Proof. The pair $(\mathcal {V},\mathcal {W})$ is clearly a torsion pair. We need to prove that $(\mathcal {U},\mathcal {V})$ is a torsion pair in $\hat {\mathcal {D}}$. The argument is standard and can be found in the literature (see [Reference Nicolás and Saorín46]). We sketch it, leaving some details to the reader. Since objects in $\mathcal {U}_0$ are compact and objects in $\mathcal {V}$ are Milnor colimits of sequences of morphisms with cones in $\text {Add}(\mathcal {V}_0)$, we see that $\mathcal {V}\subseteq \mathcal {U}^\perp$. For $M\in \mathcal {U}^\perp =(\mathcal {U}_0)^\perp$ let us consider the truncation triangle $V\longrightarrow M\longrightarrow W\stackrel {+}{\longrightarrow }$ with respect to $(\mathcal {V},\mathcal {W})$. We get $W\in \mathcal {U}^\perp$ and $W\in \mathcal {V}^\perp$. Therefore, $W\in (\mathcal {U}_0\cup \mathcal {V}_0)^\perp$. But for each $D\in \hat {\mathcal {D}}^c$ there is a triangle $U_0\longrightarrow D\longrightarrow V_0\stackrel {+}{\longrightarrow }$, whose outer terms are in $\mathcal {U}_0$ and $\mathcal {V}_0$, respectively. It follows that $\text {Hom}_{\hat {\mathcal {D}}}(D,W)=0$, for all $D\in \hat {\mathcal {D}}^c$. This implies that $W=0$, so $V\cong M$ belongs to $\mathcal {V}$. Then the pair $(\mathcal {U},\mathcal {V})$ is of the form $(\text {Loc}_{\hat {\mathcal {D}}}(\mathcal {U}_0),\text {Loc}_{\hat {\mathcal {D}}}(\mathcal {U}_0)^{\perp })$, and hence is a torsion pair. The torsion pairs $(\mathcal {U},\mathcal {V})$ and $(\mathcal {V},\mathcal {W})$ are compactly generated by sets $\mathcal {U}_0$ and $\mathcal {V}_0$, respectively.

By lemma 3.1, we know that $(\mathcal {U}\cap \hat {\mathcal {D}}^c,\mathcal {V}\cap \hat {\mathcal {D}}^c)=(\mathcal {U}^c,\mathcal {V}^c)$. Since $\mathcal {U}_0$ is a thick subcategory of a compactly generated triangulated category $\mathcal {U}$ which generates $\mathcal {U}$ and consists of compact objects of $\mathcal {U}$, we get $\mathcal {U}_0=\mathcal {U}^c$. Similarly, $\mathcal {V}_0=\mathcal {V}^c$.  □

Proof. $(1)\Longrightarrow (2)$ The functors $\hat {j}_!$, $\hat {j}^*$, $\hat {i}^*$ and $\hat {i}_*$ preserve compact objects, since they have right adjoints which preserve coproducts, because they also have right adjoints. Let us consider the TTF triple $(\mathcal {U},\mathcal {V},\mathcal {W}):=(\text {Im}(\hat {j}_!),\text {Im}(\hat {i}_*),\text {Im}(\hat {j}_*))$ associated with the recollement from assertion 1. The torsion pair $(\mathcal {U},\mathcal {V})$ is generated by $\hat {j}_!(\hat {\mathcal {X}}^c)=j_!(\hat {\mathcal {X}}^c)$. Indeed, the inclusion $(\hat {j}_!(\hat {\mathcal {X}}^c))^{\perp }\supseteq (\text {Im}(\hat {j}_!))^{\perp }$ is obvious, the inverse inclusion follows from the fact that, by infinite dévissage, $\hat {\mathcal {X}}=\text {Loc}_{\hat {\mathcal {X}}}(\hat {\mathcal {X}}^c)$ and $\hat {j}_!$ commutes with coproducts. Since $j_!(\hat {\mathcal {X}}^c)$ consists of compact objects and is skeletally small there is a set of compact objects generating $(\mathcal {U},\mathcal {V})$. Similarly, the torsion pair $(\mathcal {V},\mathcal {W})$ is generated by $\hat {i}_*(\hat {\mathcal {Y}}^c)=i_*(\hat {\mathcal {Y}}^c)$, and hence by a set of compact objects. Thus condition 2.a holds and, moreover, we have inclusions $j_!(\hat {\mathcal {X}}^c)\subseteq \mathcal {U}\cap \hat {\mathcal {D}}^c$ and $i_*(\hat {\mathcal {Y}}^c)\subseteq \mathcal {V}\cap \hat {\mathcal {D}}^c$. On the other hand, if $U\in \mathcal {U}\cap \hat {\mathcal {D}}^c$, then $\hat {j}^*U\in \hat {\mathcal {X}}^c$. Choosing now $X\in \hat {\mathcal {X}}$ such that $U=\hat {j}_!X$, we have $X\cong \hat {j}^*\hat {j}_!X\cong \hat {j}^*U\in \hat {\mathcal {X}}^c$ and, hence, $U\cong j_!X\in j_!(\hat {\mathcal {X}}^c)$. Similarly, $\mathcal {V}\cap \hat {\mathcal {D}}^c\subseteq i_*(\hat {\mathcal {Y}}^c)$.

$(2)\Longrightarrow (3)$ By condition 2.b we know that $j_!$ and $i_*$ preserve compact objects. Moreover, $j_!(\hat {\mathcal {X}}^c)=\text {Im}(j)\cap \hat {\mathcal {D}}^c$ and $i_*(\hat {\mathcal {Y}}^c)=\text {Im}(i_*)\cap \hat {\mathcal {D}}^c$, since $\mathcal {U}\cap \mathcal {D}=\text {Im}(j_!)$ and $\mathcal {V}\cap \mathcal {D}=\text {Im}(i_*)$. This implies that $j_!$ and $i_*$ also reflect compact objects, i.e. $j_!(X)\in \hat {\mathcal {D}}^c$ (resp. $i_*(Y)\in \hat {\mathcal {D}}^c$) if and only if $X\in \hat {\mathcal {X}}^c$ (resp. $Y\in \hat {\mathcal {Y}}^c$). For any $D\in \hat {\mathcal {D}}^c$ let us consider the associated triangle $j_!j^*D\longrightarrow D\longrightarrow i_*i^*D\stackrel {+}{\longrightarrow }$, we get that $j^*$ preserves compact objects if and only if so does $i^*$.

Let us prove that $i^*$ preserves compact objects. Since the semi-orthogonal decomposition $(\text {Im}(j_!),\text {Im}(i_*))$ is the restriction to $\mathcal {D}$ of $(\mathcal {U},\mathcal {V})$, the associated truncation functor $\tau :\hat {\mathcal {D}}\longrightarrow \mathcal {V}$ has the property that $\tau (\mathcal {D})=\mathcal {V}\cap \mathcal {D}=\text {Im}(i_*)$. By lemma 3.1, we know that $\tau (\hat {\mathcal {D}^c})=\mathcal {V}^c=\mathcal {V}\cap \hat {\mathcal {D}}^c=\text {Im}(i_*)\cap \hat {\mathcal {D}}^c$. We next decompose $i_*$ as $\mathcal {Y}\stackrel {\cong }{\longrightarrow }\text {Im}(i_*)=\mathcal {D}\cap \mathcal {V}\hookrightarrow \mathcal {D}$, where the first arrow $i_*$ is an equivalence of categories. Then the left adjoint $i^*$ is naturally isomorphic to the composition $\mathcal {D}\stackrel {\tau }{\longrightarrow }\text {Im}(i_*)=\mathcal {D}\cap \mathcal {V}\stackrel {i_*^{-1}}{\longrightarrow }\mathcal {Y}$. But $i_*^{-1}(\mathcal {V}\cap \hat {\mathcal {D}}^c)=\hat {\mathcal {Y}}^c$, since $i_*(\hat {\mathcal {Y}}^c)=\mathcal {V}\cap \hat {\mathcal {D}}^c$. Therefore we get that $i^*(\hat {\mathcal {D}}^c)=\hat {\mathcal {Y}}^c$.

