Silting objects and recollements of extriangulated categories

1.   Introduction

Belinson et al. ^[1] introduced the notions of recollements of abelian and triangulated categories, which were in connection with derived categories of sheaves on topological spaces, with the idea that one triangulated category may be glued together from two others. The recollements of abelian and triangulated categories are closely related, and they play an important role in algebraic geometry and representation theory; see, for instance, ^[2,3,4].

Gluing techniques concerning the recollement of triangulated or abelian categories have been investigated by more and more experts and scholars. For example, the glued cotorsion pairs ^[5], the glued torsion pairs ^[6], and so on (e.g., ^[7,8]). In particular, Liu et al. ^[9] presented explicit constructions of the gluing of silting objects for a recollement of triangulated categories.

Nakaoka and Palu ^[10] introduced an extriangulated category that extracts properties from triangulated categories and exact categories. In particular, exact categories and triangulated categories are extriangulated categories. There are a lot of examples of extriangulated categories, which are neither exact nor triangulated categories; see ^[10,11]. In the work of ^[12], Wang et al. defined the recollement of extriangulated categories, which generalized recollements of abelian categories and triangulated categories.

Motivated by the applications of gluing techniques in recollements, we want to consider gluing a silting object and a tilting object from two other silting objects and tilting objects in a recollement of extriangulated categories.

In this paper, let $(\mathscr{A}, \mathscr{C}, \mathscr{B})$ be a recollement of extriangulated categories. Our first main result describes how to glue together silting objects $\sigma$ in $\mathscr{A}$ and $\omega$ in $\mathscr{B}$ to obtain a silting object $\rho$ in $\mathscr{C}$; see Theorem 3.3. In the reverse direction, our second main result gives sufficient conditions for a silting object $\rho$ of $\mathscr{C}$, relative to the functors involved in the recollement, to induce silting objects in $\mathscr{A}$ and $\mathscr{B}$; see Theorem 3.5, which partially extends the results of [9, Theorem 3.1], [13, Theorem 2.10] and [14, Theorem 2.5] in the framework of extriangulated categories. Our third main result constructs a glued tilting object from tilting objects in $\mathscr{A}$ and in $\mathscr{B}$; see Theorems 4.2 and 4.3, which extend the result in [15, Theorem 3.5] to extriangulated categories.

2.   Preliminaries

This section briefly recalls some definitions and basic properties of extriangulated categories from ^[10,11]. We omit some details here, but we refer the readers to ^[10].

Let $\mathscr{C}$ be an additive category. All subcategories considered are full additive subcategories closed under isomorphisms. Let

be a biadditive functor, where $Ab$ is an abelian category. For any pair of objects $A, C\in\mathscr{C}$, an element $\delta\in\mathbb{E}(C, A)$ is called an $\mathbb{E}$-extension.

Let $\mathfrak{s}$ be a correspondence that associates an equivalence class

to any $\mathbb{E}$-extension $\delta\in\mathbb{E}(C, A)$. This $\mathfrak{s}$ is called a realization of $\mathbb{E}$ if it makes the diagrams in [10, Definition 2.9] commutative.

We call a triplet $(\mathscr{C}, \mathbb{E}, \mathfrak{s})$ an extriangulated category if it satisfies:

(1) $\mathbb{E}: \mathscr{C}^{op}\times \mathscr{C}\rightarrow Ab$ is a biadditive functor.

(2) $\mathfrak{s}$ is an additive realization of $\mathbb{E}$.

(3) $\mathbb{E}$ and $\mathfrak{s}$ satisfy the compatibility conditions $(ET3)$, $(ET3)^{op}$, $(ET4)$, and $(ET4)^{ op}$ in [10, Definition 2.12].

Remark 2.1. By [10, Example 2.13 and Proposition 3.22], triangulated categories and exact categories (with a condition concerning the smallness) are typical examples of extriangulated categories.

For an extriangulated category $\mathscr{C}$, we use the following notations in ^[10,11]:

● A sequence $A\stackrel{a}\rightarrow B \stackrel{b}\rightarrow C\stackrel{\delta}\dashrightarrow$ is called an $\mathbb{E}$-triangle.

