Recollements of triangulated categories play an important role in algebraic geometry (see [1]), representation theory (see [2-5]), etc. A recollement $(\mathcal C', \mathcal C, \mathcal C'')$ of triangulated categories provides a platform for various questions concerning the three terms in a recollement. For examples, given a recollement of a triangulated category $\mathcal C$ relative to $\mathcal C'$ and $\mathcal C''$, $t$-structures $(\mathcal{C'}^{\leq0}, \mathcal{C'}^{\geq0})$ and $(\mathcal{C''}^{\leq0}, \mathcal{C''}^{\geq0})$ of $\mathcal C'$ and $\mathcal C''$, respectively, Beilinson, Bernstein and Deligne [1] proved that $\mathcal C$ also has a $t$-structure $(\mathcal{C}^{\leq0}, \mathcal{C}^{\geq0})$, where

$ \begin{align*}\mathcal{C}^{\leq0}: &= \{A\in\mathcal C \ | \ j^*A \in \mathcal{C''}^{\leq0}, \ \ i^*A \in \mathcal{C'}^{\leq0}\ \}, \\ \mathcal{C}^{\geq0}: &= \{B\in\mathcal C \ | \ j^*B \in \mathcal{C''}^{\geq0}, \ \ i^!B \in \mathcal{C'}^{\geq0}\ \}.\end{align*} $

On the other hand, Lin [6] proved that certain $t$-structure on $\mathcal C$ may induce $t$-structures on $\mathcal C'$ and $\mathcal C''$. Chen [7] studied the relationship of cotorsion pairs among three triangulated categories in a recollement. She proved the following results: cotorsion pairs on $\mathcal C$ may be obtained from cotorsion pairs on $\mathcal C'$ and $\mathcal C''$ and certain cotorsion pairs on $\mathcal C$ may induce cotorsion pairs on $\mathcal C'$ and $\mathcal C''$. More relevant results can be seen in [8-11], etc.

In a viewpoint of Beilinson, Ginsburg and Schechtman (see [12]), upper and lower recollements are more fundamental than a recollement (upper and lower recollements are called steps in [8]). For a given upper (lower) recollement of $\mathcal C$ relative to $\mathcal C'$ and $\mathcal C''$, a sufficient condition that $t$-structures on $\mathcal C'$ and $\mathcal C''$ may be induced by a $t$-structure on $\mathcal C$ is given in this paper.

Recall the following definitions.

Definition 2.1  Let $\mathcal C'$, $\mathcal C$ and $\mathcal C''$ be triangulated categories.

(1) [1] A recollement of $\mathcal C$ relative to $\mathcal C'$ and $\mathcal C''$ is a diagram of triangle functors

$ {\mathcal{C}}'\ \underset{\xleftarrow{{{\mathit{i}}^{\rm{!}}}}}{\overset{\xleftarrow{{{\mathit{i}}^{\mathit{*}}}}}{\mathop{\ \xrightarrow{{{\mathit{i}}_{\mathit{*}}}}}}}\, \ \ \ \mathcal{C}\ \ \ \underset{\xleftarrow{{{\mathit{j}}_{\mathit{*}}}}}{\overset{\xleftarrow{{{\mathit{j}}_{\rm{!}}}}}{\mathop{\xrightarrow{{{\mathit{j}}^{\rm{*}}}}}}}\, \ \ {\mathcal{C}}'' $

(2.1)

such that

(R1) $(i^*, i_*), (i_*, i^!), (j_!, j^*)$ and $(j^*, j_*)$ are adjoint pairs;

(R2) $i_*, j_!$ and $j_*$ are fully faithful;

(R3) $j^*i_*=0$;

(R4) for each $X\in\mathcal C$, there are distinguished triangles

$ \begin{eqnarray*}&&j_!j^*X \stackrel{\epsilon_X}\longrightarrow X \stackrel{\eta_X}\longrightarrow i_*i^*X\longrightarrow (j_!j^*X)[1], \\ &&i_* i^!X\stackrel{\omega_X}\longrightarrow X\stackrel{\zeta_X}\longrightarrow j_*j^*X\longrightarrow (i_* i^!X)[1], \end{eqnarray*} $

where $\epsilon_X$ is the counit of $(j_!, j^*)$, $\eta_X$ is the unit of $(i^*, i_*)$, $\omega_X$ is the counit of $(i_*, i^!)$, and $\zeta_X$ is the unit of $(j^*, j_*)$.

