.1.24 in [20] (see Definition 2.1.15 ibidem). Next, it is easily seen that the category $K^{\mathrm{b}}(\operatorname{coh}(Y))$ is countable if (and only if) $\operatorname{coh}(Y)$ is. Now, $\operatorname{coh}(Y)$ is clearly countable if $Y$ is affine. Lastly, the countability of $\operatorname{coh}(Y)$ for general $Y$ (of finite type over $S$) can easily be deduced from the countability of the categories $\operatorname{coh}(Y_i)$ and $\operatorname{coh}(W_{ijk})$ if one applies the gluing lemmas (see [30], Lemmas 6.33.1 and 6.33.4 (tags 04TN and 00AN); here the $Y_i$, $1\leqslant i\leqslant n$, form an open affine cover of $Y$, and the $W_{ijk}$ form open affine covers of $Y_i\cap Y_j$ (for any fixed $(i,j)$, $1\leqslant i,j\leqslant n$). Now recall that the category $D^{\mathrm{b}}(\operatorname{coh}(X))$ is equivalent to $\mathcal{D}^{\mathrm{b}}=D^{\mathrm{b}}_{\mathrm{coh}}(\operatorname{Qcoh}(X))$ and $\mathcal{D}_{\mathrm{p}}\subset \mathcal{D}^{\mathrm{b}}$ (see Remarks 3.1 and 1.4, 3); hence $ \mathcal{D}_{\mathrm{p}}$ is also countable. Thus we can apply (Neeman’s) Proposition 1.9 to obtain that all homological functors ${(\mathcal{D}_{\mathrm{p}})^{\mathrm{op}}\to \operatorname{Ab}}$ are represented by objects of $\mathcal{D}_{\mathrm{Q}}$. Consequently, all homological functors ${(\mathcal{D}_{\mathrm{p}})^{\mathrm{op}}\to R\text{-}\mathrm{Mod}}$ are representable too; see Lemma 3.11, 2. Now we pass to the case when $R$ is self-injective. Similarly to the proof of Theorem A.1 in [6], we apply a double duality argument. The idea is to extend a homological functor $H\colon (\mathcal{D}_{\mathrm{p}})^{\mathrm{op}}\to R\text{-}\mathrm{mod}$ to a nice functor $H'\colon (\mathcal{D}_{\mathrm{Q}})^{\mathrm{op}}\to R\text{-}\mathrm{Mod}$. Consider the functors $(\,\cdot\,)^*\colon R\text{-}\mathrm{Mod}\to R\text{-}\mathrm{Mod}^{\mathrm{op}}$, $N\mapsto \operatorname{Hom}_R(N,R)$, and $\widehat{H}$: $\mathcal{D}_{\mathrm{p}}\to R\text{-}\mathrm{Mod}$, $M\mapsto H(M)^*$. Since $R$ is an injective $R$-module, $\widehat{H}$ is homological. Next, it extends to a homological functor $\widehat{H}'\colon \mathcal{D}_{\mathrm{Q}}\to R\text{-}\mathrm{Mod}$ that respects coproducts: see Proposition 2.3 of [15]. Now we take $H'\colon (\mathcal{D}_{\mathrm{Q}})^{\mathrm{op}}\to R\text{-}\mathrm{Mod}$, $M\mapsto (\widehat{H}'(M))^*$. This functor is clearly homological and respects products (that is, sends $\mathcal{D}_{\mathrm{Q}}$-coproducts into products of $R$-modules). Lemma 2.2, 2, allows us to apply the well-known Theorem 3.1 of [18] (that is, the Brown representability property of the compactly generated category $\mathcal{D}_{\mathrm{Q}}$) to obtain that all cohomological functors $\mathcal{D}_{\mathrm{Q}}\to \operatorname{Ab}$ that respects products are representable (if we consider $\mathcal{D}_{\mathrm{Q}}$ as an additive category). Consequently, $H'$ is representable too; see Lemma 3.11, 2. It remains to prove that $H'$ restricts to $\mathcal{D}_{\mathrm{p}}$ as $H$. To do this it suffices to verify that for any finitely generated $R$-module $N$ the natural map $N\to N^{**}$ is bijective, that is, $N$ is reflexive (see Remark 4.65, (a), in [17]). Now, $R$ is a quasi-Frobenius ring: see Theorem 15.1 in [17].6 Consequently, we can apply Theorem 15.11 ibidem to obtain that $N$ is reflexive indeed. 2. Since the values of $H$ belong to $R\text{-}\mathrm{mod}$, $H$ is represented by an object $N$ of $\mathcal{D}_{\mathrm{Q}}$ according to Corollary 1.10, 1. Next, $X$ is projective over $S$; hence Theorem 1.5, 2, implies that $N$ belongs to $\mathcal{D}^{\mathrm{u}}$. Lastly, if for any $M\in \operatorname{Obj} \mathcal{D}_{\mathrm{p}}$ there exists $c_M>0$ such that $H(M[i])=0$ whenever $i<-c_M$, then Theorem 1.5, 2, also says that $N$ belongs to $\mathcal{D}^+$. The corollary is proved. Remark 3.12. Recall that commutative quasi-Frobenius rings (see the proof of Corollary 1.10, 1) can be described more or less explicitly; see Theorem 15.27 in [17]. Note however that the most important (from the algebraic geometric point of view) quasi-Frobenius rings are fields, and this is the only case mentioned in [6]. § 4. Main semi-orthogonal decomposition resultsIn § 4.1 we prove our main abstract results (Proposition 4.1 and Theorem 4.4) on the existence of some new semi-orthogonal decompositions. We also deduce a simple Corollary 4.5 on the existence of adjoint functors. In § 4.2 we apply our general results to semi-orthogonal decompositions of various subcategories of $D(\operatorname{Qcoh}(X))$ (where $X$ is proper over $S=\operatorname{Spec} R$); this yields a geometric statement (Theorem 4.12) on semi-orthogonal decompositions in $D_{\mathrm{perf}}(X)$, $D^{\mathrm{b}} (\operatorname{coh}(X))$, $D^-_{\mathrm{coh}}(X)$, $D^+_{\mathrm{coh}}(\operatorname{Qcoh}(X))$ and $D_{\mathrm{coh}}(\operatorname{Qcoh}(X))$. Moreover, Proposition 4.14 says that the corresponding decompositions can be restricted to certain support subcategories of the corresponding categories (that correspond to unions of closed subsets of $S$). In § 4.3 we apply Grothendieck duality arguments to establish the regular case of Theorem 1.7, 2. We also relate semi-orthogonal decompositions to duality; see Proposition 4.19. In § 4.4 we give some definitions and establish the generalization of Theorem 4.12 to semi-orthogonal decompositions of arbitrary length. Moreover, we finish the proof of Theorem 1.3 (see Remark 4.22). 4.1. Abstract decomposition statementsNow we study certain decompositions coming from semi-orthogonal decompositions of subcategories of compact objects. Our formulations rely heavily on Definitions 1.1 and 1.8. We do not mention any decompositions of length $\neq 2$ (cf. Definition 2.9, II.2) till § 4.4. Proposition 4.1. Assume that $\mathcal{D}=\mathcal{T}^{\coprod}$, where $\mathcal{T}\subset \mathcal{D}$ is a triangulated subcategory whose objects are $\mathcal{D}$-compact and $D=(\mathcal{A},\mathcal{B})$ is a semi-orthogonal decomposition of $\mathcal{T}$. 1. Then $D^{\coprod}=(\mathcal{A}^{\coprod},\mathcal{B}^{\coprod})$ is a semi-orthogonal decomposition of $\mathcal{D}$. 2. Assume in addition that $\mathcal{T}$ is essentially small (respectively, $\mathcal{D}$ is compactly generated by $\mathcal{T}$). Then $D^{\perp}_{\mathcal{D}}$ is a semi-orthogonal decomposition of $\mathcal{D}$ too. Moreover, $D^{\perp}_{\mathcal{D}}=(D^{\coprod})^{\perp}_{\mathcal{D}}$, and $\mathcal{B}^{\perp}_{\mathcal{D}}=\mathcal{A}^{\coprod}$. 3. Assume that $\mathcal{T}_0$ is a triangulated subcategory of $\mathcal{D}$ such that $D^{\perp}_{\mathcal{D}}$ restricts to a semi-orthogonal decomposition $D_0$ on it (see Definition 2.3, 2). Then $D^{\perp}_{\mathcal{D}}$ restricts to the category $\mathcal{T}_{0}^{\coprod}$ too, and this restriction equals $D_0^{\coprod}$. Proof. 1. This statement is closely related to Proposition 4.2 in [16], and the proof of that proposition carries over to our context without any difficulty; see also Theorem 3.1.1 of [9].