$3)\Longrightarrow 2)$ (assuming any of the extra hypotheses). Setting $\mathcal {U}_0:=j_!(\hat {\mathcal {X}}^c)$ and $\mathcal {V}_0:=i_*(\hat {\mathcal {Y}}^c)$, by lemma 3.2, $(\mathcal {U},\mathcal {V},\mathcal {W}):=(\text {Loc}_{\hat {\mathcal {D}}}(j_!(\hat {\mathcal {X}}^c)),\text {Loc}_{\hat {\mathcal {D}}}(i_*(\hat {\mathcal {Y}}^c)), \text {Loc}_{\hat {\mathcal {D}}}(i_*(\hat {\mathcal {Y}}^c))^{\perp })$ is a TTF-triple with $(\mathcal {U},\mathcal {V})$ and $(\mathcal {V},\mathcal {W})$ compactly generated by $j_!(\hat {\mathcal {X}}^c)$ and $i_*(\hat {\mathcal {Y}}^c)$.

We need to check that $\mathcal {U}\cap \mathcal {D}=\text {Im}(j_!)$ and $\mathcal {V}\cap \mathcal {D}=\text {Im}(i_*)$. The equality $\text {Im}(j_*)=\mathcal {W}\cap \mathcal {D}$ will then follow automatically. Indeed, the inclusion $\text {Im}(j_*)\subseteq \mathcal {W}\cap \mathcal {D}$ is obvious, the other inclusion follows from orthogonality. By properties of recollements (see [Reference Beilinson, Bernstein and Deligne10]), $\text {Im}(i_*)=\text {Ker}(j^*)$. Since $\hat {\mathcal {X}}$ is compactly generated, an object $D$ of $\mathcal {D}$ belongs to $\text {Ker}(j^*)$ if and only if $0=\text {Hom}_\mathcal {X}(X,j^*D)\cong \text {Hom}_\mathcal {D}(j_!X,D)$, for all $X\in \hat {\mathcal {X}}^c$. This happens exactly when $D\in \text {Loc}_{\hat {\mathcal {D}}}(j_!(\hat {\mathcal {X}}^c))^\perp \cap \mathcal {D}=\mathcal {V}\cap \mathcal {D}$, and thus $\mathcal {V}\cap \mathcal {D}=\text {Im}(i_*)$.

Let us check that $\mathcal {U}\cap \mathcal {D}=\text {Im}(j_!)$. Since each object of $\mathcal {U}$ is the Milnor colimit of a sequence of morphisms in $\hat {\mathcal {D}}$ with successive cones in $\text {Add}(j_!(\hat {\mathcal {X}})^c)$ and since $\text {Hom}_{\hat {\mathcal {D}}}(j_!X,-)$ vanishes on $\text {Im}(i_*)$, for each $X\in \mathcal {X}$, we get that $\mathcal {U}\cap \mathcal {D}\subseteq \text {Im}(j_!)$. Indeed, $(\text {Im}(j_!),\text {Im}(i_*))$ is a torsion pair in $\mathcal {D}$ and hence $\text {Im}(j_!)= {}^\perp \text {Im}(i_*)\cap \mathcal {D}$. Conversely, for $D\in \text {Im}(j_!)$ let us consider the truncation triangle $U\longrightarrow D\longrightarrow V\stackrel {+}{\longrightarrow }$ in $\hat {\mathcal {D}}$ with respect to $(\mathcal {U},\mathcal {V})$. As before, $\text {Hom}_{\hat {\mathcal {D}}}(U,-)$ vanishes on $\text {Im}(i_*)$, and hence $\text {Hom}_{\hat {\mathcal {D}}}(V,-)$ vanishes on $\text {Im}(i_*)$. In the assumption that $\text {Im}(i_*)$ cogenerates $\mathcal {V}=\text {Loc}_{\hat {\mathcal {D}}}(i_*(\hat {\mathcal {Y}}^c))$, we immediately get $V=0$. In the other case, we also have that $\text {Hom}_{\hat {\mathcal {D}}}(V,-)$ vanishes on $\mathcal {W}$ and, hence, it also vanishes on $\text {Im}(j_*)$. It follows that $\text {Hom}_{\hat {\mathcal {D}}}(V,-)$ vanishes both on $\text {Im}(j_*)$ and $\text {Im}(i_*)$. This implies that $\text {Hom}_{\hat {\mathcal {D}}}(V,-)$ vanishes on $\mathcal {D}$, and hence that $V=0$ since, by hypothesis, $\mathcal {D}$ cogenerates $\hat {\mathcal {D}}$. Under both extra hypotheses, we then get that $U\cong D\in \mathcal {U}\cap \mathcal {D}$.

Let us prove the inclusions $\mathcal {U}\cap \hat {\mathcal {D}}^c\subseteq j_!(\hat {\mathcal {X}}^c)$ and $\mathcal {V}\cap \hat {\mathcal {D}}^c\subseteq i_*(\hat {\mathcal {Y}}^c)$, the inverse inclusions are obvious. For $U\in \mathcal {U}\cap \hat {\mathcal {D}}^c \subseteq \text {Im}(j_!)$ the adjunction map $j_!j^*(U)\longrightarrow U$ is an isomorphism. It follows that $U\in j_!(\hat {\mathcal {X}}^c)$, since $j^*(U)\in \hat {\mathcal {X}}^c$. The second inclusion is analogous.  □

Proof. Let us suppose that $\mathcal {D}=\hat {\mathcal {D}}^c$ in the recollement of theorem 3.3. If assertion 2 of the theorem holds so does assertion 3, and hence the functors $j_!$, $j^*$, $i^*$ and $i_*$ preserve compact objects. Hence, $\text {Im}(j^*)\subseteq \hat {\mathcal {X}}^c$ and $\text {Im}(i^*)\subseteq \hat {\mathcal {Y}}^c$. It follows that $\mathcal {Y}=\hat {\mathcal {Y}}^c$ and $\mathcal {X}=\hat {\mathcal {X}}^c$, since the functors $i^*$ and $j^*$ are dense, for any recollement. Conversely, if $\mathcal {Y}=\hat {\mathcal {Y}}^c$ and $\mathcal {X}=\hat {\mathcal {X}}^c$, then $(\text {Im}(j_!),\text {Im}(i_*))=(j_!(\hat {\mathcal {X}}^c),i_*(\hat {\mathcal {Y}}^c))$ is a semi-orthogonal decomposition of $\mathcal {D}=\hat {\mathcal {D}}^c$ and by lemma 3.2 there is a TTF triple $(\mathcal {U},\mathcal {V},\mathcal {W})=(\text {Loc}_{\hat {\mathcal {D}}}(\text {Im}(j_!)),\text {Loc}_{\hat {\mathcal {D}}}(\text {Im}(i_*)),\text {Loc}_{\hat {\mathcal {D}}}(\text {Im}(i_*))^\perp )$ in $\hat {\mathcal {D}}$, with compactly generated constituent torsion pairs, such that $(\mathcal {U}\cap \mathcal {D},\mathcal {V}\cap \mathcal {D})=(\text {Im}(j_!),\text {Im}(i_*))$. Since $\mathcal {W}\cap \mathcal {D}=\mathcal {V}^\perp \cap \mathcal {D}=\text {Im}(i_*)^\perp \cap \mathcal {D}=\text {Im}(j_*)$, the TTF triple $(\mathcal {U},\mathcal {V},\mathcal {W})$ satisfies all the conditions of assertion 2 in theorem 3.3.  □

Proof. In all the cases, it turns out that the three categories of the recollements are the subcategories of compact objects in the appropriate compactly generated triangulated categories.

