2.1. Let $\mathcal {A},\ \mathcal {B}$ be full subcategories of $\mathcal {D}$ , and define the following full subcategories:

1. a) $\mathcal {A}\ast \mathcal {B}$ has for objects all the $x\in \mathcal {D}$ , such that there exists an exact triangle $a\to x\to b$ in $\mathcal {D}$ with $a\in \mathcal {A}$ and $b\in \mathcal {B}$ .

2. b) has for objects all coproducts of objects of $\mathcal {A}$ .

3. c) has for objects all direct summands of objects of $\mathcal {A}$ .

2.2. Let G be an object in $\mathcal {D}$ , and suppose $n\in \mathbb {N}$ and $A\leq B\in \mathbb {Z}\cup \{-\infty ,+\infty \}$ . We define the following full subcategories:

1. a) If $A,B\in \mathbb {Z}$ , then $G[A,B]\subset \mathcal {D}$ has objects $\{\Sigma ^{-i} G\mid A\leq i\leq B\}$ . Similarly, $G(-\infty ,B]\subset \mathcal {D}$ has objects $\{\Sigma ^{-i} G\mid i\leq B\}$ . We define $G[A,\infty )$ and $G(-\infty ,\infty )$ analogously.

2. b) is defined inductively on the integer n by setting

   

3. c) is the smallest full subcategory $\mathcal {S}\subset \mathcal {D}$ , closed under coproducts, with $\mathcal {S}\ast \mathcal {S}\subset \mathcal {S}$ and with $G[A,B]\subset \mathcal {S}$ .

4. d) .

5. e) .

2.3 [Reference Beilinson, Bernstein and Deligne2, Definition 1.3.1]

A t-structure on a triangulated category $\mathcal {D}$ is a pair of full subcategories $(\mathcal {D}^{\leq 0}, \mathcal {D}^{\geq 0})$ satisfying

1. 1. $\Sigma {\mathcal {D}^{\leq 0}} \subset {\mathcal {D}^{\leq 0}}$ and ${\mathcal {D}^{\geq 0}} \subset \Sigma {\mathcal {D}^{\geq 0}}$ .

2. 2. .

3. 3. For every $X \in \mathcal {D}$ , there exists an exact triangle $Y \to X \to Z$ with $Y \in \Sigma {\mathcal {D}^{\leq 0}}$ and $Z\in {\mathcal {D}^{\geq 0}}$ .

2.4 [Reference Beilinson, Bernstein and Deligne2, Section 1.3]

Let $\mathcal {D}$ be a triangulated category with t-structure $(\mathcal {D}^{\leq 0}, \mathcal {D}^{\geq 0})$ . For every integer $n\in \mathbb {Z}$ , we set ${\mathcal {D}^{\leq n}}:=\Sigma ^{-n}{\mathcal {D}^{\leq 0}}$ and ${\mathcal {D}^{\geq n}}:=\Sigma ^{-n}{\mathcal {D}^{\geq 0}}$ . The inclusion ${\mathcal {D}^{\leq n}}\subset \mathcal {D}$ has a right adjoint which we denote $(-)^{\leq n}$ , whereas ${\mathcal {D}^{\geq n}}\subset \mathcal {D}$ has a left adjoint which we denote by $(-)^{\geq n}$ . For every $X\in \mathcal {D}$ , the counit and unit of these adjunctions fit into an exact triangle

$$ \begin{align*}X^{\leq n} \to X \to X^{\geq n+1}.\end{align*} $$

This is, up to unique isomorphism, the only triangle $Y \to X \to Z$ with $Y \in {\mathcal {D}^{\leq n}}$ and $Z\in {\mathcal {D}^{\geq n+1}}$ . We write $\mathcal {H}^{n}:\mathcal {D}\to \mathcal {D}$ for the functor taking X to $(X^{\leq n})^{\geq n}$ .

2.5. A recollement is a diagram of exact functors between triangulated categories,

with the following properties:

1. 1. each functor is left adjoint to the one below it;

2. 2. $j^{*} i_{*} = 0$ ;

3. 3. there exist natural transformations $d: i_{*}i^{*} \to \Sigma j_{!} j^{*}$ and $d': j_{*}j^{*} \to \Sigma i_{*}i^{!}$ , such that for any $X \in \mathcal {D},$ the following are exact triangles:

   $$\begin{align*}&j_{!}j^{*}X\xrightarrow{\epsilon} X\xrightarrow{\eta} i_{*}i^{*} X \xrightarrow{d} \Sigma j_{!}j^{*}X\\&i_{*}i^{!}X\xrightarrow{\epsilon} X\xrightarrow{\eta} j_{*}j^{*} X \xrightarrow{d'} \Sigma i_{*}i^{!}X,\end{align*}$$
   where $\eta $ and $\epsilon $ are the (co)unit maps of the adjunctions;

4. 4. $i_{*}, j_{!}, j_{*}$ are fully faithful.

We note that $i^*,\ i_{*},\ j_{!},\ j^*$ preserve coproducts, since they are left adjoints. Since $i^{*}$ and $j_{!}$ have right adjoints preserving coproducts, [Reference Neeman19, Theorem 5.1] informs us that $i^{*}$ and $j_{!}$ respect compact objects.