2. Combining assertion 1 with Proposition 2.5, 1, and Lemma 2.2, 1, we deduce that $\mathcal{A}^{\coprod}=(\mathcal{B}^{\coprod})^{\perp}_{\mathcal{D}} =\mathcal{B}^{\perp}_{\mathcal{D}}$; hence $(\mathcal{B}^{\perp}_{\mathcal{D}})^{\perp}_{\mathcal{D}} =(\mathcal{A}^{\coprod})^{\perp}_{\mathcal{D}}=\mathcal{A}^{\perp}_{\mathcal{D}}$. Hence it remains to verify that $\mathcal{A}^{\coprod}$ is right admissible in $\mathcal{D}$; see Proposition 2.5. The latter statement is given by Theorem 4.1 in [18]. 3. Since all objects of $\mathcal{A}$ and $\mathcal{B}$ are compact, both $\mathcal{A}^{\perp}_{\mathcal{D}}$ and $\mathcal{B}^{\perp}_{\mathcal{D}}$ are closed with respect to $\mathcal{D}$-coproducts. Hence if $D_0=(\mathcal{A}_0,\mathcal{B}_0)$ then $\mathcal{B}_{0}^{\coprod}\perp \mathcal{A}_{0}^{\coprod}$. Consequently, it suffices to verify that the class $\mathcal{C}_{0}$ of extensions of objects of $\mathcal{A}_{0}^{\coprod}$ by those of $\mathcal{B}_{0}^{\coprod}$ coincides with $\operatorname{Obj} \mathcal{T}_0^{\coprod}$; see Proposition 2.5, 1. Now, $\mathcal{C}_{0}$ contains $\operatorname{Obj} \mathcal{T}_{0}$, and it gives a triangulated subcategory of $\mathcal{D}$ that is closed with respect to coproducts according to Lemma 2.1. Thus, $\mathcal{C}_{0}=\operatorname{Obj} \mathcal{T}_0^{\coprod}$ indeed. The proposition is proved. To formulate one of our central theorems we need some rather technical definitions. The reader can note that Definition 4.2, 3, below axiomatizes some of the properties of the categories of the form $\mathfrak{E}^s$ that we describe in Definition 4.2, 2; cf. Remark 4.3, 1. Definition 4.2. 1. Given a ring $R$, we write $R\text{-}\mathrm{Mod}^{\mathbb{Z}}$ for the abelian category of $\mathbb{Z}$-graded $R$-modules. 2. We set $\mathfrak{E}^{\mathrm{u}}=R\text{-}\mathrm{mod}^{\mathbb{Z}}\subset R\text{-}\mathrm{Mod}^{\mathbb{Z}}$ (see Definition 1.1, 6). We write $\mathfrak{E}^-$ ($\mathfrak{E}^+$) for the following subcategories of $\mathfrak{E}^{\mathrm{u}}$: $M=\bigoplus M^i\in \operatorname{Obj} \mathfrak{E}^-$ ($\mathfrak{E}^+$, respectively) whenever $M^i=\{0\}$ for $i\gg 0$ ($i\ll 0$, respectively). We set $\mathfrak{E}^{\mathrm{b}}=\mathfrak{E}^-\cap \mathfrak{E}^+$ (cf. Definition 1.1, 2). 3. We say that $\mathfrak{E}$ is a graded weak Serre subcategory of $R\text{-}\mathrm{Mod}^{\mathbb{Z}}$ (cf. [30], Lemma 12.10.3 and Definition 12.10.1 (tags 0754 and 02MO)) if $\mathfrak{E}$ is an extension closed abelian subcategory of $R\text{-}\operatorname{Mod}^{\mathbb{Z}}$ which is stable with respect to shifts of grading on $R\text{-}\mathrm{Mod}^{\mathbb{Z}}$, that is, $M=\bigoplus M^i$ belongs to $\mathfrak{E}$ if and only if the module $M[1]=\bigoplus M^{i+1}$ does. 4. Assume that $\mathcal{T}\subset \mathcal{D}$ are triangulated categories. For $M,N\mkern-1mu\!\in\! \operatorname{Obj} \mathcal{D}$ we let $\mathcal{D}^{\bullet}(\mkern-1mu M,N)$ denote the graded module $\bigoplus_{j\in \mathbb{Z}} \mathcal{D}(M[-j],N)$. Next, for a graded weak Serre subcategory $\mathfrak{E}$ of $R\text{-}\mathrm{Mod}^{\mathbb{Z}}$ we define a full subcategory $\mathcal{T}_{\mathfrak{E}}$ of $\mathcal{D}$ as follows: $N\in \operatorname{Obj} \mathcal{D}$ is an object of $\mathcal{T}_{\mathfrak{E}}$ whenever for any $M\in \operatorname{Obj} \mathcal{T}$ the graded module $\mathcal{D}^{\bullet}(M,N)$ belongs to $\mathfrak{E}$. Remark 4.3. 1. The categories $\mathfrak{E}^{\mathrm{u}}$, $\mathfrak{E}^-$, $\mathfrak{E}^+$ and $\mathfrak{E}^{\mathrm{b}}$ are obviously graded weak Serre subcategories of $R\text{-}\mathrm{Mod}^{\mathbb{Z}}$ whenever $R$ is Noetherian. Below we use our definition only in the case when $R$ is a (commutative unital) Noetherian ring; we usually take $\mathfrak{E}$ to be one of our $\mathfrak{E}^s$ (where $s=\mathrm u,-,+,\mathrm b$). Note however that one can probably take finitely presented modules over a coherent ring instead of a Noetherian one; see Definition 2.1, Theorem 2.4, Corollary 2.7 and Lemma 2.8 in [25]. Yet the details have to be checked here. 2. One can also fix an infinite cardinal $\aleph$ and take $\mathfrak{E}$ consisting of those $M=\bigoplus M^i$ such that each $M^i$ has fewer than $\aleph$ generators over $R$. This gives us a smallness filtration on $\mathcal{D}$, which is clearly exhaustive if $\mathcal{T}$ is essentially small. We also consider certain categories $\mathfrak{E}$ related to support sets $T\subset S=\operatorname{Spec} R$ in Proposition 4.14 below. Theorem 4.4. Assume that $\mathcal{D}=\mathcal{T}^{\coprod}$, where $\mathcal{T}\subset \mathcal{D}$ is a triangulated subcategory whose objects are $\mathcal{D}$-compact, $\mathcal{D}$ is $R$-linear, and $\mathfrak{E}$ is a graded weak Serre subcategory of $R\text{-}\mathrm{Mod}^{\mathbb{Z}}$. I. Then the following hold.
II. Let $D$ be a semi-orthogonal decomposition of $\mathcal{T}$.
Proof. I.1. Since $\mathfrak{E}=\mathfrak{E}[1]$ (see Definition 4.2, 3), the same is clearly true for $\mathcal{T}_{\mathfrak{E}}$.
Next, for each $M\in \operatorname{Obj} \mathcal{T}$ the functor $\mathcal{D}(M,\,\cdot\,)$ sends $\mathcal{T}$-distinguished triangles into long exact sequences. Consequently, for any $\mathcal{D}$-distinguished triangle $B[-1] \xrightarrow{g} A \to C \to B$ there is an $R\text{-}\mathrm{Mod}^{\mathbb{Z}}$-exact sequence
$$
\begin{equation*}
0\to \operatorname{Coker} \mathcal{D}^*(M,g)\to \mathcal{D}^*(M,C)\to \operatorname{Ker} \mathcal{D}^*(M,g[1])\to 0.
\end{equation*}
\notag
$$
Since $\mathfrak{E}$ is a graded weak Serre subcategory of $R\text{-}\mathrm{Mod}^{\mathbb{Z}}$, this short exact sequence shows that if $A$ and $B$ are objects of $\mathcal{T}_{\mathfrak{E}}$, then $C$ belongs to $\mathcal{T}_{\mathfrak{E}}$ as well.
2. This is obvious. II.1. According to Proposition 2.6,
$$
\begin{equation*}
\mathcal{D}(M[-j], L_{D^{\perp}_{\mathcal{D}}}(N)) \cong \mathcal{D}(R_D(M[-j]),N)\cong \mathcal{D}(R_D(M)[-j],N).
\end{equation*}
\notag
$$
Similarly, $\mathcal{D}(M[-j],R_{D^{\perp}_{\mathcal{D}}}(N))\cong \mathcal{D}(L_D(M)[-j],N)$. Taking the (graded) direct sums for all $j\in \mathbb{Z}$ we obtain the isomorphisms in question.