● Let $A\stackrel{a}\rightarrow B \stackrel{b}\rightarrow C\stackrel{\delta}\dashrightarrow$ be an $\mathbb{E}$-triangle. $A$ is called the CoCone of $b$ and denoted by $CoCone(B, C)$; $C$ is called the Cone of $a$ and denoted by $Cone(A, B)$.

● An object $P$ is called projective if, for any $\mathbb{E}$-triangle $A\stackrel{x}\rightarrow B \stackrel{y}\rightarrow C\stackrel{\delta}\dashrightarrow$ and any morphism $c\in\mathscr{C}(P, C)$, there exists a morphism $b\in\mathscr{C}(P, B)$, satisfying $y\circ b = c$. The class of all the projective objects in $\mathscr{C}$ is denoted by $\mathcal{P}_{\mathscr{C}}$. An injective object can be dually defined, and the class of all the injective objects in $\mathscr{C}$ is denoted by $\mathcal{I}_{\mathscr{C}}$.

● We say that $\mathscr{C}$ has enough projective objects if, for any object $C \in \mathscr{C}$, there exists an $\mathbb{E}$-triangle $K\stackrel{x}\rightarrow P \stackrel{y}\rightarrow C\stackrel{\delta}\dashrightarrow$ satisfying $P\in \mathcal{P}_{\mathscr{C}}$. Dually, we can define $\mathscr{C}$ as having enough injective objects.

Let $\mathcal{X}$, $\mathcal{Y}$ be subcategories of $\mathscr{C}$.

● Denote $Cone(\mathcal{X}, \mathcal{Y})$ by the subcategory of $\mathscr{C}$ consisting of $M \in\mathscr{C}$, which admits an $\mathbb{E}$-triangle $X \rightarrow Y \rightarrow M\dashrightarrow$ in $\mathscr{C}$, with $X \in \mathcal{X}$ and $Y \in \mathcal{Y}$. We say that $\mathcal{X}$ is closed under $Cones$ if $Cone(\mathcal{X}, \mathcal{X}) \subseteq \mathcal{X}$.

● Denote $CoCone(\mathcal{X}, \mathcal{Y})$ by the subcategory of $\mathscr{C}$ consisting of $M \in\mathscr{C}$, which admits an $\mathbb{E}$-triangle $M \rightarrow X\rightarrow Y\dashrightarrow$ in $\mathscr{C}$, with $X \in \mathcal{X}$ and $Y \in \mathcal{Y}$. We say that $\mathcal{X}$ is closed under $CoCones$ if $CoCone(\mathcal{X}, \mathcal{X}) \subseteq \mathcal{X}$.

● We denote

And $\Omega$ is called the syzygy of $\mathcal{X}$. Dually, we define the cosyzygy of $\mathcal {X}$ by

see [16, Definition 4.2, and Proposition 4.3].

For any subcategory $\mathcal{X}$ of $\mathscr{C}$, put

for $k\geq1$, define $\Omega^{k}\mathcal{X}$ inductively by

$\Omega^{k}\mathcal{X}$ is called the $k$-th syzygy of $\mathcal{X}$. Dually the $k$-th cosyzygy $\Sigma^{k}\mathcal{X}$ by

for $k\geq1$ can be defined.

For any $k > 0$ and any objects $X, Y$, Liu and Nakaoka [16, Proposition 5.2] defined the higher extension groups in an extriangulated category having enough projective and injective objects as

They showed the following result:

Lemma 2.2. [16, Proposition 5.2] Let $A\stackrel{f}\rightarrow B\stackrel{g}\rightarrow C\stackrel{\delta}\dashrightarrow$ be an $\mathbb{E}$-triangle. For any object $X\in\mathscr{C}$, there are long exact sequences

and

We define

Dually, we define

Definition 2.3. [17, Definition 3.1] Let $\mathscr{C}$ be an extriangulated category, $\mathcal{X}$ and $\mathcal{Y}$ subcategories of $\mathscr{C}$. We call a pair $(\mathcal{X}, \mathcal{Y})$ a hereditary cotorsion pair if:

(1) Both $\mathcal{X}$ and $\mathcal{Y}$ are closed under direct summands;

(2) $\mathbb{E}^{k}(X, Y) = 0$ for any $k\geq1$ and $X\in \mathcal{X}$ and $Y\in \mathcal{Y}$;

(3) $\mathscr{C} = Cone(\mathcal{Y}, \mathcal{X})$;

(4) $\mathscr{C} = CoCone(\mathcal{Y}, \mathcal{X})$.