1. (1) Let $\hat {\Lambda }$ denotes the repetitive algebra of $\Lambda$. It is easy to see that the category $\text {Mod}\text {-}\hat {\Lambda }$ of unitary $\hat {\Lambda }$-modules (i.e. modules $M$ such that $M\hat {\Lambda }=M$) is a Frobenius category, so that its stable category $\underline {Mod}\text {-}\hat {\Lambda }$ is triangulated and compactly generated. Its subcategory of compact objects is precisely $\underline {mod}\text {-}\hat {\Lambda }$, where $\text {mod}\text {-}\hat {\Lambda }$ is the subcategory of $\text {Mod}\text {-}\hat {\Lambda }$ consisting of the finitely generated $\hat {\Lambda }$-modules, which coincide with the $\hat {\Lambda }$-modules of finite length.

2. (2) It is well-known that if $\Lambda$ is a self-injective Artin algebra, in particular a self-injective finite length algebra, then its module category $\text {Mod}\text {-}\Lambda$ is Frobenius and its associated triangulated stable category $\underline {Mod}\text {-}\Lambda$ has $\underline {mod}\text {-}\Lambda$ as its subcategory of compact objects.

3. (3) and (4) By [Reference Krause33, theorem 1.1], we can identify $\mathcal {D}^b(\text {coh}(\mathbf {Y}))$ with the subcategory of compact objects of $\mathcal {K}(\text {Inj}\text {-}\mathbf {Y})$, for any separated Noetherian scheme $\mathbf {Y}$, and by [Reference Krause33, proposition 2.3], we can identify $ D^b {\text ({mod-})}R$ with the subcategory of compact objects of $\mathcal {K}(\text {Inj-}R)$, for any right Noetherian ring $R$ (here $\text {mod-}R$ is the category of finitely generated $R$-modules, which coincides with that of Noetherian modules).

With all these considerations in mind, the result is now a direct consequence of the previous corollary.  □

Notation and Terminology 3.7 Given any triangulated category $\mathcal {D}$ and any class $\mathcal {X}$ of its objects, we denote by $\mathcal {D}_\mathcal {X}^-$ (resp. $\mathcal {D}_\mathcal {X}^+$ or $\mathcal {D}_\mathcal {X}^b$) the (thick) subcategory of $\mathcal {D}$ consisting of objects $M$ such that, for each $X\in \mathcal {X}$, one has $\text {Hom}_\mathcal {D}(X,M[k])=0$ for $k\gg 0$ (resp. $k\ll 0$ or $|k| \gg 0$). We denote by $\mathcal {D}_{\mathcal {X},fl}$ the (thick) subcategory of $\mathcal {D}$ consisting of objects $M$ such that, for each $X\in \mathcal {X}$ and each $k\in \mathbb {Z}$, the $K$-module $\text {Hom}_\mathcal {D}(X,M[k])$ is of finite length. We finally put $\mathcal {D}^\star _{\mathcal {X},{\dagger} }=\mathcal {D}_\mathcal {X}^\star \cap \mathcal {D}_{\mathcal {X},{\dagger} }$, for $\star \in \{\emptyset ,+,-,b\}$ and ${\dagger} \in \{\emptyset ,fl\}$. In the particular case when $\mathcal {D}$ is compactly generated and $\mathcal {X}=\mathcal {D}^c$, we will simply write $\mathcal {D}_{\dagger} ^\star$ instead of $\mathcal {D}_{\mathcal {X},{\dagger} }^\star$. Note that, in order to define $\mathcal {D}_{\dagger} ^\star$ in the latter case, one can replace $\mathcal {D}^c$ by any set $\mathcal {X}$ of compact generators of $\mathcal {D}$, since $\mathcal {D}^c=\text {thick}_\mathcal {D}(\mathcal {X})$. Let us denote by $\mathbf {P}_{\dagger} ^\star$ the property that defines the full subcategory ${\mathcal {D}}^*_{\dagger}$ of ${\mathcal {D}}$. For instance, if $\star = -$ and ${\dagger} =fl$, then, for a given $M\in {\mathcal {D}}$, we will say that $M$ satisfies property $\mathbf {P}^{\star }_{\dagger}$, for some $X\in {\mathcal {D}}^c$, when $\text {Hom}_{{\mathcal {D}}}(X,M[k])$ is zero, for $k\gg 0$, and is a $K$-module of finite length, for all $k\in \mathbb {Z}$.

Proof. Let us check that $\hat {\mathcal {D}}^b$ (resp. $\hat {\mathcal {D}}^b_{fl}$) cogenerates $\hat {\mathcal {D}}$. For this take a minimal injective cogenerator $E$ of $\text {Mod-}K$ and use the notion of Brown–Comenetz dual. Since compactly generated (or even well-generated) triangulated categories satisfy Brown representability theorem (see [Reference Neeman40, proposition 8.4.2]), for each $X\in \hat {\mathcal {D}}^c$, the functor $\text {Hom}_K(\text {Hom}_{\hat {\mathcal {D}}}(X,-),E):\hat {\mathcal {D}}^{op}\longrightarrow \text {Mod-}K$ is naturally isomorphic to the representable functor $\text {Hom}_{\hat {\mathcal {D}}}(-,D(X))$, for an object $D(X)$, uniquely determined up to isomorphism, called the Brown–Comenetz dual of $X$. It immediately follows that $\{D(X)\text {: }X\in \hat {\mathcal {D}}^c\}$ is a skeletally small cogenerating class of $\hat {\mathcal {D}}$. Our task reduces to check that $D(X)\in \hat {\mathcal {D}}^b$ (resp. $D(X)\in \hat {\mathcal {D}}^b_{fl}$), when $\hat {\mathcal {D}}$ is homologically locally bounded (resp. homologically locally finite length). But this is clear since, given any $Y,X\in \hat {\mathcal {D}}^c$, we have that $\text {Hom}_{\hat {\mathcal {D}}}(Y,D(X)[k])\cong \text {Hom}_K(\text {Hom}_{\hat {\mathcal {D}}}(Y,X[-k]),E)$ and the homologically locally bounded condition on $\hat {\mathcal {D}}$ implies that $\text {Hom}_{\hat {\mathcal {D}}}(Y,X[-k])=0$, for all but finitely many $k\in \mathbb {Z}$. When $\hat {\mathcal {D}}$ is homologically locally finite length, we have in addition that each $\text {Hom}_{\hat {\mathcal {D}}}(Y,X[-k])$ is of finite length as $K$-module, which implies that the same is true for $\text {Hom}_{\hat {\mathcal {D}}}(Y,D(X)[k])$.  □