2. Since the objects $R_D(M)$ and $L_D(M)$ belong to $\mathcal{T}$, we obtain that $L_{D^{\perp}_{\mathcal{D}}}(N)$ and $R_{D^{\perp}_{\mathcal{D}}}(N)$ belong to $\mathcal{T}_{\mathfrak{E}}$ whenever $N$ does. It remains to check that any object $N$ of $\mathcal{T}_{\mathfrak{E}}$ possesses a ${D^{\perp}_{\mathcal{D}}}$-decomposition (2.1) inside $\mathcal{T}_{\mathfrak{E}}$. This statement immediately follows from the first part of our assertion according to Proposition 2.5, 1. The theorem is proved. Theorem 4.4 easily yields the existence of certain adjoint functors. Corollary 4.5. Suppose that $\mathcal{D}$ is compactly generated by a triangulated subcategory $\mathcal{T}$, $F\colon \mathcal{D}\to \mathcal{D}'$ is an exact functor that respects coproducts, $\mathcal{D}''$ is a subcategory of $\mathcal{D}'$, and $\mathfrak{E}$ is a graded weak Serre subcategory of $R\text{-}\mathrm{Mod}^{\mathbb{Z}}$ (see Definition 4.2, 3). Moreover, assume that $\mathcal{D}$ is $R$-linear, $\mathcal{T} \subset \mathcal{T}_{\mathfrak{E}}$, and $\mathcal{D}''$ contains the essential image $F(\mathcal{T}_{\mathfrak{E}})$; take $F''\colon \mathcal{T}_{\mathfrak{E}}\to \mathcal{D}''$ to be the restriction of $F$ to $\mathcal{T}_{\mathfrak{E}}$. For any objects $M$ of $\mathcal{D}$ and $N$ of $\mathcal{D}'$ endow the abelian group $\mathcal{D}'(F(M),N)$ with the structure of $R$-module as follows: define multiplication by $r\in R$ (on $\mathcal{D}'(F(M),N)$) by means of the endomorphism $F(r\operatorname{id}_{M})$. Then the functor $F''$ possesses a right adjoint if and only if for any $M\in \operatorname{Obj} \mathcal{T}$ and $N\in \operatorname{Obj} \mathcal{D}''$ the graded $R$-module
$$
\begin{equation*}
\mathcal{D}'{}^{\bullet}(F(M),N)= \bigoplus_{j\in \mathbb{Z}}\mathcal{D}'(F(M[-j]),N)
\end{equation*}
\notag
$$
belongs to $\mathfrak{E}$. Moreover, this adjoint is exact if $\mathcal{D}''$ is a triangulated subcategory of $\mathcal{D}'$. Proof. If $F''$ possesses an adjoint functor $G''\colon \mathcal{D}''\to \mathcal{T}_{\mathfrak{E}}$, then for any $M\in \operatorname{Obj} \mathcal{T}_{\mathfrak{E}}$ and $N\in \operatorname{Obj} \mathcal{D}''$ we have
$$
\begin{equation*}
\mathcal{D}'{}^{\bullet}(F(M),N) \cong \mathcal{D}{}^{\bullet}(M,G''(N));
\end{equation*}
\notag
$$
cf. the proof of Theorem 4.4, II.1. Since $\mathcal{T}\subset \mathcal{T}_{\mathfrak{E}}$, the graded module $\mathcal{D}'{}^{\bullet}(F(M),N)$ belongs to $\mathfrak{E}$ whenever $M\in \operatorname{Obj} \mathcal{T}$.
Let us prove the converse implication. Since $\mathcal{D}$ is compactly generated and the functor $F$ respects coproducts, $F$ is well known to possess an exact right adjoint $G$; see Theorem 4.1 in [18] and Lemma 5.3.6 in [20]. Thus it suffices to verify that $G$ sends $\mathcal{D}''$ into $\mathcal{T}_{\mathfrak{E}}$. Now, for any $N\in \operatorname{Obj} \mathcal{D}''$, if $M$ belongs to $\mathcal{T}$, then $\mathcal{D}{}^{\bullet}(M,G(N)) \cong \mathcal{D}'{}^{\bullet}(F(M),N) \in \operatorname{Obj} \mathfrak{E}$; hence $G(N)$ belongs to $ \mathcal{T}_{\mathfrak{E}}$ indeed. The corollary is proved. 4.2. The main geometric applicationsUntil the end of this paper we always assume that the following condition is fulfilled. Assumption 4.6. $R$ is a Noetherian ring and $X$ is a scheme which is proper over $S=\operatorname{Spec} R$. Let us describe some examples for Theorem 4.4. Theorem 4.7. Take $\mathcal{T}=\mathcal{D}_{\mathrm{p}}$ and $\mathcal{D}=\mathcal{D}_{\mathrm{Q}}$ (see Definition 1.1, 7). 1. Then $\mathcal{D}$ is compactly generated by $\mathcal{T}$. 2. If $s$ is equal to $\mathrm{u}$, $+$, $-$ or $\mathrm b$, then $\mathcal{D}^s\subset \mathcal{T}_{\mathfrak{E}^s}$ (see Definition 4.2, 2 and 4). 3. Moreover, this inclusion is equality if either $X$ is projective over $S$ (in the weak sense specified in Definition 1.1, 8) or $s\in \{\mathrm{b},-\}$. Proof. 1. Since $R$ is Noetherian, $X$ is a Noetherian separated scheme; thus the compact generation statement is well known (see Theorem 3.1.1 in [6]).
2. This statement is easy and probably well known. It also follows easily from Lemma 3.2; see Remark 3.3, 1. 3. In the case when $X$ is projective over $S$ the assertion is just a re-formulation of Theorem 1.5, 2 (whose proof was given in § 3.1). In the cases when $s=\mathrm b$ or $s=-$ Corollary 0.5 in [21] (see Theorem 1.7, 1) implies the following: for any $N\in \operatorname{Obj} \mathcal{T}_{\mathfrak{E}^s}$ there exists $N'\in \operatorname{Obj} \mathcal{D}^s$ such that $\mathcal{Y}(N)\cong \mathcal{Y}(N')$; here we use the notation of Theorem 1.5. Lastly, $N\cong N'$ according to Corollary 3.9, 3. The theorem is proved. Remark 4.8. Clearly, one can combine Corollary 4.5 with Theorem 4.7, 3, to obtain a criterion for the existence of a right adjoint to the corresponding restriction $F''\colon \mathcal{D}^s\to \mathcal{D}''$, where $F\colon \mathcal{D}_{\mathrm{Q}}\to \mathcal{D}$ is an exact functor that respects coproducts, $F(\mathcal{D}^s)\subset \mathcal{D}''$ and $(X,s)$ is any pair satisfying the assumptions of Theorem 4.7, 3. In some statements we also need the following very common condition on $X$. Assumption 4.9. The scheme $X$ is of finite Krull dimension and either regular, or regular alterations (see Remark 4.10) exist for all of its integral closed subschemes. Remark 4.10. Recall that alterations were introduced in [11]; regular alterations generalize Hironaka’s resolutions of singularities. More precisely, a regular alteration for a scheme $Z$ is a proper surjective morphism $Y\to Z$ that is generically finite and such that $Y$ is regular and finite dimensional. Since resolutions of singularities exist for arbitrary quasi-excellent $\operatorname{Spec}\mathbb{Q}$-schemes according to Theorem 1.1 in [27], our assumption is fulfilled whenever $R$ is a quasi-excellent Noetherian $\mathbb{Q}$-algebra. Moreover, Assumption 4.6 is fulfilled if $X$ is of finite type over a scheme $B$ that is quasi-excellent of dimension at most $3$; see Theorem 1.2.5 in [28]. Proposition 4.11. Assume that $X$ satisfies Assumption 4.9. Then there exists an $R$-linear triangulated category $\mathcal{D}$ that is compactly generated by $\mathcal{T}=(\mathcal{D}^{\mathrm{b}})^{\mathrm{op}}$ and such that the corresponding category $\mathcal{T}_{\mathfrak{E}^{\mathrm{b}}}$ equals $(\mathcal{D}_{\mathrm{p}})^{\mathrm{op}}$. Proof. Clearly, to find $\mathcal{D}$ compactly generated by $\mathcal{T}$ it suffices to construct an exact fully faithful functor $F\colon \mathcal{T}\to \underline{E}$ for some $R$-linear triangulated category $\underline{E}$ that is compactly generated by the essential image of $F$. The existence of $F$ of this sort is quite simple. It is well known (see § 3.6 in [14]) and easily seen that $\mathcal{T}$ admits a dg-enhancement, that is, there exists a small $R$-linear differential graded category $C$ such that its homotopy category $H^0(C)$ (see [14], § 2.2; we take $k=R$ in the notation of [14]) is equivalent to $\mathcal{T}$. Indeed, $\mathcal{D}^{\mathrm{b}}$ is essentially small and can be embedded into the homotopy category $K(\operatorname{Inj}(X))$ of complexes of injective quasi-coherent sheaves on $X$, whereas $K(\operatorname{Inj}(X))$ is equivalent to $H^0(C')$ for some $R$-linear differential graded $C'$; see Lemma 3.3 in [14]. So we can choose $C$ to be a small subcategory of $C'$ such that the embedding $H^0(C)\to H^0(C')$ induces an equivalence $H^0(C)\to \mathcal{T}$ (that is, $C$ must be a sufficiently large small category whose objects have bounded coherent cohomology). Then Corollary 3.7 in [14] (see also the text preceding it) implies that we can take $\underline{E}=D(C)$ for our purposes (see § 3.2 ibidem).