We recall the concepts and basic properties of recollements of extriangulated categories from ^[12].

Definition 2.4. [12, Definition 3.1] Let $\mathscr{A}$, $\mathscr{C}$, and $\mathscr{B}$ be three extriangulated categories. A recollement of $\mathscr{C}$ relative to $\mathscr{A}$ and $\mathscr{B}$, denoted by $(\mathscr{A}, \mathscr{C}, \mathscr{B})$, is a diagram

given by two exact functors $i_{*}$, $j^{*}$, two right exact functors $i^{*}$, $j_{!}$, and two left exact functors $i^{!}$, $j_{*}$, which satisfies the following conditions:

$({\rm{R1}})$ $(i^{*}, i_{*}, i^{!})$ and $(j_{!}, j^{*}, j_{*})$ are adjoint triples.

$({\rm{R2}})$ Im $i_{*} =$Ker $j^{*}$.

$({\rm{R3}})$ $i_{*}$, $j_{!}$, $j_{*}$ are fully faithful.

$({\rm{R4}})$ For each $C\in\mathscr{C}$, there exists a left exact $\mathbb{E}_{\mathscr{C}}$-triangle sequence

with $A\in\mathscr{A}$.

$({\rm{R5}})$ For each $C\in\mathscr{C}$, there exists a right exact $\mathbb{E}_{\mathscr{C}}$-triangle sequence

with $A_{1}\in\mathscr{A}$.

Remark 2.5. (1) If the categories $\mathscr{A}$, $\mathscr{B}$, and $\mathscr{C}$ are triangulated, then Definition 2.4 coincides with the definition of a recollement of triangulated categories ^[1].

(2) If the categories $\mathscr{A}$, $\mathscr{B}$, and $\mathscr{C}$ are abelian, then Definition 2.4 coincides with the definition of a recollement of abelian categories ^[3,4,6].

Lemma 2.6. [12, Lemma 3.3] Let $(\mathscr{A}, \mathscr{C}, \mathscr{B})$ be a recollement of extriangulated categories as (2.1) in Definition 2.4.

(1) All the natural transformations $i^{*}i_{*}\Rightarrow Id_{\mathscr{A}}$, $Id_{\mathscr{A}} \Rightarrow i^{!}i_{*}$, $Id_{\mathscr{B}}\Rightarrow j^{*}j_{!}$, $j^{*}j_{*}\Rightarrow Id_{\mathscr{B}}$ are natural isomorphisms.

(2) $i^{*}j_{!} = 0 = i^{!}j_{*}$.

(3) $i^{*}$ preserves projective objects, and $i^{!}$ preserves injective objects.

(4) $j_{!}$ preserves projective objects, and $j_{*}$ preserves injective objects.

(5) If $i^{!}$ (resp. $j_{*}$) is exact, then $i_{*}$ (resp. $j^{*}$) preserves projective objects.

(6) If $i^{*}$ (resp. $j_{!}$) is exact, then $i_{*}$ (resp. $j^{*}$) preserves injective objects.

(7) If $\mathscr{C}$ has enough projectives, then $\mathscr{A}$ has enough projectives; if $\mathscr{C}$ has enough injectives, then $\mathscr{A}$ has enough injectives.

(8) If $\mathscr{C}$ has enough projectives and $j_{*}$ is exact, then $\mathscr{B}$ has enough projectives; if $\mathscr{C}$ has enough injectives and $j_{!}$ is exact, then $\mathscr{B}$ has enough injectives.

(9) If $i^{*}$ is exact, then $j_{!}$ is exact.

(10) If $i^{!}$ is exact, then $j_{*}$ is exact.

The following results will be used frequently in this paper; their proof is very similar to [18, Proposition 2.8], we omit it.

Lemma 2.7. Let $(\mathscr{A}, \mathscr{C}, \mathscr{B})$ be a recollement of extriangulated categories as (2.1) in Definition 2.4.

(1) If $\mathscr{A}$ has enough projectives and $i^{!}$ is exact, then

for any $k\geq1$, $X\in\mathscr{A}$ and $Y\in\mathscr{C}$.