2.6 [(Reference Beilinson, Bernstein and Deligne2, 1.4.10)]

Consider a recollement as above. Given t-structures $(\mathcal {D}^{\leq 0}_{F}, \mathcal {D}^{\geq 0}_{F})$ and $(\mathcal {D}^{\leq 0}_{U}, \mathcal {D}^{\geq 0}_{U})$ on $\mathcal {D}_{F}$ and $\mathcal {D}_{U}$ , respectively, the glued t-structure on $\mathcal {D}$ has aisle

$$ \begin{align*}{\mathcal{D}^{\leq 0}}:=\{X\in\mathcal{D}\mid i^{*}X\in {\mathcal{D}^{\leq 0}_F} \mbox{ and } j^{*}X\in{\mathcal{D}^{\leq 0}_U} \}\end{align*} $$

and coaisle

$$ \begin{align*}{\mathcal{D}^{\geq 0}}:=\{X\in\mathcal{D}\mid i^{!}X\in {\mathcal{D}^{\geq 0}_F} \mbox{ and } j^{*}X\in{\mathcal{D}^{\geq 0}_U} \}.\end{align*} $$

2.7 [(Reference Beilinson, Bernstein and Deligne2, 1.3.16)]

Let $\mathcal {D}_1$ , $\mathcal {D}_2$ be triangulated categories endowed with t-structures. A functor $f:\mathcal {D}_1\to \mathcal {D}_2$ is right t-exact if $f({\mathcal {D}^{\leq 0}_1})\subset {\mathcal {D}^{\leq 0}_2}$ , and left t-exact if $f({\mathcal {D}^{\geq 0}_1})\subset {\mathcal {D}^{\geq 0}_2}$ . We say f is t-exact if f is left and right t-exact.

Consider a recollement as in 2.5, and suppose $\mathcal {D}$ has a t-structure glued from t-structures on $\mathcal {D}_{F}$ and $\mathcal {D}_{U}$ . By [Reference Beilinson, Bernstein and Deligne2, 1.3.17(iii)], the functors $i^*,\ j_{!}$ are right t-exact, $i_*,\ j^*$ are t-exact, and $i^!,\ j_*$ are left t-exact.

2.8. Let $\mathcal {D}$ be a triangulated category with coproducts. An object $G\in \mathcal {D}$ is compact if the functor respects coproducts. We call G a compact generator for $\mathcal {D}$ if for all $X\in \mathcal {D}$ , we have that for all $i\in \mathbb {Z}$ implies $X\cong 0$ .

2.9 [Reference Neeman16, Definition 0.21]

A triangulated category $\mathcal {D}$ with coproducts is weakly approximable if there exists a compact generator $G,$ a t-structure $(\mathcal {D}^{\leq 0}, \mathcal {D}^{\geq 0}),$ and an integer $A> 0,$ such that the following hold:

1. (1) $\Sigma ^{A} G \in {\mathcal {D}^{\leq 0}}$ and

2. (2) For every $X \in {\mathcal {D}^{\leq 0}},$ there exists an exact triangle $E \to X \to D$ with $E \in \unicode{xf8} {\langle G \rangle }^{[-A,A]}$ and $D \in {\mathcal {D}^{\leq -1}}.$

If Properties $(1)$ and $(2)$ hold for a compact generator G and integer $A,$ then for every compact generator H, there is an integer $A_{H}$ for which they hold, by [Reference Neeman16, Proposition 2.6].

The category $\mathcal {D}$ is approximable if we can choose $A> 0$ , such that, moreover,

1. (3) in the exact triangle $E \to X \to D$ above, we may assume $E \in \unicode{xf8} {\langle G \rangle }^{[-A,A]}_{A}.$

If Properties $(1)$ , $(2)$ , and $(3)$ hold for a compact generator G and integer $A,$ then for every compact generator H, there is an integer $A_{H}$ for which they hold, by [Reference Neeman16, Proposition 2.6].

Proof. Write $(\mathcal {D}^{\leq 0}, \mathcal {D}^{\geq 0})$ , $(\mathcal {D}'^{\leq 0}, \mathcal {D}'^{\geq 0})$ for the t-structures on $\mathcal {D}$ obtained by gluing $(\mathcal {D}^{\leq 0}_{F}, \mathcal {D}^{\geq 0}_{F})$ , $(\mathcal {D}^{\leq 0}_{U}, \mathcal {D}^{\geq 0}_{U})$ and $({\mathcal {D}'}^{\leq 0}_{F}, {\mathcal {D}'}^{\geq 0}_{F})$ , $({\mathcal {D}'}^{\leq 0}_{U}, {\mathcal {D}'}^{\geq 0}_{U})$ , respectively. If $A>0$ is an integer such that ${\mathcal {D}^{\leq -A}_{F}}\subset {\mathcal {D}'}_{F}^{\leq 0}\subset {\mathcal {D}^{\leq A}_{F}}$ and ${\mathcal {D}^{\leq -A}_U}\subset {\mathcal {D}'}^{\leq 0}_U\subset {\mathcal {D}^{\leq A}_U}$ , then ${\mathcal {D}^{\leq -A}}\subset {\mathcal {D}'}^{\leq 0}\subset {\mathcal {D}^{\leq A}}$ .

Proof. Being nondegenerate is clearly stable under equivalence, so it suffices to show the t-structure $({\mathcal {D}^{\leq 0}_{G}}, {\mathcal {D}^{\geq 0}_{G}})$ generated by G is nondegenerate. Let X be a nonzero object in $\mathcal {D}$ . Since G is a compact generator for $\mathcal {D}$ , there exists a nonzero morphism $\Sigma ^l G\to X$ for some $l\in \mathbb {Z}$ . It follows that $X\notin {\mathcal {D}^{\geq -l+1}_G}=({\mathcal {D}^{\leq -l}_G})^{\perp }$ . Moreover, by the first part of Lemma 3.5, there exists $C>0$ , such that . Hence, , and we can conclude that $X\notin {\mathcal {D}^{\leq -C-l}_G}$ .

Proof. The t-structure gives us an exact triangle $X^{\leq 0}\to X\to X^{\geq 1}$ , and we may choose a triangle $E\to X^{\leq 0}\to D$ with $E\in \unicode{xf8} {\langle G \rangle }^{[-A,A]}$ and $D\in \mathcal {D}^{\leq -1}$ . We are assuming that $^\perp X$ contains $\{\Sigma ^iG,\,-A\leq i\leq A\}$ , and hence, $^\perp X$ must contain $\unicode{xf8} {\langle G \rangle }^{[-A,A]}$ . Therefore, the composite $E\to X^{\leq 0}\to X$ must vanish, and the map $X^{\leq 0}\to X$ must factor as $X^{\leq 0}\to D\to X$ . Applying the functor $\mathcal {H}^{0}$ , we deduce that the isomorphism $\mathcal {H}^{0}(X^{\leq 0})\to \mathcal {H}^{0}(X)$ factors through $\mathcal {H}^{0}(D)=0$ .