Now assume that $\mathcal{D}$ is compactly generated by $\mathcal{T}$ and $X$ satisfies Assumption 4.9. Then Theorem 1.7, 2, says that the objects of $(\mathcal{D}_{\mathrm{p}})^{\mathrm{op}}\subset \mathcal{T}$ represent all those homological functors $H\colon \mathcal{D}^{\mathrm{b}}\to R\text{-}\mathrm{Mod}$ for which $\bigoplus_{i\in \mathbb{Z}}H(M[i])$ is a finitely generated $R$-module for each $M\in \operatorname{Obj} \mathcal{D}^{\mathrm{b}}$. It follows that for any $N\in \operatorname{Obj} \mathcal{T}_{\mathfrak{E}^{\mathrm{b}}}$ there exists $N'\in (\mathcal{D}_{\mathrm{p}})^{\mathrm{op}}$ such that the restrictions of the functors $\mathcal{D}(\,\cdot\,,N)$ and $\mathcal{D}(\,\cdot\,,N')$ to $\mathcal{T}$ are isomorphic. Since $N'$ belongs to $\mathcal{T}$, $\operatorname{id}_{N'}$ yields a canonical morphism $f\colon N'\to N$, and we have $\operatorname{Obj} \mathcal{T}\perp \operatorname{Cone}(f)$. It easily follows that $\operatorname{Cone}(f)=0$; see Lemma 2.2, 1. Hence $\mathcal{T}_{\mathfrak{E}^{\mathrm{b}}}$ equals $(\mathcal{D}_{\mathrm{p}})^{\mathrm{op}}$ indeed. The proposition is proved. Now we pass to semi-orthogonal decompositions; see Definitions 2.3 and 1.1, parts 1, 2, 7 and 8. Theorem 4.12. Let $D$ be a semi-orthogonal decomposition of $\mathcal{D}_{\mathrm{p}}$. I. Then the following hold. 1. The pairs $D^{\perp}_{\mathcal{D}_{\mathrm{Q}}}$, $D^{\perp}_{\mathcal{D}^{\mathrm{b}}}$ and $D^{\perp}_{\mathcal{D}^-}$ give semi-orthogonal decompositions of $\mathcal{D}_{\mathrm{Q}}$, $\mathcal{D}^{\mathrm{b}}$ and $\mathcal{D}^-$, respectively. Moreover, $D^{\perp}_{\mathcal{D}_{\mathrm{Q}}} =(D^{\perp}_{\mathcal{D}^{\mathrm{b}}})^{\coprod}$; consequently, $D^{\perp}_{\mathcal{D}^-} =(D^{\perp}_{\mathcal{D}^{\mathrm{b}}})^{\coprod}_{\mathcal{D}^-}$. Furthermore, if $X$ is projective over $S$, then $D^{\perp}_{\mathcal{D}^{\mathrm{u}}}$ is a decomposition of $\mathcal{D}^{\mathrm{u}}$ and $D^{\perp}_{\mathcal{D}^+}$ is a decomposition of $\mathcal{D}^+$. 2. The map $D\mapsto D^{\perp}_{\mathcal{D}^{\mathrm{b}}}$ injects the set of all semi-orthogonal decompositions of $\mathcal{D}_{\mathrm{p}}$ into the set of semi-orthogonal decompositions of $\mathcal{D}^{\mathrm{b}}$. II. Suppose in addition that $X$ satisfies Assumption 4.9, and $E$ is a semi-orthogonal decomposition of $\mathcal{D}^{\mathrm{b}}$. 1. Then the map $D\!\mapsto\! D^{\perp}_{\mathcal{D}^{\mathrm{b}}}$ induces a bijection between the sets of semi-orthogonal decompositions of the categories $\mathcal{D}_{\mathrm{p}}$ and $\mathcal{D}^{\mathrm{b}}$. The inverse map is given by $E\mapsto {}^{\perp}_{\mathcal{D}_{\mathrm{p}}} E$. 2. Consequently, the pair $E^{\coprod}$ gives a semi-orthogonal decomposition of $\mathcal{D}_{\mathrm{Q}}$ that coincides with $({}^{\perp} E)^{\perp}_{\mathcal{D}_{\mathrm{Q}}}$, and this decomposition restricts to the semi-orthogonal decomposition $({}^{\perp} E)^{\perp}_{\mathcal{D}^-}$ of $\mathcal{D}^-$. Moreover, if $X$ is projective over $S$, then $E^{\coprod}$ also restricts to $\mathcal{D}^{\mathrm{u}}$ and $\mathcal{D}^+$. 3. The map $\mathcal{E}\mapsto \mathcal{E}\cap \mathcal{D}_{\mathrm{p}}$ induces a one-to-one correspondence between right admissible subcategories of $\mathcal{D}^{\mathrm{b}}$ and left admissible subcategories of $\mathcal{D}_{\mathrm{p}}$. Proof. I.1. According to Proposition 4.1, 2 (combined with Theorem 4.7, 1), $D^{\perp}_{\mathcal{D}_{\mathrm{Q}}}$ is a decomposition of $\mathcal{D}_{\mathrm{Q}}$. Next, Theorem 4.7, 3, allows us to apply Theorem 4.4, II.2, to obtain that $D^{\perp}_{\mathcal{D}^{\mathrm{b}}}$ and $D^{\perp}_{\mathcal{D}^-}$ give indeed semi-orthogonal decompositions of the corresponding categories. This is also true for $D^{\perp}_{\mathcal{D}^{\mathrm{u}}}$ and $D^{\perp}_{\mathcal{D}^+}$ whenever $X$ is projective over $S$.
Next, recall that $\mathcal{D}_{\mathrm{p}}\subset \mathcal{D}^{\mathrm{b}}\subset \mathcal{D}_{\mathrm{Q}}$. Hence $(\mathcal{D}_{\mathrm{p}})^{\coprod}=(\mathcal{D}^{\mathrm{b}})^{\coprod} =\mathcal{D}_{\mathrm{Q}}$ (see Theorem 4.7, 1); thus, $D^{\perp}_{\mathcal{D}_{\mathrm{Q}}} =(D^{\perp}_{\mathcal{D}^{\mathrm{b}}})^{\coprod}$ by Proposition 4.1, 3. It clearly follows that $D^{\perp}_{\mathcal{D}^-}= (D^{\perp}_{\mathcal{D}^{\mathrm{b}}})^{\coprod}_{\mathcal{D}^-}$. 2. This is immediate from Proposition 2.8, parts I.2 and II.2. II.1. Proposition 4.11 gives us a category $\mathcal{D}$ that is compactly generated by $(\mkern-1.5mu\mathcal{D}^{\mathrm{b}}\mkern-1.5mu)^{\mathrm{op}}$ such that the corresponding category $\mathcal{T}_{\mathfrak{E}^{\mathrm{b}}}$ equals $(\mathcal{D}_{\mathrm{p}})^{\mathrm{op}}$. Hence Theorem 4.4, II.2 (combined with Proposition 2.5, 3), implies that ${}^{\perp}_{\mathcal{D}_{\mathrm{p}}} E$ is a decomposition of $\mathcal{D}_{\mathrm{p}}$. Next, Proposition 2.8, II.1, implies that the decomposition $({}^{\perp}_{\mathcal{D}_{\mathrm{p}}} E)^{\perp}_{\mathcal{D}^{\mathrm{b}}}$ (provided by assertion I.1) equals $E$. Combing this statement with assertion I.2 we obtain that the map $E\mapsto {}^{\perp}_{\mathcal{D}_{\mathrm{p}}} E$ yields a bijection between the sets of semi-orthogonal decompositions of $\mathcal{D}^{\mathrm{b}}$ and $\mathcal{D}_{\mathrm{p}}$, and this map is inverse to $D\mapsto D^{\perp}_{\mathcal{D}^{\mathrm{b}}}$. 2. Assume that $D={}^{\perp}_{\mathcal{D}_{\mathrm{p}}} E$. According to assertion II.1, we have $D^{\perp}_{\mathcal{D}^{\mathrm{b}}}=E$. Hence assertion I.1 says that $E^{\coprod}=D^{\perp}_{\mathcal{D}_{\mathrm{Q}}}$ is a decomposition of $\mathcal{D}_{\mathrm{Q}}$. Moreover, $D^{\perp}_{\mathcal{D}_{\mathrm{Q}}}$ restricts to the decompositions $D^{\perp}_{\mathcal{D}^{\mathrm{b}}}$ and $D^{\perp}_{\mathcal{D}^-}$ of the corresponding categories; these decompositions exist according to assertion I.1. Furthermore, if $X$ is projective over $S$, then $E^{\coprod}=D^{\perp}_{\mathcal{D}_{\mathrm{Q}}}$ restricts to $\mathcal{D}^{\mathrm{u}}$ and $\mathcal{D}^+$ too (for the same reason). 3. Proposition 2.5, parts 1 and 2, implies easily that the correspondence $C_{\mathcal{D}_{\mathrm{p}}}$: $\mathcal{D}\mapsto \mathcal{D}^{\perp}_{\mathcal{D}_{\mathrm{p}}}$ induces a bijection between the set of right admissible subcategories of $\mathcal{D}_{\mathrm{p}}$ and the set of left admissible subcategories of $\mathcal{D}_{\mathrm{p}}$. Next, invoking assertion II.1 we also obtain that the map $C_{\mathcal{D}^{\mathrm{b}}}\colon \mathcal{D}\mapsto \mathcal{D}^{\perp}_{\mathcal{D}^{\mathrm{b}}}$ induces a bijection between the set of right admissible subcategories of $\mathcal{D}_{\mathrm{p}}$ and the set of left admissible subcategories of $\mathcal{D}^{\mathrm{b}}$. Hence $D=C_{\mathcal{D}_{\mathrm{p}}}\circ C_{\mathcal{D}^{\mathrm{b}}}^{-1}$ induces a bijection between the right admissible subcategories of $\mathcal{D}^{\mathrm{b}}$ and the left admissible subcategories of $\mathcal{D}_{\mathrm{p}}$, and it remains to note that $D(\mathcal{D}^{\perp}_{\mathcal{D}^{\mathrm{b}}}) =\mathcal{D}^{\perp}_{\mathcal{D}_{\mathrm{p}}} =(\mathcal{D}^{\perp}_{\mathcal{D}^{\mathrm{b}}})\cap \mathcal{D}_{\mathrm{p}}$. The theorem is proved. Remark 4.13. 