On tau-tilting subcategories

Abstract

The main theme of this paper is to study $\tau $-tilting subcategories in an abelian category $\mathscr {A}$ with enough projective objects. We introduce the notion of $\tau $-cotorsion torsion triples and investigate a bijection between the collection of $\tau $-cotorsion torsion triples in $\mathscr {A}$ and the collection of support $\tau $-tilting subcategories of $\mathscr {A}$, generalizing the bijection by Bauer, Botnan, Oppermann, and Steen between the collection of cotorsion torsion triples and the collection of tilting subcategories of $\mathscr {A}$. General definitions and results are exemplified using persistent modules. If $\mathscr {A}=\mathrm{Mod}\mbox {-}R$, where R is a unitary associative ring, we characterize all support $\tau $-tilting (resp. all support $\tau ^-$-tilting) subcategories of $\mathrm{Mod}\mbox {-}R$ in terms of finendo quasitilting (resp. quasicotilting) modules. As a result, it will be shown that every silting module (resp. every cosilting module) induces a support $\tau $-tilting (resp. support $\tau ^{-}$-tilting) subcategory of $\mathrm{Mod}\mbox {-}R$. We also study the theory in $\mathrm {Rep}(Q, \mathscr {A})$, where Q is a finite and acyclic quiver. In particular, we give an algorithm to construct support $\tau $-tilting subcategories in $\mathrm {Rep}(Q, \mathscr {A})$ from certain support $\tau $-tilting subcategories of $\mathscr {A}$.

1 Introduction

Tilting theory is one of the most prominent tools in the representation theory of Artin algebras. The classical tilting modules were introduced by Brenner and Butler [15], and Happel and Ringel [23] as an axiomatization of the reflection functors of Bernstein, Gelfand, and Ponomarev [12] and Auslander, Platzek, and Reiten [5]. It has been shown by Bongartz [13] that every partial tilting module can be completed to a tilting module and by Happel and Unger [24] that every almost complete tilting module can be completed in at most two ways. However, there are examples of almost complete tilting modules that have exactly one complement.

Several years later, and inspired by the cluster algebras defined by Fomin and Zelevnisky in [22], Adachi, Iyama, and Reiten introduced $\tau $ -tilting theory [1]. This is a generalization of classical tilting theory in which every almost complete support $\tau $ -tilting module has exactly two complements, allowing to introduce a notion of mutation among these objects. The success of $\tau $ -tilting theory was immediate, explaining several phenomena in the module category of Artin algebras, as well as new connections between representation theory and other areas of mathematics (see [33]).

Due to the effectiveness of $\tau $ -tilting theory for the study of the categories of finitely presented modules, many mathematicians have introduced theories generalizing $\tau $ -tilting theory, and its dual, to other contexts. For instance, there are the works of Angeleri Hügel, Marks, and Vitoria [2] and Breaz and Pop [14] for the module category of rings; Iyama, Jørgensen, and Yang [26] for functor categories; or Liu and Zhou for Hom-finite abelian categories with enough projective objects [28].

In this paper, we are interested in studying $\tau $ -tilting theory in arbitrary abelian categories with enough projective objects. Since there is no notion of Auslander–Reiten translation $\tau $ in such general categories, the definition of support $\tau $ -tilting subcategories needs to be made with no mention of it. We follow [26, 28] to define support $\tau $ -tilting subcategories.

Definition 1.1 Let $\mathscr {A}$ be an abelian category with enough projective objects. Let $\mathscr {T}$ be an additive contravariantly finite full subcategory of $\mathscr {A}$ . Then $\mathscr {T}$ is called a support $\tau $ -tilting subcategory if:

1. 1. ${\mathrm {Ext}}^1_{\mathscr {A}}(T_1,{\mathrm {Fac}}(T_2))=0,$ for all $T_1, T_2 \in \mathscr {T}$ .

2. 2. For any projective P in $\mathscr {A}$ , there exists an exact sequence

   $$\begin{align*}P \stackrel{f}{\longrightarrow} T^0 \longrightarrow T^1 \longrightarrow 0 \end{align*}$$
   such that $T^0$ and $T^1$ are in $\mathscr {T}$ and f is a left $\mathscr {T}$ -approximation of P.

By ${\mathrm {Fac}}(T)$ , for an object $T \in \mathscr {A}$ , we mean the full subcategory consisting of factor objects of finite direct sums of copies of T.

The idea of generalizing a well-behaved theory from the category of finitely presented modules of an Artin algebra to more general abelian categories is not exclusive to $\tau $ -tilting theory. Indeed, Beligiannis introduced tilting theory for arbitrary abelian categories in [10] at the beginning of the millennium (see also [11]). In recent years, the work of Bauer, Botnan, Oppermann, and Steen [7] has found a striking application of this theory of Beligiannis in persistence theory and topological data analysis (TDA). One of the main results in [7] is a bijection between tilting subcategories of an abelian category with enough projective objects and the collection of cotorsion torsion triples in the same category. The motivation that started this collaboration is to find a version of [7, Theorem 2.29] for support $\tau $ -tilting subcategories. To do that, we first need an adequate notion of triple. Inspired by [11, Lemma V.3.3], we define $\tau $ -cotorsion torsion triple as follows.