Proof. With the hypotheses as in the current lemma, Lemma 3.8 guarantees the vanishing of $\mathcal {H}^{i}(X)$ for all $-A\leq i\leq A$ . The triangles $X^{\leq i-1}\to X^{\leq i}\to \mathcal {H}^{i}(X)$ then tell us that $X^{\leq A}$ belongs to $\mathcal {D}^{\leq -A-1}$ . Consider the triangle $W\stackrel \varphi \to X\stackrel \psi \to Y$ with $W:=X^{\leq A}\in \mathcal {D}^{\leq -A-1}$ and $Y:=X^{\geq A+1}\in \mathcal {D}^{\geq A+1}$ .

The integer A is such that , and hence, vanishes for all $i\leq 1$ . But we also know that $\Sigma ^AG\in \mathcal {D}^{\leq 0}$ , and hence, . Therefore, must vanish for all $i\geq 0$ . It follows that, for every integer $i\in \mathbb {Z}$ ,

Thus, in the exact sequence

we have that, for every $i\in \mathbb {Z}$ , either the map or the map is an isomorphism. More precisely: the map is an isomorphism when $i\leq 1$ , while the map is an isomorphism when $i\geq 0$ .

We have already proved the vanishing of for all $i\leq 0$ and the vanishing of for all $i\geq 0$ . For $1\leq i\leq 2A$ , we have isomorphisms

which extend the vanishing statements of (i) and (ii) to the full range asserted. And the assertions in (i) and (ii), about the ranges in which and are isomorphisms, follow from the vanishing statements combined with the long exact sequence $(\dagger \dagger )$ above.

Now for (iv) and (v): the proof of (i) and (ii) produced an explicit triangle $W\to X\to Y$ , with $W\in \mathcal {D}^{\leq -A-1}$ and with $Y\in \mathcal {D}^{\geq A+1}$ , and showed that this particular choice of triangle satisfies (i) and (ii). If we strengthen the assumption on X as in (iv), then for all $i\in \mathbb {Z}$ , and as G is a compact generator, this forces $Y=0$ . This in turn means that $W\to X$ must be an isomorphism, but by construction, W belongs to $\mathcal {D}^{\leq -A-1}$ . Under the hypotheses of (v), we would deduce that $W=0$ , the map $X\to Y$ is an isomorphism, but by construction, Y belongs to $\mathcal {D}^{\geq A+1}$ .

Finally, assume we are given a triangle $W\to X\to Y$ , as in the hypotheses. By (iv), we have $W\in \mathcal {D}^{\leq -A-1}$ , and by (v), we have $Y\in \mathcal {D}^{\geq A+1}$ , and (iii) now follows immediately.

Proof. Write $({\mathcal {D}^{\leq 0}_{U}}, {\mathcal {D}^{\geq 0}_{U}})$ for the t-structure on $\mathcal {D}_{U}$ generated by $G_{U}$ . We note that $j_{!}G_U$ is compact in $\mathcal {D}$ because the functor $j_{!}$ preserves compactness (see 2.5). The second part of Lemma 3.5 now shows

for $n \gg 0$ . It follows that $j^{*} G'\in {\mathcal {D}^{\leq m}_{U}}$ for some $m>0$ by Corollary 3.10. Now let

which is clearly a compact generator for $\mathcal {D}$ . We write $({\mathcal {D}^{\leq 0}_{F}}, {\mathcal {D}^{\geq 0}_{F}})$ for the t-structure on $\mathcal {D}_{F}$ generated by $i^{*} G$ and $({\mathcal {D}^{\leq 0}_{\text {gl}}}, {\mathcal {D}^{\geq 0}_{\text {gl}}})$ for the t-structure on $\mathcal {D}$ obtained by gluing $({\mathcal {D}^{\leq 0}_{F}}, {\mathcal {D}^{\geq 0}_{F}})$ and $({\mathcal {D}^{\leq 0}_{U}}, {\mathcal {D}^{\geq 0}_{U}})$ . It remains to show that

$$ \begin{align*}{\mathcal{D}^{\leq 0}_G}={\mathcal{D}^{\leq 0}_{\text{gl}}},\end{align*} $$

where $({\mathcal {D}^{\leq 0}_{G}}, {\mathcal {D}^{\geq 0}_{G}})$ is the t-structure on $\mathcal {D}$ generated by G. First, we note that $i^{*}G\in {\mathcal {D}^{\leq 0}_{F}}$ and . Using that the functors $i^{*}$ and $j^{*}$ commute with coproducts, this shows $i^{*}{\mathcal {D}^{\leq 0}_{G}}\subset {\mathcal {D}^{\leq 0}_{F}}$ and $j^{*}{\mathcal {D}^{\leq 0}_{G}}\subset {\mathcal {D}^{\leq 0}_{U}}$ . Hence,

$$ \begin{align*}{\mathcal{D}^{\leq 0}_{G}}\subset {\mathcal{D}^{\leq 0}_{\text{gl}}}.\end{align*} $$

On the other hand, we clearly have $j_{!}G_U\in {\mathcal {D}^{\leq 0}_G}$ , and thus $j_{!}{\mathcal {D}^{\leq 0}_{U}}\subset {\mathcal {D}^{\leq 0}_{G}}$ , because the functor $j_{!}$ commutes with coproducts. Since $i_{*}i^{*}G$ fits in an exact triangle

$$ \begin{align*}j_{!}j^{*}G\to G\to i_{*}i^{*}G\end{align*} $$

with $j_{!}j^{*}G, \,G\in {\mathcal {D}^{\leq 0}_{G}}$ , we see that $i_{*}i^{*}G\in {\mathcal {D}^{\leq 0}_{G}}$ . It follows that $i_{*}\mathcal {D}^{\leq 0}_{F}\subset {\mathcal {D}^{\leq 0}_{G}}$ . Now, let $X\in {\mathcal {D}^{\leq 0}_{\text {gl}}}$ and consider the exact triangle

$$ \begin{align*}j_{!}j^{*}X\to X\to i_{*}i^{*}X.\end{align*} $$

By the above, $j_{!}j^{*}X,\, i_{*}i^{*}X\in {\mathcal {D}^{\leq 0}_{G}}$ , so $X\in {\mathcal {D}^{\leq 0}_{G}}$ . We conclude that ${\mathcal {D}^{\leq 0}_{G}}= {\mathcal {D}^{\leq 0}_{\text {gl}}}$ .