1. The author does not know of any direct arguments that allow one to extend arbitrary semi-orthogonal decompositions of $\mathcal{D}^{\mathrm{b}}$ to $\mathcal{D}^-\subset \mathcal{D}^{\mathrm{u}}$ (cf. part II.2 of Theorem 4.12). Recall however that Corollary 1.12 in [23] treated decompositions of $\mathcal{D}^{\mathrm{b}}$ whose components are admissible in the sense of Definition 1.1, 1. This additional restriction allowed the author to use arguments (in the proof of Proposition 1.10 ibidem) that are rather similar to ours (but avoid auxiliary categories) to obtain that decompositions of this sort restrict to $\mathcal{D}_{\mathrm{p}}$. It is worth noting that Proposition 1.10 and Corollary 1.12 in [23] extend to our $R$-linear setting without any difficulty (at least) in the case when $X$ is projective over $S$, since in this case the condition (ELF) mentioned in [23] is fulfilled for $X$ (see example (c) in [29], § 2.1.2). 2. Similarly to Theorem 4.12, I.2, Proposition 2.8 (parts I.2 and II.2) also implies that the maps $D\mapsto D^{\perp}_{\mathcal{D}^-}$ and $D\mapsto D^{\perp}_{\mathcal{D}_{\mathrm{Q}}}$ induce injections of the set of all semi-orthogonal decompositions of $\mathcal{D}_{\mathrm{p}}$ into the classes of semi-orthogonal decompositions of $\mathcal{D}^-$ and $\mathcal{D}_{\mathrm{Q}}$, respectively. Since both of these categories contain $\mathcal{D}^{\mathrm{b}}$, this statement also follows from Theorem 4.12, I.2. Let us now consider some support subcategories. Proposition 4.14. Let $\mathfrak{B}$ be an extension closed abelian subcategory of $R\text{-}\mathrm{Mod}$; let $\mathfrak{B}^\mathbb{Z}$ be the corresponding (graded) subcategory of $(R\text{-}\mathrm{Mod})^{\mathbb{Z}}$. Set $\mathcal{T}=\mathcal{D}_{\mathrm{p}}$ and $\mathcal{D}=\mathcal{D}_{\mathrm{Q}}$, where $X$ is proper over $S=\operatorname{Spec} R$, and assume that $s\in \{\mathrm u,+,-,\mathrm b\}$. 1. Then for any decomposition $D$ of $\mathcal{D}_{\mathrm{p}}$ the pair $D^{\perp}_{(\mathcal{T}_{\mathfrak{B}^\mathbb{Z}})}$ gives a decomposition of the corresponding category $\mathcal{T}_{\mathfrak{B}^\mathbb{Z}}$, and this decomposition restricts to $\mathcal{T}_{\mathfrak{E}^s}\cap \mathcal{T}_{\mathfrak{B}^\mathbb{Z}}$ (cf. Theorem 4.7 for the descriptions of these $\mathcal{T}_{\mathfrak{E}^s}$). 2. Assume that $\mathfrak{B}$ consists of $R$-modules with support on $T$, where $T$ is a subset of $S=\operatorname{Spec} R$ stable under specialization, that is, $T$ is a union of closed subsets of $S$; see [30], Definition 10.40.1 (tag 00L1). Then $\mathcal{T}_{\mathfrak{B}^\mathbb{Z}}$ consists of all those objects of $\mathcal{D}_{\mathrm{Q}}$ the sections of whose cohomology sheaves (note that these are $R$-modules) have support on $T$. Proof. 1. Clearly, $\mathfrak{B}^\mathbb{Z}$ is a graded weak Serre subcategory of $R\text{-}\mathrm{Mod}^{\mathbb{Z}}$ (in the sense of Definition 4.2, 3); hence $\mathfrak{B}^\mathbb{Z}\cap \mathfrak{E}^s$ is a graded weak Serre subcategory of $R\text{-}\mathrm{Mod}^{\mathbb{Z}}$ too (see Theorem 4.4, I.2, and Remark 4.3, 1). Thus, the assertion follows from Theorem 4.4 (parts I.2 and II.2).
2. By the definition of the support, an object $C$ of $\mathcal{D}_{\mathrm{Q}}$ belongs to $\mathcal{T}_{\mathfrak{B}^\mathbb{Z}}$ if and only if for any object $M$ of $\mathcal{D}_{\mathrm{p}}$ and any scheme point $s_0\in S\setminus T$ we have $\mathcal{D}_{\mathrm{Q}}(M,C)\otimes_R R_{s_0}\!=\!\{0\}$; here $R_{s_0}$ is the localization of $R$ at $s_0$. Next, for the dual object $M^{\bigvee}$ we have $\mathcal{D}_{\mathrm{Q}}(M,C)\!\cong\! H^0(X,M^{\bigvee}\otimes C)$; see [30], Lemma 21.48.4 (tag 08JJ). Since the ring $R_{s_0}$ is a flat $R$-module, applying the associativity of the tensor product operation we deduce
$$
\begin{equation*}
\mathcal{D}_{\mathrm{Q}}(M,C)\otimes_R R_{s_0}\cong \mathcal{D}_{\mathrm{Q}}(M,C\otimes_R R_{s_0}).
\end{equation*}
\notag
$$
Combining Theorem 4.7, 1, with Lemma 2.2, 1, we obtain that $C$ belongs to $\mathcal{T}_{\mathfrak{B}^\mathbb{Z}}$ if and only if $C\otimes_R R_{s_0}=0$ for all ${s_0}\in S\setminus T$.
Moreover, the flatness of $R_{s_0}$ implies that for any $n\in \mathbb{Z}$ and any open $U\subset X$ we have $H^n(C\otimes_R R_{s_0})(U)\cong H^n(C)(U)\otimes_R R_{s_0}$. Since a complex of sheaves is acyclic if and only if all the sections of its cohomology sheaves are trivial, we conclude that $C\otimes_R R_{s_0}=0$ if and only if the sections of all sheaves $H^n(C)$ have support on $T$. The proposition is proved. Remark 4.15. Clearly, all categories $\mathcal{T}_{\mathfrak{E}^s}\cap \mathcal{T}_{\mathfrak{B}^\mathbb{Z}}$ as in Proposition 4.14, 1, depend on the category $\mathfrak{B}\cap (R\text{-}\mathrm{mod})$ only. Now, any extension closed abelian subcategory of $R\text{-}\mathrm{mod}$ consists of finitely generated $R$-modules with support on some $T$ as in Proposition 4.14, 2; see Theorem A in [26].7 4.3. Some duality argumentsNow let us pass to arguments related to Grothendieck duality; see Definition 1.1, 7, for the notation. Remark 4.16. Note that of greatest interest in algebraic geometry are schemes of finite type over regular schemes of finite Krull dimension (say, over the spectra of fields or Dedekind domains). In this case $X$ admits a dualizing complex automatically; see Proposition 4.17, 3. Proposition 4.17. Assume that $X$ admits a dualizing complex $K\in \mathcal{D}^{\mathrm{u}}$ in the sense of Definition 48.2.2 and Lemma 48.2.5 (tags 0A87 and 0A89) of [30] (take Remark 3.1 into account). Then the following hold. 1. An exact Grothendieck duality functor $D_X\colon \mathcal{D}^{\mathrm{u}}\to (\mathcal{D}^{\mathrm{u}})^{\mathrm{op}}$ is defined (uniquely up to an isomorphism).8 The functor $(D_X)^{\mathrm{op}}\circ D_X$ is isomorphic to the identity; correspondingly, $D_X$ is an equivalence. 2. The functor $D_X$ switches $\mathcal{D}^-$ and $\mathcal{D}^+$ and sends $\mathcal{D}^{\mathrm{b}}$ into itself. 3. If $Y'$ is a scheme of finite type over a Gorenstein scheme $Y$ (see [30], Definition 48.24.1 (tag 0AWW)) of finite Krull dimension, then $Y'$ admits a dualizing complex. Moreover, if $X$ is Gorenstein of finite Krull dimension, then $D_X$ restricts to an equivalence $\mathcal{D}_{\mathrm{p}}\to (\mathcal{D}_{\mathrm{p}})^{\mathrm{op}}$. In particular, this is the case when $X$ is regular and of finite Krull dimension. 4. Assume that $D_0=(\mathcal{A}_0,\mathcal{B}_0)$ is a semi-orthogonal decomposition of a triangulated subcategory $\mathcal{T}_0$ of $\mathcal{D}^{\mathrm{u}}$; set $D_X(D_0)$ to be the pair $(D_X(\mathcal{B}_0),D_X(\mathcal{A}_0))$. Then $D_X(D_0)$ is a semi-orthogonal decomposition of the subcategory $(D_X)^{\mathrm{op}}(\mathcal{T}_0^{\mathrm{op}})$ of $\mathcal{D}^{\mathrm{u}}$. Proof. All statements in assertions 1–3 follow easily from the properties of Grothendieck duality listed in [30], the beginning of § 48.19 (tag 0AU3) and Lemmas 48.24.3 and 48.24.4 (tags 0DWG and 0BFQ), and in [12]; see the sufficient condition 2 in [12], Ch. V, § 10.