(2) If $\mathscr{A}$ has enough injectives and $i^{*}$ is exact, then

for any $k\geq1$, $X\in\mathscr{C}$ and $Y\in\mathscr{A}$.

(3) If $\mathscr{B}$ has enough projectives and $j_{!}$ is exact, then

for any $k\geq1$, $Z\in\mathscr{B}$ and $Y\in\mathscr{C}$.

3.   Gluing silting objects

In this section, we always assume $\mathscr{A}, \mathscr{C}$ and $\mathscr{B}$ are three extriangulated categories with enough projective and injective objects.

A subcategory $\mathcal{X}$ of $\mathscr{C}$ is called thick if it is closed under extensions, $Cones$, $CoCones$, and direct summands. Let $thick\mathcal{X}$ denote the smallest thick subcategory containing $\mathcal{X}$. When $\mathcal{X}$ is a single object $M$, then $thick (M) = thick$ (add $M$).

Definition 3.1. An object $\sigma$ in an extriangulated category $\mathscr{C}$ is silting if

(1) $\mathbb{E}^{k}(\sigma, \sigma) = 0$ for all $k\geq1$;

(2) $thick (\sigma) = \mathscr{C}$.

For an object $\sigma$, add $\sigma$ denotes the closure of $\sigma$ under finite direct sums and summands. Clearly, when $\sigma$ is a silting object, then add $\sigma$ is a silting subcategory in the sense of [17, Definition 5.1]. When $\mathscr{C}$ is a well generated triangulated category, Definition 3.1 coincides with [13, Definition 1.11].

Example 3.2. Let $R$ be any ring, and let $\mathcal{P}^{\wedge}(R)$ denote the category of right $R$-modules of finite projective dimension. Since $\mathcal{P}^{\wedge}(R)$ is closed under extensions, it is an extriangulated category. We can easily check that $R$ is a silting object of $\mathcal{P}^{\wedge}(R)$.

For any subcategory $\mathcal{X}$ of $\mathscr{C}$, let

We denote the full subcategories $\mathcal{X}_{i}^{\vee}$ and $\mathcal{X}_{i}^{\wedge}$ for $i > 0$ inductively by

Put

Proposition 3.3. Assume that $\mathscr{C}$ admits a recollement relative to $\mathscr{A}$ and $\mathscr{B}$ as (2.1) in Definition 2.4 and $\sigma$, $\omega$ are silting objects in $\mathscr{A}$ and $\mathscr{B}$, respectively. If $i^{*}$ is exact, then

Proof. Let $C\in\mathscr{C}$. As $i^{*}$ is exact, there exists an $\mathbb{E}$-triangle by [12, Proposition 3.4],

Clearly, $i^{*}C\in\mathscr{A}$ and $j^{*}C\in\mathscr{B}$. By [17, Theorem 5.7], there exist two hereditary cotorsion pairs ((add $\sigma)^{\vee},$ (add $\sigma)^{\wedge})$ in $\mathscr{A}$ and ((add $\omega)^{\vee},$ (add $\omega)^{\wedge})$ in $\mathscr{B}$, respectively. So we get two $\mathbb{E}$-triangles associated with the above two hereditary cotorsion pairs:

where $N\in$(add $\sigma)^{\wedge}$, $M\in$(add $\sigma)^{\vee}$ and $V\in$(add $\omega)^{\wedge}$, $U\in$(add $\omega)^{\vee}$.

Since $i_{*}$ is exact, $i_{*}$(add $\sigma)^{\wedge}\subseteq$ (add $i_{*}\sigma)^{\wedge}$ and $i_{*}$(add $\sigma)^{\vee}\subseteq$ (add $i_{*}\sigma)^{\vee}$. So $i_{*}i^{*}C\in thick (i_{*}\sigma)$ by the $\mathbb{E}$-triangle (2). Similarly, since $i^{*}$ is exact, $j_{!}$ is exact. So $j_{!}j^{*}C\in thick (j_{!}\omega)$ by the $\mathbb{E}$-triangle (3). Hence $C\in thick (j_{!}\omega\oplus i_{*}\sigma)$ by the $\mathbb{E}$-triangle (1). So

This completes the proof. □

Theorem 3.4. Let $(\mathscr{A}, \mathscr{C}, \mathscr{B})$ be a recollement as (2.1) in Definition 2.4. Assume $\sigma$ and $\omega$ are silting objects in $\mathscr{A}$ and $\mathscr{B}$, respectively. If

for any $k\geq1$ and $i^{*}$ is exact, then $j_{!}w\oplus i_{*}\sigma$ is a silting object in $\mathscr{C}$.