(2) [5, 12, 13] Let $\mathcal C'$, $\mathcal C$ and $\mathcal C''$ be triangulated categories. An upper recollement of $\mathcal C$ relative to $\mathcal C'$ and $\mathcal C''$ is a diagram of triangle functors

$ {\mathcal{C}}'\ \ \ \ \begin{matrix} \xleftarrow{{{\mathit{i}}^{\mathit{*}}}} \\ \xrightarrow{{{\mathit{i}}_{\mathit{*}}}} \\ \end{matrix}\ \ \mathcal{C}\ \ \begin{matrix} \xleftarrow{{{\mathit{j}}_{\rm{!}}}} \\ \xrightarrow{{{\mathit{j}}^{\rm{*}}}} \\ \end{matrix}\ \ {\mathcal{C}}'' $

(2.2)

such that the conditions involved $i^\ast, i_\ast, j_!, j^\ast$ in (1) are satisfied.

(3) [5, 12, 13] Let $\mathcal C'$, $\mathcal C$ and $\mathcal C''$ be triangulated categories. An lower recollement of $\mathcal C$ relative to $\mathcal C'$ and $\mathcal C''$ is a diagram of triangle functors

$ {\mathcal{C}}'\ \ \ \ \begin{matrix} \xrightarrow{{{\mathit{i}}_{\mathit{*}}}} \\ \xleftarrow{{{\mathit{i}}^{\rm{!}}}} \\ \end{matrix}\ \ \ \ \mathcal{C}\ \ \begin{matrix} \xrightarrow{{{\mathit{j}}^{\rm{*}}}} \\ \xleftarrow{{{\mathit{j}}_{\mathit{*}}}} \\ \end{matrix}\ \ \ {\mathcal{C}}'' $

(2.3)

such that the conditions involved $i_\ast, i^!, j^\ast, j_\ast$ in (1) are satisfied.

For short, we denote respectively the recollement $(2.1)$, upper recollement $(2.2)$ and lower recollement $(2.3)$ by $(\mathcal C', \mathcal C, \mathcal C'', i^\ast, i_\ast$, $i^!, j_!, j^\ast, j_\ast)$, $(\mathcal C', \mathcal C, \mathcal C''$, $i^\ast, i_\ast, j_!, j^\ast)$ and $(\mathcal C', \mathcal C, \mathcal C''$, $i_\ast, i^!, j^\ast, j_\ast)$, or uniformly by $(\mathcal C', \mathcal C, \mathcal C'')$.

We need the following fact.

Lemma 2.2(see [14])  Let $\rm {(}\mathcal{C'}, \mathcal{C}, \mathcal{C''}{\rm )}$ be an upper recollement. Then there exists a triangle-equivalence $\widetilde{j^{*}}:{\mathcal C}/{i_{*}{\mathcal C'}} \cong \mathcal C''$ such that $\widetilde{j^{*}}V=j^{*}$, where $V: \mathcal C\to \mathcal C/{i_{*}{\mathcal C'}}$ is the Verdier functor.

The subcategories in this section are full subcategories closed under isomorphisms.