Assertion 4 follows easily from Proposition 2.5, 3; recall that $D_X$ is fully faithful. These statements allow us to finish the proof of Theorem 1.7, 2. Corollary 4.18. Assume that $X$ is regular of finite Krull dimension. Then $\mathcal{D}_{\mathrm{p}}=\mathcal{D}^{\mathrm{b}}$, and to a functor $F\colon \mathcal{D}^{\mathrm{b}}\to R\text{-}\mathrm{Mod}$ is corepresented by an object of $\mathcal{D}_{\mathrm{p}}$ if and only if $\bigoplus_{i\in \mathbb{Z}}F(N[i])$ is a finitely generated $R$-module for each $N\in \operatorname{Obj} \mathcal{D}^{\mathrm{b}}$ (cf. Theorem 1.7, 2). Proof. It is well known that $\mathcal{D}_{\mathrm{p}}=\mathcal{D}^{\mathrm{b}}$ if $X$ is regular.
Next, $X$ admits a dualizing complex according to Proposition 4.17, 3. Hence we can apply $D_X$ and reduce the second statement to (the corresponding case of) Theorem 1.7, 1 (cf. Theorem 4.7, 3). The corollary is proved. Let us also make some observations that relate semi-orthogonal decompositions to duality. Proposition 4.19. Suppose that $X$ admits a dualizing complex. I. Assume that $X$ satisfies Assumption 4.9 and $E$ is a semi-orthogonal decomposition of $\mathcal{D}^{\mathrm{b}}$.
Consequently, the above decomposition $E^+$ equals $E^{\coprod}_{\mathcal{D}^+}$. II. Assume that $X$ is a Gorenstein scheme (cf. Proposition 4.17, 3) of finite Krull dimension and $D=(\mathcal{A},\mathcal{B})$ is a semi-orthogonal decomposition of $\mathcal{D}_{\mathrm{p}}$. Then ${}^{\perp}_{\mathcal{D}^{\mathrm{b}}}D =({}^{\perp}_{\mathcal{D}^{\mathrm{b}}}\mathcal{A}, {}^{\perp}_{\mathcal{D}^{\mathrm{b}}}\mathcal{B})$ is a semi-orthogonal decomposition of $\mathcal{D}^{\mathrm{b}}$ and ${}^{\perp}_{\mathcal{D}^+}D$ is a semi-orthogonal decomposition of $\mathcal{D}^+$. Moreover, if $X$ is projective over $S$, then ${}^{\perp}_{\mathcal{D}^{\mathrm{u}}}D$ is a semi-orthogonal decomposition of $\mathcal{D}^{\mathrm{u}}$. Proof. I. According to Proposition 4.17, parts 1, 2 and 4, $D_X(E)$ is a semi-orthogonal decomposition of $\mathcal{D}^{\mathrm{b}}$. Hence Theorem 4.12, II.1, implies that $D={}^{\perp}_{\mathcal{D}_{\mathrm{p}}} (D_X(E))$ is a decomposition of $\mathcal{D}_{\mathrm{p}}$ and $D^{\perp}_{\mathcal{D}^{\mathrm{b}}}=D_X(E)$. Moreover, $(D_X(E))^{\coprod}_{\mathcal{D}^-}=D^{\perp}_{\mathcal{D}^-}$ is a decomposition of $\mathcal{D}^-$ that restricts to $\mathcal{D}^{\mathrm{b}}$ as $D_X(E)$.
1. Applying $D_X$ to the decomposition $(D_X(E))^{\coprod}_{\mathcal{D}^-}$ we obtain that $E^+$ is a decomposition of $\mathcal{D}^+$ that restricts indeed to $\mathcal{D}^{\mathrm{b}}$ as $D_X(D_X(E))=E$ indeed. 2. In this case Theorem 4.12, II.1, yields in a similar way that $E^{\mathrm{u}}$ is a decomposition of $\mathcal{D}^{\mathrm{u}}$ that extends both $E^+$ and $E=(\mathcal{A}^{\mathrm{b}},\mathcal{B}^{\mathrm{b}})$. Next, Theorem 4.12, II.1, also implies that $D^{\perp}_{\mathcal{D}^{\mathrm{u}}}=(D_X(E))^{\coprod}_{\mathcal{D}^{\mathrm{u}}}$ (recall that $D={}^{\perp}_{\mathcal{D}_{\mathrm{p}}} (D_X(E))$); hence $D^{\perp}_{\mathcal{D}^{\mathrm{u}}} =D_X(D_X(D_X(E)^{\coprod}_{\mathcal{D}^{\mathrm{u}}})) =D_X(E^{\mathrm{u}})$. Let $D=(\mathcal{A},\mathcal{B})$. Applying Proposition 4.17, 4, we obtain
$$
\begin{equation*}
E^{\mathrm u}=D_X(D_X(E^{\mathrm u}))=D_X(D^{\perp}_{\mathcal{D}^{\mathrm u}})=D_X((\mathcal{A}^{\perp}_{\mathcal{D}^{\mathrm u}},\mathcal{B}^{\perp}_{\mathcal{D}^{\mathrm u}}))= ({}^{\perp}_{\mathcal{D}^{\mathrm u}}D_X(\mathcal{B}),{}^{\perp}_{\mathcal{D}^{\mathrm u}}D_X(\mathcal{A})).
\end{equation*}
\notag
$$
Since the decomposition $E^{\mathrm{u}}=(\mathcal{A}^{\mathrm{u}},\mathcal{B}^{\mathrm{u}})$ extends $E$, we have $\mathcal{A}^{\mathrm{b}}\subset {}^{\perp}_{\mathcal{D}^{\mathrm{u}}}D_X(\mathcal{B})$ and $\mathcal{B}^{\mathrm{b}}\subset {}^{\perp}_{\mathcal{D}^{\mathrm{u}}}D_X(\mathcal{A})$. Now, representable functors are cohomological and convert coproducts into products; hence both ${}_{\mathcal{D}_{\mathrm{Q}}}^{\perp}D_X(\mathcal{B})$ and ${}_{\mathcal{D}_{\mathrm{Q}}}^{\perp}D_X(\mathcal{A})$ are closed under coproducts. Consequently, $(\mathcal{A}^{\mathrm{b}})^{\coprod}_{\mathcal{D}^{\mathrm{u}}} =(\mathcal{A}^{\mathrm{b}})^{\coprod}\cap\mathcal{D}^{\mathrm{u}}\subset {}^{\perp}_{\mathcal{D}^{\mathrm{u}}}D_X(\mathcal{B}) =\mathcal{A}^{\mathrm{u}}$ and $(\mathcal{B}^{\mathrm{b}})^{\coprod}_{\mathcal{D}^{\mathrm{u}}}\subset \mathcal{B}^{\mathrm{u}}$.