Proof. The fact that the recollement is the upper part of a ladder of recollements of height two implies that the functors $\hat {i}^*,\hat {i}_*,\hat {j}_!,\hat {j}^*$ preserve compact objects, since their right adjoints preserve coproducts. Note that if in an adjoint pair of triangulated functors $(F:\mathcal {D}\longrightarrow \mathcal {E},G:\mathcal {E}\longrightarrow \mathcal {D})$ between compactly generated triangulated categories the functor $F$ preserves compact objects, then $G(\mathcal {E}^\star _{\dagger} )\subseteq \mathcal {D}^\star _{\dagger}$. Indeed, let $E\in \mathcal {E}^\star _{\dagger}$ be any object. Then $G(E)\in \mathcal {D}^\star _{\dagger}$ if and only if $G(E)$ satisfies property $\mathbf {P}_{\dagger} ^*$ for all $X\in \mathcal {D}^c$. Due to the natural isomorphism $\text {Hom}_\mathcal {E}(F(X),E[n])\cong \text {Hom}_\mathcal {D}(X,G(E)[n])$, for all $X\in \mathcal {D}^c$ and $n\in \mathbb {Z}$, and since $F(X)\in \mathcal {E}^c$ we get that $G(E)$ satisfies property $\mathbf {P}_{\dagger} ^*$ for all $X\in \mathcal {D}^c$, since $E$ satisfies property $\mathbf {P}_{\dagger} ^*$, for all $Z\in \mathcal {E}^c$. This implies that all the functors $\hat {i}_*$, $\hat {i}^!$, $\hat {j}^*$ and $\hat {j}_*$ restrict to the $(-)^\star _{\dagger}$-level. In particular, the triangle $\hat {i}_*\hat {i}^!D\longrightarrow D\longrightarrow \hat {j}_*\hat {j}^*D\stackrel {+}{\longrightarrow }$, which is the truncation triangle with respect to $(\mathcal {V},\mathcal {W}):=(\text {Im}(\hat {i}_*),\text {Im}(\hat {j}_*))$, belongs to $\hat {\mathcal {D}}^\star _{\dagger}$ for each $D\in \hat {\mathcal {D}}^\star _{\dagger}$. Therefore the semi-orthogonal decomposition $(\mathcal {V},\mathcal {W})$ restricts to $\hat {\mathcal {D}}^\star _{\dagger}$. As a consequence, assertion 2 holds if and only if the triangle $\hat {j}_!\hat {j}^*D\longrightarrow D\longrightarrow \hat {i}_*\hat {i}^*D\stackrel {+}{\longrightarrow }$ (I), which is the truncation triangle with respect to $(\mathcal {U},\mathcal {V}):=(\text {Im}(\hat {j}_!),\text {Im}(\hat {i}_*))$, belongs to $\hat {\mathcal {D}}^\star _{\dagger}$ for each $D\in \hat {\mathcal {D}}^\star _{\dagger}$.

Since $\hat {i}_*$ and $\hat {j}_*$ are fully faithful, the counits of the adjoint pairs $(\hat {i}^*,\hat {i}_*)$ and $(\hat {j}^*,\hat {j}_*)$ are natural isomorphisms, and, hence $\hat {\mathcal {Y}}^\star _{\dagger} \subseteq \hat {i}^*(\hat {\mathcal {D}}^\star _{\dagger} )$ and $\hat {j}^*(\hat {\mathcal {D}}^\star _{\dagger} )=\hat {\mathcal {X}}^\star _{\dagger}$. Using this, we get that the triangle (I) belongs to $\hat {\mathcal {D}}^\star _{\dagger}$, for each $D\in \hat {\mathcal {D}}^\star _{\dagger}$ if and only if $\hat {j}_!(\hat {\mathcal {X}}^\star _{\dagger} )\subseteq \hat {\mathcal {D}}^\star _{\dagger}$. Thus, assertions 2 and 4 are equivalent.

On the other hand, we have $\hat {i}^*(\hat {\mathcal {D}}^c)=\hat {\mathcal {Y}}^c$. Indeed, we only need to check the inclusion $\supseteq$, since $\hat {i}^*$ preserves compact objects. If $Y\in \hat {\mathcal {Y}}^c$ then $Y\cong \hat {i}^*\hat {i}_*(Y)$, and $\hat {i}_*Y\in \hat {\mathcal {D}}^c$. The equality $\hat {i}^*(\hat {\mathcal {D}}^c)=\hat {\mathcal {Y}}^c$ implies, that for $Y\in \hat {\mathcal {Y}}$, one has that $\hat {i}_*Y\in \hat {\mathcal {D}}^\star _{\dagger}$ if and only if $Y\in \hat {\mathcal {Y}}^\star _{\dagger}$. Indeed $\hat {i}_*Y\in \hat {\mathcal {D}}^\star _{\dagger}$ if and only if $\hat {i}_*Y$ satisfies property $\mathbf {P}^\star _{\dagger}$, for all $D_0\in \hat {\mathcal {D}}^c$, which, by adjunction, is equivalent to saying that $Y$ satisfies property $\mathbf {P}^\star _{\dagger}$ for all $\hat {i}^*(D_0)$, with $D_0\in \hat {\mathcal {D}}^c$. That is, if and only if $Y$ satisfies property $\mathbf {P}^\star _{\dagger}$, for all $Y_0\in \hat {\mathcal {Y}}^c$, if and only if $Y\in \hat {\mathcal {Y}}^\star _{\dagger}$. Thus, triangle (I) belongs to $\hat {\mathcal {D}}^\star _{\dagger}$, for each $D\in \hat {\mathcal {D}}^\star _{\dagger}$, if and only if $\hat {i}_*\hat {i}^*D\in \hat {\mathcal {D}}^\star _{\dagger}$ if and only if $\hat {i}^*D\in \hat {\mathcal {Y}}^\star _{\dagger}$, if and only if $\hat {i}^*(\hat {\mathcal {D}}^\star _{\dagger} )\subseteq \hat {\mathcal {Y}}^\star _{\dagger}$. Therefore assertions 2 and 3 are also equivalent. Finally, taking into account the first paragraph of this proof, it is clear that if the equivalent assertions 3 and 4 hold, then assertion 1 holds.  □

Proof. We are going to use the notation from the proof of theorem 3.3. To prove the first assertion, we are going to check that $\mathcal {D}(\mathcal {B}) \stackrel {\cong }{\longrightarrow }\mathcal {V}$ and that this equivalence restricts to $\mathcal {D}^{\star }_{\dagger} (\mathcal {B})\stackrel {\cong }{\longrightarrow } \text {Im}(i_*)$. Note that $\mathcal {B}$ is homologically locally bounded considered as a dg category. When ${\dagger} =fl$, due to the inclusion $\mathcal {D}^c(\mathcal {B})\subseteq \mathcal {D}^{\star }_{\dagger} (\mathcal {B})$, the dg category $\mathcal {B}$ is also homologically locally finite length. By lemma 3.2, $(\mathcal {U},\mathcal {V},\mathcal {W})=(\text {Loc}_{\hat {\mathcal {D}}}(j_!(\hat {\mathcal {X}}^c)),\text {Loc}_{\hat {\mathcal {D}}}(i_*(\mathcal {D}^c(\mathcal {B}))),\text {Loc}_{\hat {\mathcal {D}}}(i_*(\mathcal {D}^c(\mathcal {B})))^\perp )$ is a TTF triple in $\hat {\mathcal {D}}$ and ${\mathcal {V}}^c=\mathcal {V}\cap \hat {\mathcal {D}}^c$. By the proof of the implication $(3)\Longrightarrow (2)$ of theorem 3.3, $\text {Im}(i_*)=\mathcal {V}\cap \hat {\mathcal {D}}^{\star }_{\dagger}$. Indeed, the additional condition on $\text {Im}(i_*)$ was used only to check that $\text {Im}(j_!)=\mathcal {U}\cap \hat {\mathcal {D}}^{\star }_{\dagger}$.