Definition 2.3[1]  Let $\mathcal C$ be a triangulated category with the shift functor [1]. A $t$-structure on $\mathcal{D}$ is a pair of full subcategories $(\mathcal{D}^{\leq0}, \mathcal{D}^{\geq0})$ with the following properties:

If we put $\mathcal{D}^{\leq n}:=\mathcal{D}^{\leq0}[-n]$ and $\mathcal{D}^{\geq n}:=\mathcal{D}^{\geq0}[-n], \ \forall \ n\in\Bbb Z, $ we have

(t1) ${\rm{Hom}}_{\mathcal{D}}(X, Y)=0, \ \forall \ X\in \mathcal{D}^{\leq0}, \ Y\in\mathcal{D}^{\geq1};$

(t2) $\mathcal{D}^{\leq0}\subseteq\mathcal{D}^{\leq1}$ and $\mathcal{D}^{\geq1}\subseteq\mathcal{D}^{\geq0}$;

(t3) For each $X\in\mathcal{D}$, there is a distinguished triangle

$ A\longrightarrow X\longrightarrow B\longrightarrow A[1], $

where $A\in\mathcal{D}^{\leq0}, $ $B\in\mathcal{D}^{\geq1}$.

Let $(\mathcal{U}, \mathcal{V})$ be a $t$-structure on $\mathcal{C}$. We call $(\mathcal{U}, \mathcal{V})$ a stable $t$-structure, if $\mathcal{U}$ and $\mathcal{V}$ are triangulated subcategories of $\mathcal{C}$ (see [15, Definition 0.2]).

Here are basic properties of stable $t$-structures.

Lemma 2.4(see [15])  Let $\mathcal D$ be a triangulated category, $\mathcal C$ a thick subcategory of $\mathcal D$, and $Q: \mathcal{D}\rightarrow \mathcal{D}/{\mathcal C}$ the canonical quotient. For a stable $t$-structure $(\mathcal U, \mathcal V)$ on $\mathcal D$, the following are equivalent.

(ⅰ) $(Q(\mathcal U), Q(\mathcal V))$ is a stable $t$-structure on $\mathcal D/{\mathcal C}$, where $Q(\mathcal U)$ (resp. $Q(\mathcal V))$ is the full subcategory of $\mathcal D/{\mathcal C}$ consisting of objects $Q(\mathcal X)$ for $X\in \mathcal U$ (resp. $Q(\mathcal Y)$ for $Y\in \mathcal V)$;

(ⅱ) $(\mathcal U \cap \mathcal C, \mathcal V\cap \mathcal C)$ is a stable $t$-structure on $\mathcal C$.

Definition 2.5[1]  Let $\mathcal C$ and $\mathcal D$ be two triangulated categories with $t$-structures $(\mathcal{C}^{\leq0}, \mathcal{C}^{\geq0})$ and $(\mathcal{D}^{\leq0}, \mathcal{D}^{\geq0})$. An triangle functor $F:\mathcal{C}\longrightarrow \mathcal{D}$ is

(ⅰ) left $t$-exact if $F(\mathcal{C}^{\geq0})\subset \mathcal{D}^{\geq0};$

(ⅱ) right $t$-exact if $F(\mathcal{C}^{\leq0})\subset \mathcal{D}^{\leq0}.$

This section aims to prove the main result of this paper. Let $\mathcal C', \mathcal C$ and $\mathcal C''$ be triangulated categories. Given a upper recollement of $\mathcal C$ relative to $\mathcal C'$ and $\mathcal C''$, a $t$-structure on $\mathcal C$ induces $t$-structures on $\mathcal C'$ and $\mathcal C''$ under some conditions.

Proposition 3.6  Let $\mathcal C'$, $\mathcal C$ and $\mathcal C''$ be triangulated categories, let diagram $(2.2)$ be an upper recollement of $\mathcal C$ relative to $\mathcal C'$ and $\mathcal C''$, and let $(\mathcal{C}^{\leq0}, \mathcal{C}^{\geq0})$ be a $t$-structure on $\mathcal C$. If $i_*i^*$ is left $t$-exact and $j_!j^*$ is right $t$-exact, then

(ⅰ) $(i^*(\mathcal{C}^{\leq0}), i^*(\mathcal{C}^{\geq0}))$ is a $t$-structure on $\mathcal C'$;

(ⅱ) $(j^*(\mathcal{C}^{\leq0}), j^*(\mathcal{C}^{\geq0}))$ is a $t$-structure on $\mathcal C''$;

(ⅲ) If $(\mathcal{C}^{\leq0}, \mathcal{C}^{\geq0})$ and $(i^*(\mathcal{C}^{\leq0}), i^*(\mathcal{C}^{\geq0}))$ are stable $t$-structures on $\mathcal C$ and $\mathcal C'$, respectively, then $(j^*(\mathcal{C}^{\leq0}), j^*(\mathcal{C}^{\geq0}))$ is a stable $t$-structure on $\mathcal C''$.