Proof. For the categories $(\mathcal {D}_U)^-_c$ , $(\mathcal {D}_U)^b_c$ , $\mathcal {D}^-_c$ , and $\mathcal {D}_c^b$ , the assertion is immediate from Remark 3. What we will prove is that the category $\mathcal {D}_F$ has a compact generator H with for $n \gg 0$ .

Choose for $\mathcal {D}_F,\ \mathcal {D}_U$ t-structures in the preferred equivalence class, and glue them to form a t-structure on $\mathcal {D}$ . By Corollary 3.12, the glued t-structure on $\mathcal {D}$ is in the preferred equivalence class. Pick a compact generator $G\in \mathcal {D}$ .

Lemma 3.5(1) allows us to choose an integer $C>0$ with . As $i_*$ is t-exact, we have $i_*\mathcal {D}_F^{\leq -C}\subset \mathcal {D}^{\leq -C}$ , and hence

But the t-structure on $\mathcal {D}_F$ is in the preferred equivalence class, and [Reference Neeman16, Observation 0.20(ii)] informs us that the compact object $i^*G\in \mathcal {D}_F$ must be contained in $\mathcal {D}_F^-$ . Hence, we may choose an integer $B>0$ with $i^*G\in \mathcal {D}_F^{\leq B}$ .

But now it’s immediate that the compact generator $i^*G\in \mathcal {D}_F$ is such that for $n\geq B+C$ .

Proof. Choose t-structures for each of $\mathcal {D}_F,\ \mathcal {D}_U$ , in the preferred equivalence classes, and glue them to form a t-structure on $\mathcal {D}$ . By Corollary 3.12, the glued t-structure on $\mathcal {D}$ is in the preferred equivalence class. We are assuming that, for the compact generator $G\in \mathcal {D}$ , we have $j^{*} G \in (\mathcal {D}_{U})^{-}_{c}$ . As $ (\mathcal {D}_{U})^{-}_{c}$ is a thick triangulated subcategory which contains $j^{*}G$ , it must also contain $j^*\mathcal {D}^c$ , since $\mathcal {D}^c$ is the smallest thick triangulated subcategory of $\mathcal {D}$ containing G.

We first show the left sides are contained in the right sides. Let X be in $\mathcal {D}^{-}_{c},$ fix $m> 0,$ and let $E \to X \to D$ be an exact triangle in $\mathcal {D}$ with $E \in \mathcal {D}^{c}$ and $D \in {\mathcal {D}^{\leq -m}}.$ Since $i^{*} D \in \mathcal {D}^{\leq -m}_{F}$ and $i^{*}$ preserves compactness (see 2.5), the exact triangle $i^{*} E \to i^{*} X \to i^{*} D$ shows that $i^{*} X$ is in $(\mathcal {D}_{F})_{c}^{-}.$ Now consider the exact triangle $j^{*} E \to j^{*} X \to j^{*} D$ with $j^{*} D \in \mathcal {D}_{U}^{\leq -m}$ . By the first paragraph of the proof, $j^*E \in (\mathcal {D}_{U})^{-}_{c}$ , hence, we can find an exact triangle ${\widetilde {E}} \to j^{*} E \to {\widetilde {D}}$ with ${\widetilde {E}}$ in $(\mathcal {D}_{U})^{c}$ and ${\widetilde {D}}$ in $\mathcal {D}_{U}^{\leq -m}.$ The octahedral axiom, applied to ${\widetilde {E}} \to j^{*}E \to j^{*}X$ , gives an object $D'$ and exact triangles ${\widetilde {E}} \to j^{*} X \to D'$ and ${\widetilde {D}} \to D' \to j^{*} D$ in $\mathcal {D}_U$ , such that the following diagram is commutative:

Since ${\widetilde {D}}$ and $j^{*} D$ are in $\mathcal {D}_{U}^{\leq -m},$ we know $D'$ is also in $\mathcal {D}_{U}^{\leq -m}$ . The exact triangle ${\widetilde {E}} \to j^{*} X \to D'$ now shows that $j^{*} X$ is in $(\mathcal {D}_{U})_{c}^{-}$ . If we assume further that X is in $\mathcal {D}^{b}_{c} = \mathcal {D}^{-}_{c} \cap \mathcal {D}^{b},$ then we have $j^{*} X \in \mathcal {D}^{b}_{U}$ and $i^{!} X \in (\mathcal {D}_{F})^{+}$ because $j^*$ is t-exact and $i^!$ is left t-exact.

We now show the right sides are contained in the left sides. Assume $i^{*} X \in (\mathcal {D}_{F})^{-}_{c}$ and $j^{*} X \in (\mathcal {D}_{U})^{-}_{c}$ , and fix $m> 0$ . By definition, there exists an exact triangle

$$\begin{align*}E' \to i^{*} X \to D'\end{align*}$$

with $E' \in (\mathcal {D}_{F})^{c}\subset \mathcal {D}_{F}^-$ and $D ' \in \mathcal {D}_{F}^{\leq -m}$ . Choose an odd integer $n>0$ with $\Sigma ^n E'\in \mathcal {D}_F^{\leq -m}$ . The object vanishes in $K_0((\mathcal {D}_{F})^{c})$ , and [Reference Neeman20, Corollary 4.5.14] tells us that must therefore lie in the image $i^*(\mathcal {D}^{c})$ . And then [Reference Neeman20, Proposition 4.4.1], applied to the composite , allows us to find a morphism $E" \to X$ in $\mathcal {D},$ with $E"$ in $\mathcal {D}^{c},$ and such that $i^{*}(E" \to X)$ is isomorphic to Complete $E" \to X$ to an exact triangle

$$ \begin{align*}E" \to X \to D".\end{align*} $$

Since , we deduce that $i^*D"\in \mathcal {D}_F^{\leq -m}$ .