By hypothesis, we have $i_*(\mathcal {D}^c(\mathcal {B}))\subseteq \mathcal {V}\cap \hat {\mathcal {D}}^c$. For $V\in \mathcal {V}\cap \hat {\mathcal {D}}^c \subseteq \text {Im}(i_*)$ take $W$ such that $i_*W\simeq V$, since $i^*$ preserves compact objects $W\simeq i^*i_*W$ is compact and $V\in i_*(\mathcal {D}^c(\mathcal {B}))$. So ${\mathcal {V}}^c=\mathcal {V}\cap \hat {\mathcal {D}}^c=i_*(\mathcal {D}^c(\mathcal {B}))$. Hence, also $i_*i^*\hat {\mathcal {D}}^c=i_*\mathcal {D}^c(\mathcal {B}).$

Note that $\mathcal {V}$ is a quotient of an algebraic compactly generated triangulated category by a localizing subcategory generated by a set of compact objects. Then $\mathcal {V}$ is compactly generated by [Reference Neeman39, theorem 2.1] (using the description of compact objects and the right adjoint to the localization functor), and it is also algebraic as a triangulated subcategory of an algebraic triangulated category. Furthermore, by [Reference Keller24, theorem 9.2] there is a triangulated equivalence $F:\mathcal {D}(\mathcal {B})\stackrel {\cong }{\longrightarrow }\mathcal {V}$ such that $F(B^\wedge )\cong i_*(B^\wedge )$, for each $B\in \mathcal {B}$. Since $\mathcal {D}^c(\mathcal {B})=\text {thick}_{\mathcal {D}(\mathcal {B})}(B^\wedge \text {: }B\in \mathcal {B})$, we get $F(\mathcal {D}^c(\mathcal {B}))= i_*(\mathcal {D}^c(\mathcal {B}))$.

Consider the canonical triangle $j_!j^*Z\longrightarrow Z\longrightarrow i_*i^*Z\stackrel {+}{\longrightarrow }$, for each $Z\in \hat {\mathcal {D}}^c$. Since $j_!j^*(Z)$ is compact $\text {Hom}_{\hat {\mathcal {D}}}(j_!j^*(Z),-)$ vanishes on $\text {Im}(F)=\mathcal {V}$. For $Y\in \mathcal {D}(\mathcal {B})$ we get that $F(Y)$ belongs to $\hat {\mathcal {D}}^*_{\dagger}$ iff it satisfies property $\mathbf {P}^\star _{\dagger}$, for all $Z\in \hat {\mathcal {D}}^c$, iff it satisfies property $\mathbf {P}^\star _{\dagger}$, for all $Z'\in \hat {\mathcal {D}}^c$ such that $Z'\cong i_*i^*(Z)$. Since $i_*i^*\hat {\mathcal {D}}^c=i_*\mathcal {D}^c(\mathcal {B})$, we get that $F(Y)$ is in $\hat {\mathcal {D}}^*_{\dagger}$ iff it satisfies property $\mathbf {P}^{\star }_{\dagger}$, for all $Z=i_*(M)\cong F(M')$, with $M, M'\in \mathcal {D}^c(\mathcal {B})$. The fact that $F$ is an equivalence implies that $F(Y)\in \hat {\mathcal {D}}^*_{\dagger}$ iff $Y$ satisfies property $\mathbf {P}^\star _{\dagger}$ in $\mathcal {D}(\mathcal {B})$ for all $M'\in \mathcal {D}^c (\mathcal {B})$. Thus, $F(Y)\in \hat {\mathcal {D}}^*_{\dagger}$ iff $Y\in \mathcal {D}^*_{\dagger} \mathcal {B}$. This means that $F(\mathcal {D}^*_{\dagger} (\mathcal {B}))=\hat {\mathcal {D}}^*_{\dagger} \cap \text {Im}(F)=\hat {\mathcal {D}}^*_{\dagger} \cap \mathcal {V}= \text {Im}(i_*)$. Therefore $F$ induces an equivalence of triangulated categories $F:\mathcal {D}^*_{\dagger} (\mathcal {B})\stackrel {\cong }{\longrightarrow }\text {Im}(i_*)$. Note that, this equivalence need not be naturally isomorphic to the one induced by $i_*$. By the proof of corollary 3.11, $\mathcal {D}^*_{\dagger}$ cogenerates $\mathcal {D}(\mathcal {B})$, and hence $\text {Im}(i_*)$ cogenerates $\mathcal {V}$.

Let us prove the second assertion of the proposition. From the TTF triple constructed above we get a recollement:

(3.1)

where $\hat {j}_*:\mathcal {W}\hookrightarrow \hat {\mathcal {D}}$ is the inclusion, such that the associated TTF triple $(\text {Im}(\hat {j}_!),\text {Im}(\hat {i}_*),\text {Im}(\hat {j}_*))$ in $\hat {\mathcal {D}}$ restricts to the TTF triple $(\text {Im}(j_!),\text {Im}(i_*),\text {Im}(j_*))$ in $\hat {\mathcal {D}}^*_{\dagger}$. In particular, there is an equivalence of categories $\mathcal {U}\stackrel {\cong }{\longrightarrow }\mathcal {W}$ which restricts to the canonical equivalence $\text {Im}(j_!)\stackrel {\cong }{\longrightarrow }\text {Im}(j_*)$ given by $j_*j_!^{-1}$. As a quotient of an algebraic triangulated category $\mathcal {W}$ is algebraic. Since $\mathcal {W} \stackrel {\cong }{\longrightarrow }\mathcal {U}=\text {Loc}_{\hat {\mathcal {D}}}(j_!(\mathcal {D}^c(\mathcal {C})))=\text {Loc}_{ \hat {\mathcal {D}}}(j_!(C^\wedge )\text {: }C\in \mathcal {C})$, we conclude that $\mathcal {W}$ is compactly generated by $\{j_*(C^\wedge )\text {: }C\in \mathcal {C}\}$.