Proof. Since $i^{*}$ is exact, $j_{!}$ is exact by Lemma 2.6(9). So

and

for any $k\geq1$ by Lemmas 2.6(1) and 2.7(3). Since $i^{*}$ is exact, $i_{*}$ preserves injective objects. So

by Lemmas 2.6(1) and 2.7(2). So for any $k\geq1$, we get

Hence $j_{!}w\oplus i_{*}\sigma$ is a silting object in $\mathscr{C}$ by Proposition 3.2 and Definition 3.1. □

Let $\sigma$ and $\omega$ be silting objects; then there exist hereditary cotorsion pairs ((add$\sigma)^{\vee},$ (add$\sigma)^{\wedge})$ and ((add $\omega)^{\vee},$ (add $\omega)^{\wedge})$ in $\mathscr{A}$ and $\mathscr{B}$ by [17, Theorem 5.7], respectively. Define

and

By [12, Theorem 4.4], $(\mathcal{U}, \mathcal{V})$ is a glued hereditary cotorsion pair in $\mathscr{C}$.

On the other hand, by Theorem 3.3, we obtain a silting object

in $\mathscr{C}$. So there exists a hereditary cotorsion pair ((add $\rho)^{\vee},$ (add $\rho)^{\wedge})$ in $\mathscr{C}$.

The following result indicates that our gluing preserves the cotorsion pair:

Theorem 3.5. In the situation above, let

Then

Proof. Since $\sigma$ and $\omega$ are silting objects,

and

by [17, Lemma 4.11]. By Theorem 3.3, $\rho$ is a silting object in $\mathscr{C}$. So

Hence

by [17, Lemma 4.11]. So $C\in \mathcal{V}$ if and only if $i^{!}C\in \sigma^{\bot}$ and $j^{*}C\in\omega^{\bot}$. By Lemma 2.7, the latter holds if and only if $C\in (i_{*}\sigma)^{\bot}$ and $C\in (j_{!}\omega)^{\bot}$ if and only if

So

□

For the converse of Theorem 3.3, we have

Theorem 3.6. Let $(\mathscr{A}, \mathscr{C}, \mathscr{B})$ be a recollement as (2.1) in Definition 1.4. Assume that $\rho$ is a silting object in $\mathscr{C}$ and $i^{*}$ is exact,

(1) If $i_{*}i^{*}\rho\in$ add $\rho$, then $i^{*}\rho$ is a silting object in $\mathscr{A}$.

(2) If $j_{!}j^{*}\rho\in$ add $\rho$, then $j^{*}\rho$ is a silting object in $\mathscr{B}$.

In this case, $\rho$ is a glued silting object with respect to two silting objects, $\sigma$ in $\mathscr{A}$ and $\omega$ in $\mathscr{B}$.

Proof. (1) Since $i^{*}$ is exact,

for any $k\geq1$ by Lemma 2.11. Since $i_{*}i^{*}\rho\in$ add $\rho$,

by Definition 3.1. So

For any $A\in \mathscr{A}$, then $i_{*}A\in\mathscr{C}$. Since $\rho$ is a silting object in $\mathscr{C}$, there exists a hereditary cotorsion pair ((add $\rho)^{\vee},$ (add $\rho)^{\wedge})$ induced by $\rho$. So there exists an $\mathbb{E}$-triangle $V_{1}\rightarrow U_{1}\rightarrow i_{*}A\dashrightarrow$ with $U_{1}\in$(add $\rho)^{\vee}$ and $V_{1}\in$(add $\rho)^{\wedge}$. Since $i^{*}$ is exact, we get another $\mathbb{E}$-triangle: $i^{*}V_{1}\rightarrow i^{*}U_{1}\rightarrow i^{*}i_{*}A\dashrightarrow$, Clearly, $i^{*}V_{1}\in (i^{*}($add $\rho))^{\wedge}\subseteq thick (i^{*}\rho)$ and $i^{*}U_{1}\in (i^{*}$(add $\rho))^{\vee}\subseteq thick (i^{*}\rho)$. Also

so $A\in thick (i^{*}\rho)$ and

Hence, $i^{*}\rho$ is a silting object in $\mathscr{A}$.