Proof  (ⅰ) For $X \in \mathcal{C}^{\leq0}$, $Y\in \mathcal{C}^{\geq1}$, since $(i^\ast, i_\ast)$ is an adjoint pair and $i_*i^*$ is left $t$-exact, we have ${\rm Hom}_{\mathcal{C'}}(i^*X, i^*Y)\cong {\rm Hom}_{\mathcal{C}}(X, i_*i^*Y)=0$. Thus (t1) hold.

Condition (t2) follows from the closure of $\mathcal{C}^{\leq0}$ and $\mathcal{C}^{\geq0}$ under the shifts [1] and [-1], respectively.

Let $X^{'}\in \mathcal{C'}$. There is a distinguished triangle $A\rightarrow i_*X'\rightarrow B\rightarrow A$[1] in $\mathcal{C}$, where $A \in \mathcal{C}^{\leq0}$, $B\in \mathcal{C}^{\geq1}.$ Applying $i^*$ to this triangle, we have $i^*A\rightarrow i^*i_*X'\rightarrow i^*B\rightarrow i^*A $[1], where $i^*A \in i^*(\mathcal{C}^{\leq0})$, $i^*B\in i^*(\mathcal{C}^{\geq1}).$ Since $i_*$ is fully faithful and $(i^\ast, i_\ast)$ is an adjoint pair, we have $i^*i_*X'\cong X'$. Therefore, the distinguished triangle $i^*A\rightarrow X'\rightarrow i^*B\rightarrow i^*A$[1] is the $t$-decomposition of $X'$. We have condition (t3).

(ⅱ) Similarly, we obtain argument (ⅱ).

(ⅲ) We prove the last statement by three steps.

Step 1  $j_!j^*$ is right $t$-exact $\Rightarrow$ $i_*i^*$ is right $t$-exact.

Let $X \in \mathcal{C}^{\leq0}$, for $Y\in \mathcal{C}^{\geq1}$. Applying cohomological functor ${\rm Hom}_{\mathcal{C}}(-, Y)$ to the distinguished triangle

$ j_!j^*X \stackrel{\epsilon_X}\longrightarrow X \stackrel{\eta_X}\longrightarrow i_*i^*X\longrightarrow (j_!j^*X)[1], $

we get an exact sequence

$ \cdots \rightarrow{\rm Hom}_{\mathcal{C}}(X[1], Y)\rightarrow {\rm Hom}_{\mathcal{C}}(j_{!}j^{*}X[1], Y)\rightarrow\\ {\rm Hom}_{\mathcal{C}}(i_*i^*X, Y)\rightarrow {\rm Hom}_{\mathcal{C}}(X, Y)\rightarrow \cdots. $

Since ${\rm Hom}_{\mathcal{C}}(X, Y)={\rm Hom}_{\mathcal{C}}(X[1], Y)=0$, we get ${\rm Hom}_{\mathcal{C}}(i_*i^*X, Y)\cong{\rm Hom}_{\mathcal{C}}(j_{!}j^{*}X[1], Y)=0$.

Step 2  We claim $i_*i^*(\mathcal{C}^{\leq0})=i_*\mathcal{C'}\cap \mathcal{C}^{\leq0}$ and $i_*i^*(\mathcal{C}^{\geq0})=i_*\mathcal{C'}\cap \mathcal{C}^{\geq0}$.