As before, by [Reference Keller24, theorem 9.2], there is a triangulated equivalence $G:\mathcal {D}(\mathcal {C})\stackrel {\cong }{\longrightarrow }\mathcal {W}$ such that $G(C^\wedge )\cong j_*(C^\wedge )$, for each $C\in \mathcal {C}$. Let $X\in \mathcal {D}(\mathcal {C})$ be any object. We claim that $G(X)\in \text {Im}(j_*)=\mathcal {W}\cap \hat {\mathcal {D}}^*_{\dagger}$ iff $X\in \mathcal {D}_{\dagger} ^*(\mathcal {C})$. Indeed, $G(X)\in \hat {\mathcal {D}}^*_{\dagger}$ iff $\text {Hom}_{\hat {\mathcal {D}}}(Z,G(X)[k])$ satisfies property $\mathbf {P}^{\star }_{\dagger}$, for all $Z\in \hat {\mathcal {D}}^c$. Using the triangle $i_*i^!Z\longrightarrow Z\longrightarrow j_*j^*Z\stackrel {+}{\longrightarrow }$ and the fact that $\text {Hom}_{\hat {\mathcal {D}}}(i_*i^!Z,-)$ vanishes on $\mathcal {W}=\text {Im}(G)$, we get that $G(X)$ is in $\hat {\mathcal {D}}^*_{\dagger}$ iff $\text {Hom}_{\hat {\mathcal {D}}}(j_*j^*Z,G(X)[k])$ satisfies $\mathbf {P}^{\star }_{\dagger}$, for all $Z\in \hat {\mathcal {D}}^c$. By hypothesis, we have an inclusion $j^*(\hat {\mathcal {D}}^c)\subseteq \mathcal {D}^c(\mathcal {C})$. Conversely, if $X'\in \mathcal {D}^c(\mathcal {C})$ then $j_!(X')\in \hat {\mathcal {D}}^c$, so that $X'\cong j^*j_!(X')\in j^*(\hat {\mathcal {D}}^c)$. Thus, when $Z$ runs through the objects of $\hat {\mathcal {D}}^c$, the object $j^*Z$ runs through the objects of $\mathcal {D}^c(\mathcal {C})$. Since $\mathcal {D}^c(\mathcal {C})=\text {thick}_{\mathcal {D}(\mathcal {C})}(C^\wedge \text {: }C\in \mathcal {C})$, we easily conclude that $G(X)\in \hat {\mathcal {D}}^*_{\dagger}$ iff $\text {Hom}_{\hat {\mathcal {D}}}(j_*(C^\wedge ),G(X)[k])$ satisfies $\mathbf {P}^{\star }_{\dagger}$, for all $C\in \mathcal {C}$. Since $G$ is an equivalence of categories and $j_*(C^\wedge )\cong G(C^\wedge )$, we have $G(X)\in \hat {\mathcal {D}}^*_{\dagger}$ iff $\text {Hom}_{\mathcal {D}(\mathcal {C})}(C^\wedge ,X[k])$ satisfies $\mathbf {P}^{\star }_{\dagger}$, for all $C\in \mathcal {C}$. That is, iff $X\in \mathcal {D}^*_{\dagger} (\mathcal {C})$.

The previous paragraph yields an equivalence of categories $G:\mathcal {D}(\mathcal {C})\stackrel {\cong }{\longrightarrow }\mathcal {W}$ which induces by restriction another equivalence $\mathcal {D}^*_{\dagger} (\mathcal {C})\stackrel {\cong }{\longrightarrow }\text {Im}(j_*)$. This implies that we can replace $\mathcal {W}$ by $\mathcal {D}(\mathcal {C})$ in the recollement (5), thus obtaining a recollement as in the final assertion of the proposition, which in turn restricts to a recollement whose associated TTF triple is $(\text {Im}(j_!),\text {Im}(i_*),\text {Im}(j_*))$. This last recollement is then equivalent to the original one.

It remains to prove that the obtained recollement

is the upper part of a ladder of recollements of height two. This is a direct consequence of [Reference Gao and Psaroudakis19, proposition 3.4] since $\hat {i}_*$ preserves compact objects.  □

Proof. (1) By [Reference Rouquier54, lemma 7.46], for an arbitrary projective scheme $\mathbb {X}$ over $K$, we have $\mathcal {D}^b(\text {coh}(\mathbb {X}))=\mathcal {D}(\text {Qcoh}(\mathbb {X}))^b_{fl}$, which is well-known to be Hom-finite over $K$. By [Reference Rouquier54, lemma 7.49], $\mathcal {D}(\text {Qcoh}(\mathbb {X}))^c$ consists of $X\in \mathcal {D}(\text {Qcoh}(\mathbb {X}))$ such that, for each $M\in \mathcal {D}^b(\text {coh}(\mathbb {X}))$, the $K$-vector space $\oplus _{k\in \mathbb {Z}}\text {Hom}_{\mathcal {D}(\text {Coh}(\mathbb {X}))}(X,M[k])$ is finite dimensional. But if $X,M\in \mathcal {D}^b(\text {coh}(\mathbb {X}))$, then $\text {Hom}_{\mathcal {D}(\text {Qcoh}(\mathbb {X}))}(X,M[k])=0$ for $k\ll 0$. We also know that $\mathcal {D}^b(\text {coh}(\mathbb {X}))$ is $\text {Hom}$-finite. Thus, the subcategory consisting of $X\in \mathcal {D}^b(\text {coh}(\mathbb {X}))$ such that, for each $M\in \mathcal {D}^b(\text {coh}(\mathbb {X}))$, one has $\text {Hom}_{\mathcal {D}(\text {Qcoh}(\mathbb {X}))}(X,M[k])=0$ for $k\gg 0$, coincides with the subcategory of $X\in \mathcal {D}^b(\text {coh}(\mathbb {X}))$ such that $\oplus _{k\in \mathbb {Z}}\text {Hom}_{\mathcal {D}(\text {Qcoh}(\mathbb {X}))}(X,M[k])$ is finite dimensional. This subcategory is precisely $\mathcal {D}(\text {Qcoh}(\mathbb {X}))^c\cap \mathcal {D}^b(\text {coh}(\mathbb {X}))=\mathcal {D}(\text {Qcoh}(\mathbb {X}))^c$.

(2) By [Reference Nicolás and Saorín46, § 3.7] or [Reference Keller and Nicolás28, example 6.1], there is a canonical t-structure $(\mathcal {D}^{\leq 0}A,\mathcal {D}^{\geq 0}A)$ in $\mathcal {D}(A)$, where $\mathcal {D}^{\leq 0}(A)$ (resp. $\mathcal {D}^{\geq 0}(A)$) consists of the dg ${A-modules}$ $M$ such that $H^k(M)=0$, for all $k>0$ (resp. $k<0$). Moreover, $\mathcal {D}^{\leq 0}(A)=\text {Susp}_{\mathcal {D}(A)}(A)$. By corollary 4.8 below for $\mathcal {T}=\{A\}$, this t-structure restricts to $\mathcal {D}^b_{fl}(A)$.