Thus, $E^{\coprod}_{\mathcal{D}^{\mathrm{u}}}\leqslant_{R} E^{\mathrm{u}}$, and applying Proposition 2.8, I.1 we also obtain $E^{\mathrm{u}}\leqslant_{R} E^{\coprod}_{\mathcal{D}^{\mathrm{u}}}$; hence $E^{\mathrm{u}}= E^{\coprod}_{\mathcal{D}^{\mathrm{u}}}$ by Proposition 2.8, I.2. II. Applying Proposition 4.17, parts 3 and 4, we obtain that $D_X(D)$ is also a semi-orthogonal decomposition of $\mathcal{D}_{\mathrm{p}}$. Hence Theorem 4.12, I.1, yields that $(D_X(D))^{\perp}_{\mathcal{D}^{\mathrm{b}}}$ and $(D_X(D))^{\perp}_{\mathcal{D}^-}$ are decompositions of $\mathcal{D}^{\mathrm{b}}$ and $\mathcal{D}^-$, respectively; if $X$ is projective over $S$, then the pairs $(D_X(D))^{\perp}_{\mathcal{D}^+}$ and $(D_X(D))^{\perp}_{\mathcal{D}^{\mathrm{u}}}$ give decompositions of the corresponding categories too. It remains to apply Proposition 4.17, parts 4 and 2, to obtain all the assumptions in question. The proposition is proved. 4.4. Generalizations to arbitrary length decompositionsUntil the end of the paper we assume that $n$ is a positive integer. We return to semi-orthogonal decompositions of length $n+1$; consult § 2.3 for the corresponding definitions. Once again, we need certain orthogonals; pay attention to the numerations of subcategories in these families! Definition 4.20. Let $\mathcal{D}$ be a triangulated category; assume that $\mathcal{T}$ and $\mathcal{T}'$ are triangulated subcategories of $\mathcal{D}$. 1. Assume that $C=(C_{<i})$, $1\leqslant i\leqslant n$, is a family of subcategories of $\mathcal{T}$. We define the family $C[^{\perp}_{\mathcal{T}'}]=(C[^{\perp}_{\mathcal{T}'}]_{<i})$ (${[^{\perp}_{\mathcal{T}'}]}C=({[^{\perp}_{\mathcal{T}'}]}C_{<i})$; in both cases we assume that $1\leqslant i\leqslant n$) by setting $C[^{\perp}_{\mathcal{T}'}]_{<i}=(C_{<n+1-i}){}^{\perp}_{\mathcal{T}'}$ (${[^{\perp}_{\mathcal{T}'}]}C_{<i}={}^{\perp}_{\mathcal{T}'}(C_{<n+1-i})$, respectively). 2. Assume that $D=(D_i)$, $0\leqslant i\leqslant n$, is a system of triangulated subcategories of $\mathcal{T}$. We write $D[^{\perp}_{\mathcal{T}'}]$ ($[^{\perp}_{\mathcal{T}'}]D$) for the family ($([^{\perp}_{\mathcal{T}'}]D_i)= (\bigcap_{0\leqslant j\leqslant n,\ j\neq n- i}{}^{\perp}_{\mathcal{T}'}D_j)$, respectively).Firstly we note that the following length $n$ version of Theorem 1.3 holds. Proposition 4.21. Assume that $R$ is a Noetherian ring and $X$ is projective over $S=\operatorname{Spec} R$. 1. Let $C$ be a length $n$ left admissible chain in $\mathcal{D}_{\mathrm{p}}$ (see Definition 2.9, I.1). Then the families $C[^{\perp}_{\mathcal{D}^{\mathrm{b}}}]$, $C[^{\perp}_{\mathcal{D}^-}]$, $C[^{\perp}_{\mathcal{D}^+}]$, $C[^{\perp}_{\mathcal{D}^{\mathrm{u}}}]$ and $C[^{\perp}_{\mathcal{D}_{\mathrm{Q}}}]$ are length $n$ left admissible chains in $\mathcal{D}^{\mathrm{b}}$, $\mathcal{D}^-$, $\mathcal{D}^+$, $\mathcal{D}^{\mathrm{u}}$ and $\mathcal{D}_{\mathrm{Q}}$, respectively. Moreover, $C[^{\perp}_{\mathcal{D}_{\mathrm{Q}}}] =C[^{\perp}_{\mathcal{D}^{\mathrm{b}}}]^{\coprod}$ (see Definition 1.1, 5). Furthermore, the map $C\mapsto C[^{\perp}_{\mathcal{D}^{\mathrm{b}}}]$ injects the set of all length $n$ left admissible chains in $\mathcal{D}_{\mathrm{p}}$ into the set of length $n$ left admissible chains in $\mathcal{D}^{\mathrm{b}}$. 2. Assume that $X$ satisfies Assumption 4.9 and $B=(B_{<i})$ is a length $n$ left admissible chain in $\mathcal{D}^{\mathrm{b}}$. Then the map $C\mapsto C[^{\perp}_{\mathcal{D}^{\mathrm{b}}}]$ induces a one-to-one correspondence between the set of length $n$ left admissible chains in $\mathcal{D}_{\mathrm{p}}$ and the set of length $n$ left admissible chains in $\mathcal{D}^{\mathrm{b}}$; the inverse map is given by $B\mapsto {[^{\perp}_{\mathcal{D}_{\mathrm{p}}}]}B$. Moreover, the families $B^{\coprod}_{\mathcal{D}^-}=(B_{<i}{}^{\coprod}_{\mathcal{D}^-})$ (see Definition 1.1, 5), $B^{\coprod}_{\mathcal{D}^+}$, $B^{\coprod}_{\mathcal{D}^{\mathrm{u}}}$ and $B^{\coprod}_{\mathcal{D}_{\mathrm{Q}}}$ are length $n$ left admissible chains in $\mathcal{D}^-$, $\mathcal{D}^+$, $\mathcal{D}^{\mathrm{u}}$ and $\mathcal{D}_{\mathrm{Q}}$, respectively. Furthermore, for any $j$, $1\leqslant j\leqslant n$, we have $ (B_{\geqslant j})^{\coprod}\perp (B_{<j})^{\coprod}$. Proof. All these statements reduce easily to their $n=1$ cases (note that most of these versions are contained in Theorem 1.3). Now, if $n=1$, then it suffices to combine Theorem 4.12 with Proposition 2.5, parts 1 and 2.
The proposition is proved. Remark 4.22. 1. Similarly to the argument above, it suffices to assume that $X$ is proper over $S$ to prove that the families $C[^{\perp}_{\mathcal{D}^{\mathrm{b}}}]$, $C[^{\perp}_{\mathcal{D}^-}]$ and $C[^{\perp}_{\mathcal{D}_{\mathrm{Q}}}]$ are length $n$ left admissible chains in $\mathcal{D}^{\mathrm{b}}$, $\mathcal{D}^-$ and $\mathcal{D}_{\mathrm{Q}}$, respectively (provided that $C$ is a length $n$ left admissible chain in $\mathcal{D}_{\mathrm{p}}$). Moreover, if $B=(B_{<i})$ is a length $n$ left admissible chain in $\mathcal{D}^{\mathrm{b}}$ and $X$ satisfies Assumption 4.9, then $B^{\coprod}_{\mathcal{D}^-}$ and $B^{\coprod}_{\mathcal{D}_{\mathrm{Q}}}$ are length $n$ left admissible chains in $\mathcal{D}^-$ and $\mathcal{D}_{\mathrm{Q}}$, respectively. 2. The obvious right admissible versions of Proposition 4.21 and of its modification formulated in Remark 4.22, 1, can also be proved similarly. 3. Note that Theorem 4.12, II.3, proves the ‘furthermore statement’ in Theorem 1.3, II.1. All the remaining statements in that theorem are provided by the cases $n=1$ of Proposition 4.21 and Remark 4.22, parts 1 and 2. Now recall that for a length $n$ left admissible chain $C$ in $\mathcal{T}$ the family $D=\mathcal{D}ec(C)$ is a length $n+1$ semi-orthogonal decomposition of $\mathcal{T}$; see Definition 2.9, I.2, and Proposition 2.10, 2. Theorem 4.23. Assume that $\mathcal{T},\mathcal{T}'\subset \mathcal{D}$, $C$ is a length $n$ left admissible chain in $\mathcal{T}$, and $D=(D_i)$ is the semi-orthogonal decomposition $\mathcal{D}ec(C)$ (of length $n+1$). Then the following results hold. 1. For each $j$, $0\leqslant j\leqslant n$, we have $D[^{\perp}_{\mathcal{T}'}]_j= C[^{\perp}_{\mathcal{T}'}]_{<j+1}\cap (D_{\geqslant n+1-j}){}^{\perp}_{\mathcal{T}'}$ and $(D[^{\perp}_{\mathcal{T}'}])_{<j}\subset C[^{\perp}_{\mathcal{T}'}]_{<j}$ (see Definition 2.9, II.1). 2. Assume that for any $j$, $1\leqslant j \leqslant n$, the pair $((C_{<j}){}^{\perp}_{\mathcal{T}'}, (C_{\geqslant j}){}^{\perp}_{\mathcal{T}'})$ (see Definition 2.9, I.2) is a semi-orthogonal decomposition of $\mathcal{T}'$. Then $C[^{\perp}_{\mathcal{T}'}]=(D[^{\perp}_{\mathcal{T}'}])_{<}$. 3. Similarly, if for each $j$, $1\leqslant j \leqslant n$, the pair $({}^{\perp}_{\mathcal{T}'}(C_{<j}),{}^{\perp}_{\mathcal{T}'}(C_{\geqslant j}))$ is a semi-orthogonal decomposition of $\mathcal{T}'$, then $[^{\perp}_{\mathcal{T}'}]C=([^{\perp}_{\mathcal{T}'}]D)_{<}$. 4. Assume that $\mathcal{D}$ is closed with respect to coproducts and $(C_{\geqslant j})^{\coprod}\perp (C_{<j})^{\coprod}$ for any $j$, $1\leqslant j\leqslant n$. Then $((C_{<i})^{\coprod}_{\mathcal{T}'})=(D^{\coprod}_{\mathcal{T}'})_{<}$. Proof. 1. Both statements are immediate consequences of the following trivial observation: if $A_i$, $i\in I$, is a family of subcategories of $\mathcal{T}$, $A$ is the smallest triangulated subcategory of $\mathcal{T}$ that contains all the $A_i$, and $A_i=A_i[1]$ for each $ i\in I$, then $A{}^{\perp}_{\mathcal{T}'}=\cap_{ i\in I}((A_i){}^{\perp}_{\mathcal{T}'})$.