The first paragraph of the proof tells us that $j^{*} E" \in (\mathcal {D}_{U})_{c}^{-}$ , while $j^{*} X \in (\mathcal {D}_{U})_{c}^{-}$ by hypothesis. Thus, $j^{*} D"$ is in $(\mathcal {D}_{U})_{c}^{-}$ . Fix an exact triangle ${\widetilde {E}} \to j^{*} D" \to {\widetilde {D}}$ with ${\widetilde {E}} \in (\mathcal {D}_{U})^{c}$ and ${\widetilde {D}} \in \mathcal {D}^{\leq -m}_{U}.$ We now have the following diagram

with $E"\in \mathcal {D}^{c}, \ {\widetilde {E}} \in (\mathcal {D}_{U})^{c}$ , and ${\widetilde {D}} \in \mathcal {D}_{U}^{\leq -m}$ . Applying the octahedral axiom, to $j_{!} {\widetilde {E}} \to j_{!}j^{*} D" \to D"$ , we find an object D and exact triangles $j_{!} {\widetilde {E}} \to D"\to D$ and $j_{!}{\widetilde {D}} \to D \to i_{*} i^{*} D"$ in $\mathcal {D}$ that fit into the following commutative diagram:

Since $i^{*} D" \in \mathcal {D}_{F}^{\leq -m}$ and $i_{*}$ is t-exact, we know $i_{*}i^{*} D" \in {\mathcal {D}^{\leq -m}}$ . Since ${\widetilde {D}} \in {\mathcal {D}^{\leq -m}_{U}}$ and $j_{!}$ is right t-exact, we know $j_{!} {\widetilde {D}} \in {\mathcal {D}^{\leq -m}}.$ Thus, $D \in {\mathcal {D}^{\leq -m}} .$

Next, we complete the octahedron on $X \to D" \to D$ and find an object E with exact triangles $E \to X \to D$ and $E" \to E \to j_{!} {\widetilde {E}}$ in $\mathcal {D}$ , fitting into the following commutative diagram:

Note that $j_{!}{\widetilde {E}}$ is compact because $j_{!}$ preserves compactness. Since $E"$ is also compact, so is E. Hence, the exact triangle $E \to X \to D$ shows that X is in $\mathcal {D}^{-}_{c}.$ If we now further assume that $i^{!} X \in (\mathcal {D}_{F})^{+}$ and $j^{*} X \in (\mathcal {D}_{U})^{b},$ then $X \in \mathcal {D}^{b}$ by the definition of glued t-structure.

Proof. Choose t-structures for each of $\mathcal {D}_F,\ \mathcal {D}_U$ , in the preferred equivalence classes, and glue them to form a t-structure on $\mathcal {D}$ . By Corollary 3.12, the glued t-structure on $\mathcal {D}$ is in the preferred equivalence class, hence, equivalent to the given t-structure on $\mathcal {D}$ . Since replacing the given t-structure by the equivalent glued one is harmless, let us assume that the t-structure on $\mathcal {D}$ is the one obtained from the gluing.

Now choose a compact generator $G\in \mathcal {D}$ and a compact generator $H\in \mathcal {D}_U$ . The object is a compact generator for $\mathcal {D}$ , and Lemma 3.5 permits us to choose an integer $C>0$ with . Assume also that $C\geq A$ , where $A>0$ is the integer given in the lemma. By the weak approximability of $\mathcal {D}_{U}$ and Lemma 3.7, we may, possibly at the cost of increasing C, assume that every object $Z \in \mathcal {D}^{\leq 0}_{U}$ admits an exact triangle $E \to Z \to D$ with $E \in \unicode{xf8} {\langle H \rangle }^{[-C,C]}$ and $D \in \mathcal {D}^{\leq -1}_{U}$ . We assert that in the lemma, we may set $B=3C-1$ .

By [Reference Neeman20, Proposition 4.4.1], we can represent $f':i^{*}K\rightarrow i^{*}X$ in $\mathcal {D}_F$ by a roof in $\mathcal {D}$ ,

such that Y is compact, $i^{*}(\varphi )$ is an isomorphism in $\mathcal {D}_F$ , and $f' i^{*}(\varphi )=i^{*}({\widetilde {f}})$ . Since $i^{*}(\varphi )$ is an isomorphism, we can find an exact triangle $j_{!}Z\rightarrow Y\xrightarrow {\varphi } K$ for some $Z\in \mathcal {D}_U$ . Moreover, we know $Y\in \mathcal {D}^c\subset \mathcal {D}^-$ and $K\in {\mathcal {D}^{\leq A}}$ . Hence, $j_{!}Z\in {\mathcal {D}^{\leq N}}$ for some N, and if $N\leq 3C-1$ , we are done. Assume, therefore, $N\geq 3C$ . The isomorphism $Z\cong j^{*}j_{!}Z$ tells us that $Z\in \mathcal {D}^{\leq N}_U$ . By [Reference Neeman16, 2.2.1] (with $m =N+1-3C$ and $F = \Sigma ^{N} Z$ ), there is an exact triangle $E\rightarrow Z\rightarrow D$ in $\mathcal {D}_U$ with $E \in \unicode{xf8} {\langle H\rangle }^{[2C,N+C]}$ and $D \in \mathcal {D}^{\leq 3C-1}_{U}$ . We have the diagram

which we complete to an octahedron, giving an object $Y'$ and exact triangles $j_{!}E\rightarrow Y\rightarrow Y'$ and $j_{!}D\rightarrow Y'\xrightarrow {\psi } K$ in $\mathcal {D}$ , making the following diagram commutative:

Using that , we see that there are no nonzero maps from $j_{!}E\in \unicode{xf8} {\langle j_{!}H\rangle }^{[2C,N+C]}$ to $X\in {\mathcal {D}^{\leq A}}\subset {\mathcal {D}^{\leq C}}$ . Hence, the morphism ${\widetilde {f}}: Y \rightarrow X$ factors via $Y\rightarrow Y'$ :

We have thus found morphisms

such that $i^{*}(\psi )$ is an isomorphism in $\mathcal {D}_F$ and $f' i^{*}(\psi )=i^{*}(f)$ . Furthermore, the exact triangle $j_{!}D\rightarrow Y'\rightarrow K$ with $j_{!}D \in {\mathcal {D}^{\leq 3C-1}}$ and $K\in {\mathcal {D}^{\leq A}}$ shows $Y'~\in ~{\mathcal {D}^{\leq 3C-1}}$ . This proves the lemma.