Note that there is a dg subalgebra $\tilde {A}$ of $A$, given by $\tilde {A}^n=A^n$, for $n<0$, $\tilde {A}^0=Z^0(A)=\{0\text {-cycles of }A\}$, and $\tilde {A}^n=0$, for $n>0$. The inclusion $\lambda :\tilde {A}\hookrightarrow A$ is a quasi-isomorphism, and the associated restriction of scalars $\lambda _*:\mathcal {D}(A)\longrightarrow \mathcal {D}(\tilde {A})$ is a triangulated equivalence which takes $A$ to $\tilde {A}$. As a consequence, this equivalence preserves the canonical t-structure, and hence it induces an equivalence between the corresponding hearts. The heart of $(\mathcal {D}^{\leq 0}(\tilde {A}),\mathcal {D}^{\geq 0}(\tilde {A}))$ is known to be equivalent to the category of modules over $H^0(\tilde {A})\cong H^0(A)$ (see [Reference Keller and Nicolás28, example 6.1]). This equivalence is given by $H^0:\mathcal {H}\stackrel {\cong }{\longrightarrow }\text {Mod-}H^0(A)$ ($M\rightsquigarrow H^0(M)$). Putting $\mathcal {H}_{fl}:=\mathcal {H}\cap \mathcal {D}^b_{fl}(A)$, which is the heart of the restricted t-structure $(\mathcal {D}^{\leq 0}(A)\cap \mathcal {D}^b_{fl}(A),\mathcal {D}^{\geq 0}(A)\cap \mathcal {D}^b_{fl}(A))$ in $\mathcal {D}^b_{fl}(A)$, we deduce an equivalence of categories $H^0:\mathcal {H}_{fl}\stackrel {\cong }{\longrightarrow }\text {mod-}H^0(A)$, bearing in mind that $H^0(A)$ is a finite length $K$-algebra.

Let now $X\in \mathcal {D}^b_{fl}(A)$ be any object. By the proof of [Reference Nicolás, Saorín and Zvonareva47, theorem 2] and the fact that $\text {Hom}_{\mathcal {D}(A)}(A[k],M)\cong H^{-k}M$ is a $K$-module of finite length, for each $M\in \mathcal {D}^b_{fl}(A)$ and each $k\in \mathbb {Z}$, we know that $X$ is the Milnor colimit of a sequence $0=X_{-1}\stackrel {f_0}{\longrightarrow }X_0\stackrel {f_1}{\longrightarrow }X_1\longrightarrow \cdots \longrightarrow {X_n}\stackrel {f_n}\longrightarrow \cdots$ such that $\text {cone}(f_n)\in \text {add}(A)[n]$, for each $n\in \mathbb {N}$. Let $r>0$ be arbitrary and, for each $n> r$, put $u_n:=f_n\circ \dots \circ f_{r+1}:X_{r}\longrightarrow X_n$ and $C_n=\text {cone}(u_n)$. By Verdier's $3\times 3$ lemma (see [Reference May37, lemma 1.7]), we have a commutative diagram, where all rows and columns are triangles

Since each $C_n$ is a finite iterated extension of objects in $\text {add}(A)[n]$, with $n>r$, we get $C_n\in \mathcal {D}^{<-r}(A)$, for each $n>r$. It follows that $C\in \mathcal {D}^{\leq -r}(A)$. But $C\in \mathcal {D}^b_{fl}(A)$, since the left two terms of the triangle in the bottom row of the diagram are in $\mathcal {D}^b_{fl}(A)$. We then get a triangle $X_r\longrightarrow X\longrightarrow C\stackrel {+}{\longrightarrow }$ in $\mathcal {D}^b_{fl}(A)$ such that $X_r\in \mathcal {D}^c(A)$ and $C\in \mathcal {D}^{\leq -r}(A)$. If now $Y\in \mathcal {D}^b_{fl}(A)$ is any object, then we know that $Y\in \mathcal {D}^{>-r}(A)$, for some integer $r>0$. For this integer, we then get a monomorphism $\text {Hom}_{\mathcal {D}(A)}(X,Y)\longrightarrow \text {Hom}_{\mathcal {D}(A)}(X_r,Y)$ whose target is a $K$-module of finite length. This proves that $\mathcal {D}^b_{fl}(A)$ is Hom-finite.

On the other hand, by [Reference Bridgeland15, lemma 3.7], we know that

(3.2)\begin{equation} \mathcal{D}^{{\leq} 0}(A)\cap\mathcal{D}^b_{fl}(A)={\bigcup}_{n\in\mathbb{N}}\text{add}(\mathcal{H}_{fl}[n]\star\mathcal{H}_{fl}[n-1]\star\cdots\star\mathcal{H}_{fl}[0]). \end{equation}

Let $X\in \mathcal {D}^b_{fl}(A)$ be an object such that, for each $M\in \mathcal {D}^b_{fl}(A)$, one has $\text {Hom}_{\mathcal {D}(A)}(X,M[k])=0$ for $k\gg 0$ (and, hence, also for $|k| \gg 0$). Let us choose a (necessarily finite) set $\mathcal {S}$ of representatives of the isoclasses of simple objects of $\mathcal {H}_{fl}$. If $m$ is the Loewy length of $H^0(A)$, then $\mathcal {H}_{fl}\subset \text {add}(\mathcal {S})\star \stackrel {m}{\dots }\star \text {add}(\mathcal {S})$. Fixing $r\in \mathbb {N}$ such that $\text {Hom}_{\mathcal {D}(A)}(X,\mathcal {S}[k])=0$, for $k\geq r$, we get that $\text {Hom}_{\mathcal {D}(A)}(X,-[k])$ vanishes on $\mathcal {H}_{fl}$, for all $k\geq r$. By (6) above, $\text {Hom}_{\mathcal {D}(A)}(X,-)$ vanishes on $\mathcal {D}^{\leq -r}(A)\cap \mathcal {D}^b_{fl}(A)$. If for this $r$ we consider the triangle $X_r\longrightarrow X\longrightarrow C\stackrel {+}{\longrightarrow }$ constructed above, then the arrow $X\longrightarrow C$ is the zero map, and hence $X$ is isomorphic to a direct summand of $X_r$ and $X\in \mathcal {D}^c(A)$.  □

Proof. It is clear that if $\mathcal {D}$ and $\mathcal {E}$ are triangulated categories which are compact-detectable in finite length and $F:\mathcal {D}^b_{fl}\longrightarrow \mathcal {E}^b_{fl}$ is a functor that has a right adjoint, then $F$ preserves compact objects. Therefore $j_!$, $j^*$, $i^*$ and $i_*$ preserve compact objects. Now corollary 3.11 says that assertion 2 of theorem 3.3 holds. The last assertion of the proposition is a direct consequence of the last assertion of theorem 3.13.  □

Proof. Assume $\mathcal {T}$ is partial silting in $\mathcal {E}$ and the associated t-structure $\tau$ in $\mathcal {E}$ restricts to $\mathcal {D}$. The restricted t-structure in $\mathcal {D}$ is $( {}^\perp (\mathcal {T}^{\perp _{\leq 0}})\cap \mathcal {D},\mathcal {T}^{\perp _{<0}}\cap \mathcal {D})$, where the orthogonals are taken in $\mathcal {E}$. Since $\text {Hom}_\mathcal {E}(T,-)$ vanishes on $( {}^\perp \mathcal {T}^{\perp _{\leq 0}})[1]$, it vanishes on $( {}^\perp (\mathcal {T}^{\perp _{\leq 0}})\cap \mathcal {D})[1]$. It remains to see that the restricted t-structure is generated by $\mathcal {T}$ in $\mathcal {D}$. That is, that $( {}^\perp (\mathcal {T}^{\perp _{\leq 0}})\cap \mathcal {D},\mathcal {T}^{\perp _{<0}}\cap \mathcal {D})=( {}^\perp (\mathcal {T}^{\perp _{\leq 0}}\cap \mathcal {D})\cap \mathcal {D},\mathcal {T}^{\perp _{<0}}\cap \mathcal {D})$. Right parts of these pairs coincide and we clearly have the inclusion $\subseteq$ on the left parts. If $X\in {}^\perp (\mathcal {T}^{\perp _{\leq 0}}\cap \mathcal {D})\cap \mathcal {D}$ and $U\stackrel {f}{\longrightarrow } X\stackrel {g}{\longrightarrow }V\stackrel {+}{\longrightarrow }$ is the truncation triangle with respect to the restricted t-structure, then $g=0$ and hence $X$ is a direct summand of $U\in {}^\perp (\mathcal {T}^{\perp _{\leq 0}})\cap \mathcal {D}$. This implies that $X$ belongs to ${}^\perp (\mathcal {T}^{\perp _{\leq 0}})\cap \mathcal {D}$ since this class is closed under direct summands.