2. Since $(D[^{\perp}_{\mathcal{T}'}])_{<j}\subset C[^{\perp}_{\mathcal{T}'}]_{<j}$ by assertion 1, it suffices to verify that $C[^{\perp}_{\mathcal{T}'}]_{<j}\subset (D[^{\perp}_{\mathcal{T}'}])_{<j}$. We use induction on $j$ to prove this statement. If $j=1$, then $(D[^{\perp}_{\mathcal{T}'}])_{<j}=D[^{\perp}_{\mathcal{T}'}]_0$. According to the previous assertion, $D[^{\perp}_{\mathcal{T}'}]_0= C[^{\perp}_{\mathcal{T}'}]_{<1}\cap (D_{\geqslant n+1}){}^{\perp}_{\mathcal{T}'}=(C[^{\perp}_{\mathcal{T}'}])_{<1}$ indeed. Next, assume that $2\leqslant j\leqslant n$, $C[^{\perp}_{\mathcal{T}'}]_{<j-1}\subset (D[^{\perp}_{\mathcal{T}'}])_{<j-1}$ and $M\in C[^{\perp}_{\mathcal{T}'}]_{<j}$. Recall that $((C_{<n+2-j})^{\perp}_{\mathcal{T}'}, (C_{\geqslant n+2-j})^{\perp}_{\mathcal{T}'})$ gives a decomposition of $\mathcal{T}'$. Hence there exists a distinguished triangle with $B\in (C_{\geqslant n+2-j})^{\perp}_{\mathcal{T}'}= (D_{\geqslant n+2-j})^{\perp}_{\mathcal{T}'}$ (see Proposition 2.10, 2) and $A\in (C_{<n+2-j})^{\perp}_{\mathcal{T}'}$. Now, $(C_{<n+2-j})^{\perp}_{\mathcal{T}'}=C[^{\perp}_{\mathcal{T}'}]_{<j-1}$; hence $A\in (D[^{\perp}_{\mathcal{T}'}])_{<j-1}$. Since $M,A\in C[^{\perp}_{\mathcal{T}'}]_{<j}$, we obtain $B\in (D_{\geqslant n+2-j})^{\perp}_{\mathcal{T}'}\cap C[^{\perp}_{\mathcal{T}'}]_{<j}=D[^{\perp}_{\mathcal{T}'}]_{j-1}$. Thus, $M\in (D[^{\perp}_{\mathcal{T}'}])_{<j}$; this yields the inductive step.3. One needs just a few modifications (of ‘replacing right $\mathcal{T}'$-orthogonals by left ones’) to convert the proof of assertion 2 into the proof of assertion 3. 4. It clearly suffices to consider the case $\mathcal{T}'=\mathcal{D}$. Next, for any $j$, $1\leqslant j\leqslant n$, the category $(C_{<j})^{\coprod}_{\mathcal{D}}=(D_{<j})^{\coprod}_{\mathcal{D}}$ is triangulated, closed with respect to $\mathcal{D}$-coproducts and contains $D_i$ for $0\leqslant i<j$. Hence $(D^{\coprod}_{\mathcal{T}'})_{<j}\subset (C_{<j})^{\coprod}_{\mathcal{D}}$. We prove the reverse inclusion by induction on $j$. It is trivial in the case $j=1$. So we assume that $j\geqslant 2$ and $(C_{<j-1})^{\coprod}_{\mathcal{D}}\subset (D^{\coprod}_{\mathcal{T}'})_{<j-1}$. If suffices to verify that the class $\mathcal{C}$ of extensions of objects of $(C_{<j-1})^{\coprod}_{\mathcal{D}}$ by objects of $(D_{j-1})^{\coprod}_{\mathcal{D}}$ contains $(C_{<j})^{\coprod}_{\mathcal{D}}$. Now recall that $(C_{\geqslant j-1})^{\coprod}\perp (C_{<j-1})^{\coprod}$; hence $(D_{j-1})^{\coprod}\perp (C_{<j-1})^{\coprod}$. According to Lemma 2.1, $\mathcal{C}$ gives a triangulated subcategory of $\mathcal{D}$ that is closed with respect to $\mathcal{D}$-coproducts. Moreover, $\mathcal{C}$ contains $C_{<j}$ (see Proposition 2.10, 1); hence it contains $(C_{<j})^{\coprod}_{\mathcal{D}}$ indeed. The theorem is proved. Remark 4.24. 1. Clearly, one can dualize the notion of left admissible chains (cf. Proposition 2.5, 3); this yields chains of right admissible subcategories. For this reason parts 2 and 3 of Theorem 4.23 are not dual to each other. 2. The assumption $ (C_{\geqslant j})^{\coprod}\perp (C_{<j})^{\coprod}$ is obviously equivalent to $ C_{\geqslant j}\perp (C_{<j})^{\coprod}$. Now we are able to generalize (some statements in) Theorem 4.12. Corollary 4.25. Assume that $R$ is a Noetherian ring, $X$ is projective over $S=\operatorname{Spec} R$ and $D$ is a length $n+1$ semi-orthogonal decomposition of $\mathcal{D}_{\mathrm{p}}$. 1. Then the families $D[^{\perp}_{\mathcal{D}^{\mathrm{b}}}]$, $D[^{\perp}_{\mathcal{D}^-}]$, $D[^{\perp}_{\mathcal{D}^+}]$, $D[^{\perp}_{\mathcal{D}^{\mathrm{u}}}]$ and $D[^{\perp}_{\mathcal{D}_{\mathrm{Q}}}]$ are length $ n+1$ semi-orthogonal decompositions of $\mathcal{D}^{\mathrm{b}}$, $\mathcal{D}^-$, $\mathcal{D}^+$, $\mathcal{D}^{\mathrm{u}}$ and $\mathcal{D}_{\mathrm{Q}}$, respectively. Moreover, $D[^{\perp}_{\mathcal{D}_{\mathrm{Q}}}] =(D[^{\perp}_{\mathcal{D}^{\mathrm{b}}}])^{\coprod}$. 2. Suppose in addition that $X$ satisfies Assumption 4.9, and $E$ is a length $ n+1 $ semi-orthogonal decomposition of $\mathcal{D}^{\mathrm{b}}$. Then the map $D \mapsto D[^{\perp}_{\mathcal{D}^{\mathrm{b}}}]$ induces a one-to-one correspondence between the set of all length $ n+1 $ semi-orthogonal decompositions of $\mathcal{D}_{\mathrm{p}}$ and the set of length $n+1$ semi-orthogonal decompositions of $\mathcal{D}^{\mathrm{b}}$. The inverse map is given by $E\mapsto {[^{\perp}_{\mathcal{D}_{\mathrm{p}}}]}E$. Moreover, the families $E^{\coprod}_{\mathcal{D}^-}=(E_{<i}{}^{\coprod}_{\mathcal{D}^-})$, $E^{\coprod}_{\mathcal{D}^+}$, $E^{\coprod}_{\mathcal{D}^{\mathrm{u}}}$ and $E^{\coprod}_{\mathcal{D}_{\mathrm{Q}}}$ are length $ n+1 $ semi-orthogonal decompositions of $\mathcal{D}^-$, $\mathcal{D}^+$, $\mathcal{D}^{\mathrm{u}}$ and $\mathcal{D}_{\mathrm{Q}}$, respectively. Proof. All these statements can easily be deduced from their left admissible chain analogues provided by Proposition 4.21. For this purpose one can apply Theorem 4.23; note here that the assumptions of Theorem 4.23, parts 2 and 3, are provided by Theorem 4.12, parts I.1 and II.1 (combined with Proposition 2.5, 2).
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