Proof. Corollary 3.12 allows us to choose t-structures for $\mathcal {D}_F,\ \mathcal {D}_U$ in the preferred equivalence classes, glue them to form a t-structure on $\mathcal {D}$ , and replace the given t-structure on $\mathcal {D}$ by the equivalent glued one we have just constructed.

Now choose a compact generator $H\in \mathcal {D}_U$ . The object G is a compact generator for $\mathcal {D}$ , and Lemma 3.5(1) permits us to choose an integer $C>0$ with . Since G is a compact generator in $\mathcal {D}$ and $j_!H$ is compact, [Reference Neeman16, Lemma 0.9(i)] allows us to increase C so that $j_!H\in \unicode{xf8} {\langle G\rangle }^{[-C,C]}_C$ . Let $A>0$ be the integer given in the lemma, and let $A'>0$ be an integer with $G\in \mathcal {D}^{\leq A'}$ ; Lemma 5.2 permits us to produce an integer $B'>0$ so that any morphism $i^*K\to i^*X$ , with $K\in \mathcal {D}^c\cap \mathcal {D}^{\leq A+A'}$ and $X\in \mathcal {D}^{\leq A+A'}$ , can be represented by a roof with $Y\in \mathcal {D}^{\leq B'}$ . At the cost of possibly increasing C, assume $B'\leq C$ and $A+A'\leq C$ . Finally, the weak approximability (respectively, approximability) of $\mathcal {D}_{U}$ and Lemma 3.7 permits us, possibly at the cost of increasing C, to assume that every object $Z \in \mathcal {D}^{\leq 0}_{U}$ admits an exact triangle $E \to Z \to D$ with $D \in \mathcal {D}^{\leq -1}_{U}$ and $E \in \unicode{xf8} {\langle H \rangle }^{[-C,C]}$ (respectively, $E \in \unicode{xf8} {\langle H \rangle }_C^{[-C,C]}$ ). We assert that in the lemma, we may set $B=\max (4C,3C^3+1)$ .

It suffices to show the lemma for $E' = \Sigma ^{n} i^*G$ with $n \in [-A, A]$ . Consider a morphism $f':\Sigma ^{n} i^*G\to i^{*} X$ in $\mathcal {D}_F$ with $X\in {\mathcal {D}^{\leq A}}$ . By the choice of C, there exists an object Y in ${\mathcal {D}^{\leq C}}$ and morphisms

such that $i^{*}(\psi )$ is an isomorphism in $\mathcal {D}_F$ and $f'i^{*}(\psi )=i^{*}({\widetilde {f}})$ . In particular, we can find an exact triangle $Y\xrightarrow {\psi } \Sigma ^n G\to j_{!} Z$ for some $Z\in \mathcal {D}_U$ . Since $\Sigma ^{n} G$ and Y are both in ${\mathcal {D}^{\leq C}} ,$ so is $j_{!} Z$ . Hence, $Z\cong j^{*} j_{!} Z\in \mathcal {D}^{\leq C}_{U}$ . By [Reference Neeman16, 2.2.1 and 2.2.2] (with $m = 3C$ and $F=\Sigma ^CZ$ ), there exists an exact triangle in $\mathcal {D}_U$

$$ \begin{align*} {\widetilde{E}} \to Z \to {\widetilde{D}}, \end{align*} $$

with ${\widetilde {D}} \in \mathcal {D}^{\leq -2C}_{U}$ and ${\widetilde {E}}\in \unicode{xf8} {\langle H \rangle }^{[1-3C, 2C]}$ , or if $\mathcal {D}_U$ is approximable with ${\widetilde {E}}\in \unicode{xf8} {\langle H \rangle }^{[1-3C, 2C]}_{3C^2}$ . We thus get the following diagram in $\mathcal {D}$ :

Since $j_{!}$ is right t-exact, we have $j_{!} {\widetilde {D}} \in {\mathcal {D}^{\leq -2C}} $ . Since , there are no nonzero maps $\Sigma ^{n}G \to j_{!} {\widetilde {D}}$ . It follows that $\Sigma ^{n}G \to j_!Z$ factors via $j_{!} {\widetilde {E}} \to j_!Z$ . We thus find a morphism $\Sigma ^{n} G \to j_{!} {\widetilde {E}}$ , an exact triangle $E\xrightarrow {\varphi } \Sigma ^{n} G \to j_{!} {\widetilde {E}}$ , and a morphism $E\xrightarrow {g} Y$ , such that the following diagram commutes:

Since $j_{!}H\in \unicode{xf8} {\langle G\rangle }^{[-C,C]}_C$ and ${\widetilde {E}}$ is in $\unicode{xf8} {\langle H \rangle }^{[1-3C, 2C]}$ (respectively, ${\widetilde {E}}\in \unicode{xf8} {\langle H \rangle }^{[1-3C, 2C]}_{3C^2}$ ), we see that $j_{!}{\widetilde {E}}\in \unicode{xf8} {\langle G\rangle }^{[1-4C,3C]}$ (respectively, $j_!{\widetilde {E}}\in \unicode{xf8} {\langle G \rangle }^{[1-4C, 3C]}_{3C^3}$ ). The triangle

$$ \begin{align*}E\to \Sigma^n G\to j_{!}{\widetilde{E}}\end{align*} $$

now shows that $E \in \unicode{xf8} {\langle G\rangle }^{[-4C, 4C]}$ (respectively, $E \in \unicode{xf8} {\langle G\rangle }^{[-4C, 4C]}_{3C^3+1}$ ). Finally, we let $f:={\widetilde {f}} g: E\to X$ . Now $i^{*}(\varphi )$ is an isomorphism in $\mathcal {D}_F$ and

$$ \begin{align*}f'i^{*}(\varphi)=f'i^{*}(\psi)i^{*}(g)=i^{*}({\widetilde{f}g})=i^{*}(f).\\[-37pt]\end{align*} $$