For the second part of the statement note that by [Reference Nicolás, Saorín and Zvonareva47, theorem 1], the set $\mathcal {T}$ is partial silting in $\mathcal {E}$ and ${}^\perp (\mathcal {T}^{\perp _{\leq 0}})=\mathcal {T}^{\perp _{>0}}$. On the other hand, the t-structure in $\mathcal {D}$ generated by $\mathcal {T}$ is $\tau '=({}^\perp (\mathcal {T}^{\perp _{\leq 0}}\cap \mathcal {D})\cap \mathcal {D},\mathcal {T}^{\perp _{<0}}\cap \mathcal {D})$, and the partial silting condition of $\mathcal {T}$ in $\mathcal {D}$ gives ${}^\perp (\mathcal {T}^{\perp _{\leq 0}}\cap \mathcal {D})\cap \mathcal {D}\subseteq \mathcal {T}^{\perp _{>0}}$, so ${}^\perp (\mathcal {T}^{\perp _{\leq 0}}\cap \mathcal {D})\cap \mathcal {D}\subseteq \mathcal {T}^{\perp _{>0}}\cap \mathcal {D}$. Thus, we get a chain of inclusions ${}^\perp (\mathcal {T}^{\perp _{\leq 0}})\cap \mathcal {D}\subseteq {}^\perp (\mathcal {T}^{\perp _{\leq 0}}\cap \mathcal {D})\cap \mathcal {D} \subseteq \mathcal {T}^{\perp _{>0}}\cap \mathcal {D}$, all of which must be equalities. Hence, $\tau '=(\mathcal {T}^{\perp _{>0}}\cap \mathcal {D},\mathcal {T}^{\perp _{<0}}\cap \mathcal {D})$ is the restriction of $\tau$ to $\mathcal {D}$.  □

Proof. Let $\mathcal {C}:={}^\perp (\mathcal {D}^{\leq 0}[1])\cap \mathcal {D}^{\leq 0}$ be the co-heart of $(\mathcal {D}^{\leq 0},\mathcal {D}^{\geq 0})$ and let $\mathcal {T}' \subset \mathcal {D}^c$ be any partial silting set in $\mathcal {D}$ which generates this t-structure. Since $\text {Hom}_\mathcal {D}(T',-)$ vanishes on $\mathcal {D}^{\leq 0}[1]$, for all $T'\in \mathcal {T}'$, we have $\mathcal {T}'\subset \mathcal {C}$, and hence $\text {add}(\mathcal {T}')\subseteq \mathcal {C}\cap \hat {\mathcal {D}}^c$. Let $\mathcal {T}$ be a set of representatives of isomorphism classes of objects of $\mathcal {C}\cap \hat {\mathcal {D}}^c$. Let us check that $\mathcal {T}$ generates $(\mathcal {D}^{\leq 0},\mathcal {D}^{\geq 0})$.

Without loss of generality, we can assume that $\mathcal {T}'\subseteq \mathcal {T}$. This implies that $\mathcal {T}^{\perp _{\leq 0}}\cap \mathcal {D}\subseteq \mathcal {T}'^{\perp _{\leq 0}}\cap \mathcal {D}=\mathcal {D}^{>0}:=\mathcal {D}^{\geq 0}[-1]$. Since $\mathcal {T}\subset \mathcal {D}^{\leq 0}$, we have that $\text {Hom}_\mathcal {D}(T,Y)=0$, for all $T\in \mathcal {T}$ and $Y\in \mathcal {D}^{>0}$. Hence, the inclusion $\mathcal {D}^{>0}=\mathcal {T}'^{\perp _{\leq 0}}\cap \mathcal {D}\subseteq \mathcal {T}^{\perp _{\leq 0}}\cap \mathcal {D}$ also holds and $\mathcal {T}$ generates the t-structure $(\mathcal {D}^{\leq 0},\mathcal {D}^{\geq 0})$.

Let us prove the last assertion of the proposition. Suppose that $(\mathcal {D}^{\leq 0},\mathcal {D}^{\geq 0})=({}^\perp (\mathcal {T}_0^{\perp _{\leq 0}})\cap \mathcal {D},\mathcal {T}_0^{\perp _{<0}}\cap \mathcal {D})$, for some non-positive set $\mathcal {T}_0\subset \hat {\mathcal {D}}^c$. Recall that ${}^\perp (\mathcal {T}_0^{\perp _{\leq 0}})=\text {Susp}_{\hat {\mathcal {D}}}(\mathcal {T}_0)$ (see [Reference Nicolás, Saorín and Zvonareva47, theorem 2]). If $C\in \mathcal {C}\cap \hat {\mathcal {D}}^c$, then $\text {Hom}_{\hat {\mathcal {D}}}(C,-)$ vanishes on $\text {Susp}_{\hat {\mathcal {D}}}(\mathcal {T}_0)[1]={}^\perp (\mathcal {T}_0^{\perp _{\leq 0}})[1]$. Indeed, ${\bigcup} _{k>0}\mathcal {T}_0[k]\subset \mathcal {D}^{\leq 0}[1]$, $\text {Hom}_{\hat {\mathcal {D}}}(C,-)$ vanishes on $\mathcal {D}^{\leq 0}[1]$ and $C$ is compact. Hence, $\mathcal {C}\cap \hat {\mathcal {D}}^c$ belongs to the co-heart $\hat {\mathcal {C}}:={}^\perp \text {Susp}_{\hat {\mathcal {D}}}(\mathcal {T}_0)[1]\cap \text {Susp}_{\hat {\mathcal {D}}}(\mathcal {T}_0)$ of the t-structure $({}^\perp (\mathcal {T}_0^{\perp _{\leq 0}}),\mathcal {T}_0^{\perp _{<0}})$ in $\hat {\mathcal {D}}$. By [Reference Nicolás, Saorín and Zvonareva47, lemma 6], we conclude that $\mathcal {C}\cap \hat {\mathcal {D}}^c\subseteq \text {Add}(\mathcal {T}_0)$ and, since $\mathcal {C}\cap \hat {\mathcal {D}}^c$ consists of compact objects, $\mathcal {C}\cap \hat {\mathcal {D}}^c\subseteq \text {add}(\mathcal {T}_0)$. On the other hand, by proposition 4.4, $\mathcal {T}_0$ is a partial silting set in $\mathcal {D}$ which generates $(\mathcal {D}^{\leq 0},\mathcal {D}^{\geq 0})$. By the first paragraph of the proof, $\text {add}(\mathcal {T}_0)\subseteq \mathcal {C}\cap \hat {\mathcal {D}}^c$, and hence $\text {add}(\mathcal {T}_0)=\text {add}(\mathcal {T})$.  □