(2) Since $i^{*}$ is exact, $j^{*}$ preserves injective objects by Lemma 2.10(6). So

for any $k\geq1$, by Lemma 2.7. Since $j_{!}j^{*}\rho\in$ add $\rho$,

For any $B\in \mathscr{B}$, using the same proof method, we can also obtain

Hence, $j^{*}\rho$ is a silting object in $\mathscr{B}$.

Denote $i^{*}\rho$ by $\sigma$ and $j^{*}\rho$ by $\omega$. Then $\sigma$ is a silting object in $\mathscr{A}$, and $\omega$ is a silting object in $\mathscr{B}$. Moreover, $i_{*}\sigma\in$ add $\rho$ and $j_{!}\omega\in$add $\rho$. So

By Theorem 3.3, $j_{!}\omega\oplus i_{*}\sigma$ is a silting object. Since $i^{*}$ is exact, there exists an $\mathbb{E}_{\mathscr{C}}$-triangle $j_{!}\omega\rightarrow \rho\rightarrow i_{*}\sigma\dashrightarrow$ by [12, Proposition 3.4]. So

and $\rho$ glued from $\sigma$ and $\omega$. □

By applying Theorem 3.5 to triangulated categories and using the fact that any triangulated category can be viewed as an extriangulated category, we get the following result:

Corollary 3.7. When ($\mathscr{A}, \mathscr{C}, \mathscr{B}$) is a recollement of triangulated categories, assume that $i^{*}$ is exact. The silting object $\rho$ glued from two silting objects $\sigma$ in $\mathscr{A}$ and $\omega$ in $\mathscr{B}$ if and only if $i_{*}i^{*}\rho\in$ add $\rho$ and $j_{!}j^{*}\rho\in$ add $\rho$, which is a special case of [13, Theorem 2.26].

4.   Gluing tilting objects

In this section, we restrict the gluing procedures to tilting objects. We always assume that $\mathscr{A}, \mathscr{C}, \mathscr{B}$ are three extriangulated categories with enough projective and injective objects.

The notion of tilting objects in an extriangulated category was introduced in [19, Definition 5.2], and we recall it.

Definition 4.1. Let $T$ be an object in an extriangulated category $\mathscr{C}$. It is called tilting if

(1) $T\in(\mathcal{P}_{\mathscr{C}})^{\wedge}$;

(2) $\mathbb{E}^{k}(T, T) = 0$, for all $k\geq1$;

(3) $\mathcal{P}_{\mathscr{C}}\subseteq$ (add $T)^{\vee}$.

Clearly, if $T$ is a tilting object, then

By [17, Proposition 4.10],

and by [19, Lemma 5.4], any tilting object of $\mathscr{C}$ is silting of $(\mathcal{P}_{\mathscr{C}})^{\wedge}$ and they coincide when every object in $\mathscr{C}$ has finite projective dimension by [19, Theorem 5.6].

Theorem 4.2. Let ($\mathscr{A}, \mathscr{C}, \mathscr{B}$) be a recollement of extriangulated categories as (2.1) in Definition 2.4 and $i^{*}$, $i^{!}$ exact. Assume that $T_{1}$ and $T_{2}$ are tilting objects in $\mathscr{A}$ and $\mathscr{B}$, respectively, such that

for all $k\geq1$. Then there exists a glued tilting object $\rho$ with respect to $T_{1}$ and $T_{2}$ in $\mathscr{C}$.

Proof. By [19, Lemma 5.4], $T_{1}$ and $T_{2}$ are silting objects of $(\mathcal{P}_{\mathscr{A}})^{\wedge}$ and $(\mathcal{P}_{\mathscr{B}})^{\wedge}$, respectively, and we have

and

Moreover, there exist two cotorsion pairs ((add $T_{1})^{\vee},$ (add $T_{1})^{\wedge})$ in $(\mathcal{P}_{\mathscr{A}})^{\wedge}$ and ((add $T_{2})^{\vee},$ (add $T_{2})^{\wedge})$ in $(\mathcal{P}_{\mathscr{B}})^{\wedge}$, respectively. Denote $j_{!}T_{2}\oplus i_{*}T_{1}$ by $\rho$. Since $i^{*}$ is exact, for any $k\geq1$,

and

by Lemmas 2.6 and 2.7.