By Step 1 we have $i_*i^*$ is right $t$-exact, i.e. $i_*i^*(\mathcal{C}^{\leq0})\subseteq \mathcal{C}^{\leq0}$. Therefore, $i_*i^*(\mathcal{C}^{\leq0})\subseteq i_*\mathcal{C'}\cap \mathcal{C}^{\leq0}$. Conversely, for $X\in i_*\mathcal{C'}\cap \mathcal{C}^{\leq0}$, there exists a distinguished triangle $j_{!}j^{*}X\rightarrow X\rightarrow i_{*}i^{*}X\rightarrow (j_{!}j^{*}X)$[1]. Since $ X\in i_*\mathcal{C'}$, it follows $j_{!}j^{*}X=0$. Since $X$ is in $\mathcal{C}^{\leq0}$, we have $X\cong i_{*}i^{*}X\subseteq i_*i^*(\mathcal{C}^{\leq0})$.

Similarly we have $i_*i^*(\mathcal{C}^{\geq0})=i_*\mathcal{C'}\cap \mathcal{C}^{\geq0}$.

Therefore, $(i_*\mathcal{C'}\cap \mathcal{C}^{\leq0}, i_*\mathcal{C'}\cap \mathcal{C}^{\geq0})$=$(i_*i^*(\mathcal{C}^{\leq0}), i_*i^*(\mathcal{C}^{\geq0}))$.

Step 3  Assume that $(i^*(\mathcal{C}^{\leq0}), i^*(\mathcal{C}^{\geq0}))$ is a stable $t$-structure on $\mathcal C'$. Since $i_*$ is fully faithful, $(i_*i^*(\mathcal{C}^{\leq0}), i_*i^*(\mathcal{C}^{\geq0}))$ is a stable $t$-structure on $i_*\mathcal C'$. By Step 2, $(i_*\mathcal{C'}\cap \mathcal{C}^{\leq0}, i_*\mathcal{C'}\cap \mathcal{C}^{\geq0})$ is a stable $t$-structure on $i_*\mathcal C'$. Hence $(Q(\mathcal{C}^{\leq0}), Q(\mathcal{C}^{\geq0}))$ is a stable $t$-structure on $\mathcal C/{i_*{\mathcal C'}}$ by Lemma 2.4. There exists a triangle-equivalence $\widetilde{j^{*}}:{\mathcal C}/{i_{*}{\mathcal C'}} \cong \mathcal C''$ such that $j^{*}=\widetilde{j^{*}}Q$, so $(j^*(\mathcal{C}^{\leq0}), j^*(\mathcal{C}^{\geq0}))$ is a stable $t$-structure on $\mathcal C''$. The proof is completed.

By the similar argument we have statements for lower recollements.

Corollary 3.7  Let $\mathcal C'$, $\mathcal C$ and $\mathcal C''$ be triangulated categories, let diagram $(2.3)$ be a lower recollement of $\mathcal C$ relative to $\mathcal C'$ and $\mathcal C''$, and let $(\mathcal{C}^{\leq0}, \mathcal{C}^{\geq0})$ a $t$-structure on $\mathcal C$. If $i_*i^!$ is right $t$-exact and $j_*j^*$ is left $t$-exact, then

(ⅰ) ($i^!(\mathcal{C}^{\leq0}), i^!(\mathcal{C}^{\geq0}))$ is a $t$-structure on $\mathcal C'$;

(ⅱ) ($j^*(\mathcal{C}^{\leq0}), j^*(\mathcal{C}^{\geq0}))$ is a $t$-structure on $\mathcal C''$;

(ⅲ) If $(\mathcal{C}^{\leq0}, \mathcal{C}^{\geq0})$ and ($i^!(\mathcal{C}^{\leq0}), i^!(\mathcal{C}^{\geq0}))$ are stable $t$-structures on $\mathcal C$ and $\mathcal C'$, respectively, then ($j^*(\mathcal{C}^{\leq0}), j^*(\mathcal{C}^{\geq0}))$ is a stable $t$-structure on $\mathcal C''$.