Proof. Corollary 3.12 allows us to choose t-structures for $\mathcal {D}_F,\ \mathcal {D}_U$ in the preferred equivalence classes, glue them to form a t-structure on $\mathcal {D}$ , and replace the given t-structure on $\mathcal {D}$ by the equivalent glued one we have just constructed.

The case $n=1$ is shown in Lemma 5.3. We proceed by induction on n, and suppose the lemma holds for some $n\geq 1$ . Fix $A>0$ , choose ${\widetilde {B}}$ as in Lemma 5.3, meaning any morphism $f':E'\to i^*X$ in the category $\mathcal {D}_F$ , with and $X\in {\mathcal {D}^{\leq A}}$ , is isomorphic to $i^*f$ for some morphism $f:E\rightarrow X$ in $\mathcal {D}$ with $E\in \unicode{xf8} {\langle G \rangle }_{{\widetilde {B}}}^{[-{\widetilde {B}},{\widetilde {B}}]}$ . Increasing ${\widetilde {B}}$ if necessary, assume ${\widetilde {B}}\geq A$ and $G\in \mathcal {D}^{\leq {\widetilde {B}}}$ .

Using the induction hypothesis, choose $B'$ , such that, if $f':E'\rightarrow i^{*}X $ is a morphism in $\mathcal {D}_{F}$ with and $X\in {\mathcal {D}^{\leq 2{\widetilde {B}}}}$ , then there exists a morphism $f:E\rightarrow X$ in $\mathcal {D}$ with $E\in \unicode{xf8} {\langle G \rangle }_{B'}^{[-B',B']}$ , such that $i^{*}E\cong E'$ and $i^{*}f\cong f'$ .

Consider a morphism $f':E'\to i^{*}X$ with and $X\in {\mathcal {D}^{\leq A}}.$ By definition, there exists an exact triangle $E_{1}' \to E' \to E^\prime _{n} $ with and . We can complete the octahedron on $E_1'\rightarrow E'\rightarrow i^{*}X$ to find exact triangles $E'\to i^*X\to Z'$ , $E^\prime _1\to i^*X\to Y'$ , and $E^\prime _n\to Y'\to Z'$ , such that the diagram

commutes. Since $E^\prime _{1}$ is in and X is in ${\mathcal {D}^{\leq A}}$ , we can apply our choices to find an exact triangle $E_1\to X\to Y$ in $\mathcal {D}$ with $E_1\in \unicode{xf8} {\langle G \rangle }_{{\widetilde {B}}}^{[-{\widetilde {B}},{\widetilde {B}}]}$ and such that $i^{*}(E_1 \to X \to Y)\cong E_1'\to i^{*}X\to Y'$ . Note that

$$ \begin{align*}Y\in{\mathcal{D}^{\leq A}}*\unicode{xf8}{\langle G \rangle}_{{\widetilde{B}}}^{[-{\widetilde{B}}-1,{\widetilde{B}}-1]}\subseteq {\mathcal{D}^{\leq 2{\widetilde{B}}}}.\end{align*} $$

Next, we consider the morphism $E_n'\to Y'\cong i^{*}Y$ with and $Y\in {\mathcal {D}^{\leq 2{\widetilde {B}}}}$ . By our choices of integers, there exists an exact triangle $E_n\to Y\to Z$ in $\mathcal {D}$ with $E_n\in \unicode{xf8} {\langle G \rangle }_{B'}^{[-B',B']}$ and such that $i^{*}(E_n\to Y\to Z)\cong E_n'\to Y'\to Z'$ . We now have the following diagram:

Completing the octahedron, we find an object E and exact triangles $E_{1} \to E \to E_{n}$ and $E \xrightarrow {f} X \to Z$ in $\mathcal {D}$ , so that the following diagram is commutative:

Since there are isomorphisms $i^{*}(X\to Y)\cong i^{*}X\to Y'$ and $i^{*}(Y\to Z)\cong Y'\to Z'$ , there is an isomorphism $i^{*}f\cong f'$ . The lemma follows since

$$ \begin{align*}E \in \unicode{xf8}{\langle G \rangle}_{{\widetilde{B}}}^{[-{\widetilde{B}},{\widetilde{B}}]}*\unicode{xf8}{\langle G \rangle}_{B'}^{[-B',B']} \subset \unicode{xf8}{\langle G \rangle}_{{\widetilde{B}}+B'}^{[-{\widetilde{B}}-B',{\widetilde{B}}+B']}.\end{align*} $$

Proof. Fix $A>0$ . In [Reference Neeman15, Corollary 1.12], it is shown that

By Lemma 5.4, there exists $B>0$ , such that if $f':E'\rightarrow i^{*}X $ is a morphism in $\mathcal {D}_{F}$ with and $X\in {\mathcal {D}^{\leq 2A}}$ , then there exists $E~\in ~\unicode{xf8} {\langle G \rangle }_B^{[-B,B]}$ and a morphism $f:E\rightarrow X$ in $\mathcal {D}$ with $i^{*}E\cong E'$ and $i^{*}f\cong f'$ .

Proof of Theorem 5.1

Let $G_{F}$ and $G_{U}$ be compact generators of $\mathcal {D}_F$ and $\mathcal {D}_U$ , respectively. By [Reference Neeman20, Theorem 4.4.9], there exists a compact object $G'$ in $\mathcal {D}$ , such that $G_{F}$ is a direct summand of $i^{*} G'$ . It follows that is a compact generator for $\mathcal {D}$ .