So

and $\rho$ is self-orthogonal. Since $i^{*}$ is exact, $j_{!}$ is exact. Since $i^{!}$ is exact, $i_{*}$ preserves projective objects. So

by [19, Lemma 5.1].

Since $i^{*}$ is exact, for any $C\in(\mathcal{P}_{\mathscr{C}})^{\wedge}$, we have the following $\mathbb{E}$-triangle by [12, Proposition 3.4],

Since both $i^{*}$ and $j^{*}$ preserves projective objects, $i^{*}C\in(\mathcal{P}_{\mathscr{A}})^{\wedge}$ and $j^{*}C\in(\mathcal{P}_{\mathscr{B}})^{\wedge}$. So we get two $\mathbb{E}$-triangles associated with the above two cotorsion pairs:

where $N\in$(add $T_{1})^{\wedge}$, $M\in$(add $T_{1})^{\vee}$ and $V\in$(add $T_{2})^{\wedge}$, $U\in$(add $T_{2})^{\vee}$.

Since $i_{*}$ is exact, $i_{*}i^{*}C\in thick (i_{*}T_{1})$ by the $\mathbb{E}$-triangle (2). Similarly, $j_{!}j^{*}C\in thick (j_{!}T_{2})$ by the $\mathbb{E}$-triangle (3). Hence $C\in thick (j_{!}T_{2}\oplus i_{*}T_{1})$ by the $\mathbb{E}$-triangle (1). So

and thus $\rho$ is a silting object in $(\mathcal{P}_{\mathscr{C}})^{\wedge}$. Hence ((add $\rho)^{\vee},$ (add $\rho)^{\wedge})$ is a cotorsion pair in $(\mathcal{P}_{\mathscr{C}})^{\wedge}$. Since $\mathcal{P}_{\mathscr{C}}\subseteq (\mathcal{P}_{\mathscr{C}})^{\wedge}$, for any $M\in \mathcal{P}_{\mathscr{C}}$, there exists an $\mathbb{E}$-triangle $V\rightarrow U\rightarrow M\dashrightarrow,$ where $U\in$ (add $\rho)^{\vee}$ and $V\in$ (add $\rho)^{\wedge}$. Since $M$ is a project object, the $\mathbb{E}$-triangle splits. So $M$ is a direct summand of $U$, and thus $\mathcal{P}_{\mathscr{C}}\subseteq$ (add $\rho)^{\vee}$. So $\rho$ is a tilting object in $\mathscr{C}$ by Definition 4.1. □

For the converse direction,

Theorem 4.3. Let ($\mathscr{A}, \mathscr{C}, \mathscr{B}$) be a recollement of extriangulated categories as (2.1) in Definition 2.4 and $i^{*}$, $i^{!}$ are exact. Assume that $\rho$ is a tilting object in $\mathscr{C}$.

(1) If $i_{*}i^{*}\rho\in$ add $\rho$, then $i^{*}\rho$ is a tilting object in $\mathscr{A}$.

(2) If $j_{!}j^{*}\rho\in$ add $\rho$, then $j^{*}\rho$ is a tilting object in $\mathscr{B}$.

In this case, $\rho$ glued from two tilting objects in $\mathscr{A}$ and $\mathscr{B}$.

Proof. By Theorem 3.5, both $i^{*}\rho$ and $j^{*}\rho$ are self-orthogonal. Since $i^{*}$ is exact and $i^{*}$ preserves projective objects,

Since $i^{!}$ is exact, $j^{*}$ preserves projective objects. So

On the other hand, since $\rho$ is a tilting object in $\mathscr{C}$, $\mathcal{P}_{\mathscr{C}}\subseteq (\mbox{add} \rho)^{\vee}$. For any $M\in\mathcal{P}_{\mathscr{A}}$, then $i_{*}M\in \mathcal{P}_{\mathscr{C}}$. So

For any $N\in\mathcal{P}_{\mathscr{B}}$, then $j_{!}N\in \mathcal{P}_{\mathscr{C}}$. So

Hence, $i^{*}\rho$ is a tilting object in $\mathscr{A}$, and $j^{*}\rho$ is a tilting object in $\mathscr{B}$, by Definition 4.1.