Suppose now that for $n \gg 0$ ; we’re in the situation of Corollary 3.12, allowing us to choose t-structures for $\mathcal {D}_F,\ \mathcal {D}_U$ in the preferred equivalence classes, glue them to form a t-structure on $\mathcal {D}$ , and we’re guaranteed that the glued t-structure on $\mathcal {D}$ is in the preferred equivalence class. Choose a compact generator $H\in \mathcal {D}_U$ , and choose an integer $A>0$ so that $G\in \mathcal {D}^{\leq A}$ , , and $j_!H\in {\langle G \rangle }_{A}^{[-A,A]}$ . Recalling that $\mathcal {D}_U$ and $\mathcal {D}_F$ are approximable and remembering Lemma 3.7, we may, by increasing A if necessary, also assume that every object $X\in \mathcal {D}^{\leq 0}_U$ admits an exact triangle $E\to X\to D$ with $E\in \unicode{xf8} {\langle H \rangle }_{A}^{[-A,A]}$ and $D\in \mathcal {D}^{\leq -1}_U$ , and every object $X\in \mathcal {D}^{\leq 0}_F$ admits an exact triangle $E\to X\to D$ with $E\in \unicode{xf8} {\langle i^*G \rangle }_{A}^{[-A,A]}$ and $D\in \mathcal {D}^{\leq -1}_F$ . By Lemma 5.5, there exists an integer $B>0$ , such that, if $f':E'\rightarrow i^{*}X $ is any morphism in $\mathcal {D}_{F}$ with $E'\in \unicode{xf8} {\langle i^*G \rangle }_{A}^{[-A,A]}$ and $X\in {\mathcal {D}^{\leq A}}$ , then there exists a morphism $f: E \to X$ in $\mathcal {D}$ with $E\in \unicode{xf8} {\langle G \rangle }_{B}^{[-B,B]}$ , such that $i^{*} f \cong f'.$

Let X be an object of ${\mathcal {D}^{\leq 0}}$ , hence, $i^{*} X \in \mathcal {D}^{\leq 0}_{F}.$ We find an exact triangle

$$ \begin{align*}E'\xrightarrow{f'} i^{*}X\to D'\end{align*} $$

with $E'\in \unicode{xf8} {\langle i^*G \rangle }_{A}^{[-A,A]}$ and $D'\in \mathcal {D}_F^{\leq -1}$ . By the choices of integers above, there exists a morphism $E \xrightarrow {f} X $ in $\mathcal {D}$ with $E\in \unicode{xf8} {\langle G \rangle }_{B}^{[-B,B]}$ , such that $i^{*}f\cong f' $ . Complete f to a triangle

$$ \begin{align*}E \xrightarrow{f} X \to D".\end{align*} $$

Since $E \in \unicode{xf8} {\langle G \rangle }_{B}^{[-B,B]}$ and $X\in {\mathcal {D}^{\leq 0}},$ we have $D" \in {\mathcal {D}^{\leq A+B}},$ and thus, $j^{*} D" \in \mathcal {D}^{\leq A+B}_{U}.$ By [Reference Neeman16, 2.2.1] (with $F=\Sigma ^{A+B}j^*D"$ and $m=A+B+1$ ), we find an exact triangle

$$ \begin{align*}{\widetilde{E}}\to j^{*} D"\to {\widetilde{D}}\end{align*} $$

in $\mathcal {D}_U$ , such that ${\widetilde {E}}\in \unicode{xf8} {\langle H \rangle }_{(A+B+1)A}^{[-A,2A+B]}$ and ${\widetilde {D}}\in \mathcal {D}^{\leq -1}_U$ , obtaining a diagram

Completing the octahedron on $j_{!} {\widetilde {E}} \to j_{!}j^{*} D" \to D"$ , we find an object D with exact triangles $j_{!} {\widetilde {E}} \to D"\to D$ and $j_{!}{\widetilde {D}} \to D \to i_{*} i^{*} D"$ in $\mathcal {D}$ , that fit into the following commutative diagram:

We note that $j_{!} {\widetilde {D}} \in {\mathcal {D}^{\leq -1}}$ since ${\widetilde {D}} \in \mathcal {D}^{\leq -1}_{U}$ and $j_{!}$ is right t-exact. Furthermore, $i_{*}i^{*} D" \in {\mathcal {D}^{\leq -1}} ,$ since $i^{*} D" \cong D' \in \mathcal {D}^{\leq -1}_{F}$ and $i_{*}$ is t-exact. Thus, $D \in {\mathcal {D}^{\leq -1}} .$

Next, we complete the octahedron on $X \to D" \to D$ , finding an object F with exact triangles $F \to X \to D$ and $E \to F \to j_{!} {\widetilde {E}}$ in $\mathcal {D}$ , that fit into the following commutative diagram:

Since $E\in \unicode{xf8} {\langle G \rangle }_{B}^{[-B,B]}$ and

$$ \begin{align*}j_{!}{\widetilde{E}}\in \unicode{xf8}{\langle j_{!}H \rangle}_{(A+B+1)A}^{[-A,2A+B]}\subset \unicode{xf8}{\langle G \rangle}_{(A+B+1)A^2}^{[-2A,3A+B]},\end{align*} $$

we see that $F \in \unicode{xf8} {\langle G \rangle }_{(A+B+1)A^2+B}^{[-3A-B,3A+B]}$ . Thus, the exact triangle $F \to X \to D$ proves the approximability of  $\mathcal {D}$ .

Proof. By [Reference Maycock12, Section 3] (see also [Reference Jorgensen6]), there is a recollement:

Note that B and C are cohomologically bounded above, since their derived categories are approximable, and therefore A is also cohomologically bounded above. Thus, we may apply Theorem 5.1, with $A = G,$ to see $D(A)$ is approximable. By [Reference Maycock12, Lemma 3.11], $j^{*}(A) \cong C,$ and thus is in $D(C)^{-}_{c},$ so we may apply Proposition 4.3.