Since $i^{*}$ is exact, we have an $\mathbb{E}$-triangle by [12, Proposition 3.4],

Since $i_{*}i^{*}\rho\in$ add $\rho$ and $j_{!}j^{*}\rho\in$ add $\rho$,

So

and thus $\rho$ glued from two tilting objects $i^{*}\rho$ in $\mathscr{A}$ and $j^{*}\rho$ in $\mathscr{B}$. □

Corollary 4.4. (1) If we let $\mathscr{C}$ be a triangulated category that is $K$-linear over a field $K$, $\mathscr{B}$ a localizing subcategory of $\mathscr{C}$, and

Then Theorem 4.2 is exactly [14, Theorem 2.5] when viewing triangulated categories as extriangulated categories.

(2) Let $\Lambda_{1}$, $\Lambda$, and $\Lambda_{2}$ be artin algebras,

Then Theorem 4.2 is [15, Theorem 3.5].

5.   Conclusions

In this section, we present the conclusions of the paper.

Let $(\mathscr{A}, \mathscr{C}, \mathscr{B})$ be a recollement as (2.1) in Definition 2.4. Our first main result describes how to glue together silting objects $\sigma$ in $\mathscr{A}$ and $\omega$ in $\mathscr{B}$ to obtain a silting object $\rho$ in $\mathscr{C}$; see Theorem 3.3.

Theorem 5.1. Assume $\sigma$ and $\omega$ are silting objects in $\mathscr{A}$ and $\mathscr{B}$, respectively. If

for any $k\geq1$ and $i^{*}$ is exact, then $j_{!}w\oplus i_{*}\sigma$ is a silting object in $\mathscr{C}$.

In the reverse direction, our second main result gives sufficient conditions for a silting object $\rho$ of $\mathscr{C}$, relative to the functors involved in the recollement, to induce silting objects in $\mathscr{A}$ and $\mathscr{B}$; see Theorem 3.5, which partially extends the results of [9, Theorem 3.1], [13, Theorem 2.10] and [14, Theorem 2.5] in the framework of extriangulated categories.

Theorem 5.2. Assume that $\rho$ is a silting object in $\mathscr{C}$ and $i^{*}$ is exact,

(1) If $i_{*}i^{*}\rho\in$ add $\rho$, then $i^{*}\rho$ is a silting object in $\mathscr{A}$.

(2) If $j_{!}j^{*}\rho\in$ add $\rho$, then $j^{*}\rho$ is a silting object in $\mathscr{B}$.

In this case, $\rho$ glued from two silting objects, $\sigma$ in $\mathscr{A}$ and $\omega$ in $\mathscr{B}$.

Our third main result constructs a glued tilting object from tilting objects in $\mathscr{A}$ and in $\mathscr{B}$; see Theorems 4.2 and 4.3, which extend the result in [15, Theorem 3.5] to extriangulated categories.

Theorem 5.3. Let $i^{*}$ and $i^{!}$ be exact. Assume that $T_{1}$ and $T_{2}$ are tilting objects in $\mathscr{A}$ and $\mathscr{B}$, respectively, such that

for all $k\geq1$. Then there exists a glued tilting object $\rho$ with respect to $T_{1}$ and $T_{2}$ in $\mathscr{C}$.

Theorem 5.4. Let $i^{*}$, $i^{!}$ be exact. Assume that $\rho$ is a tilting object in $\mathscr{C}$.

(1) If $i_{*}i^{*}\rho\in$ add $\rho$, then $i^{*}\rho$ is a tilting object in $\mathscr{A}$.

(2) If $j_{!}j^{*}\rho\in$ add $\rho$, then $j^{*}\rho$ is a tilting object in $\mathscr{B}$.

In this case, $\rho$ glued from two tilting objects in $\mathscr{A}$ and $\mathscr{B}$.

Author contributions

Zhen Zhang: contributed the creative ideals and proof techniques for this paper; Shance Wang: consulted the relevant background of the paper and composed the article, encompassing the structure of the article and the modification of grammar. All authors have read and agreed to the published version of the manuscript.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (No. 12101344) and Shan Dong Provincial Natural Science Foundation of China (No. ZR2015PA001).

Conflict of interest

The authors declared that they have no conflicts of interest.