Definition 1.2 Let $\mathscr {A}$ be an abelian category with enough projective objects. A triple of full subcategories $(\mathscr {C}, \mathscr {D}, \mathscr {F})$ of $\mathscr {A}$ is called a $\tau $ -cotorsion torsion triple if:

1. 1. $\mathscr {C} = {}^{\perp _1}\mathscr {D}$ .

2. 2. For every projective object $P \in \mathscr {A}$ , there exists an exact sequence

   $$\begin{align*}P \stackrel{f}\longrightarrow {D}\longrightarrow C \longrightarrow 0, \end{align*}$$
   where $D \in \mathscr {C}\cap \mathscr {D}$ , $C \in \mathscr {C}$ , and f is a left $\mathscr {D}$ -approximation.

3. 3. $\mathscr {C}\cap \mathscr {D}$ is a contravariantly finite subcategory of $\mathscr {A}$ .

4. 4. $(\mathscr {D}, \mathscr {F})$ is a torsion pair in $\mathscr {A}$ .

Having the notion of $\tau $ -cotorsion torsion triple, we can show the desired bijection in an abelian category with enough projective objects.

Theorem 1.3 (Theorem 5.7)

Let $\mathscr {A}$ be an abelian category with enough projective objects. Then there are bijections

$$ \begin{align*} \Phi: \{\mbox{support } \tau\mbox{-tilting subcategories}\}& \longrightarrow \{ \tau\mbox{-cotorsion torsion triples}\}\\ \mathscr{T}& \longmapsto ({}^{\perp_1}{(\mathrm{Fac}(\mathscr{T}))}, \mathrm{Fac}(\mathscr{T}), \mathscr{T}^{\perp_0}),\\[5pt] \Psi: \{ \tau\mbox{-cotorsion torsion triples}\} & \longrightarrow \{\mbox{support } \tau\mbox{-tilting subcategories}\}\\ (\mathscr{C}, \mathscr{D}, \mathscr{F}) & \longmapsto \mathscr{C}\cap\mathscr{D} \end{align*} $$

which are mutually inverse. Moreover, these bijections restrict to bijections between the collection of tilting subcategories of $\mathscr {A}$ and the collection of cotorsion torsion triples in  $\mathscr {A}$ .

We note that Buan and Zhou [16] introduced the notion of left weak cotorsion torsion triple to provide a version of [7, Theorem 2.29] for support $\tau $ -tilting modules in ${\mathrm {{mod\mbox {-}}}}\Lambda $ , where $\Lambda $ is an Artin algebra. In Theorem 4.10, we show that a triple $(\mathscr {C}, \mathscr {D}, \mathscr {F})$ of subcategories of ${\mathrm {{mod\mbox {-}}}}\Lambda $ is a left weak cotorsion torsion triple if and only if it is a $\tau $ -cotorsion torsion triple.

One of the main properties of support $\tau $ -tilting modules is that they generate functorially finite torsion classes in module categories. As a consequence of our results, we obtain the following corollary (see Theorem 5.7 and Corollary 4.8).

Corollary 1.4 Let $\mathscr {A}$ be an abelian category with enough projective objects, and let $\mathscr {T}$ be a support $\tau $ -tilting subcategory of $\mathscr {A}$ . Then ${\mathrm {Fac}}(\mathscr {T})$ is a functorially finite torsion class in $\mathscr {A}$ .

For ${\mathrm {{mod\mbox {-}}}}\Lambda $ , using results of [1], it is shown that a module T in ${\mathrm {{mod\mbox {-}}}}\Lambda $ is a support $\tau $ -tilting module if and only if ${\mathrm {add}}(T)$ is a support $\tau $ -tilting subcategory of ${\mathrm {{mod\mbox {-}}}} \Lambda $ . On the other hand, T is a support $\tau ^-$ -tilting module if and only if ${\mathrm {add}}(T)$ is a support $\tau ^-$ -tilting subcategory of ${\mathrm {{mod\mbox {-}}}} \Lambda $ (see Propositions 5.9 and 6.11).

A prominent example of an abelian category with enough projective objects is $\mathrm{Mod}\mbox {-}R$ , the category of modules over a unitary associative ring R. Angeleri Hügel, Marks, and Vitoria [2] introduced the concept of silting modules in $\mathrm{Mod}\mbox {-}R$ , as a generalization of support $\tau $ -tilting modules. They showed that if R is an Artin algebra, finitely presented silting modules coincide with the support $\tau $ -tilting modules of [1]. As the categorical dual of silting modules, Breaz and Pop [14] introduced the notion of cosilting modules in $\mathrm{Mod}\mbox {-}R$ and showed that over Artin algebras finitely copresented cosilting modules coincide with the support $\tau ^-$ -tilting modules.

We study the connection between silting (resp. cosilting) modules in $\mathrm{Mod}\mbox {-}R$ with the $\tau $ -tilting (resp. $\tau ^-$ -tilting) subcategories of $\mathrm{Mod}\mbox {-}R$ . Furthermore, we characterize all support $\tau $ -tilting subcategories of $\mathrm{Mod}\mbox {-}R$ . In particular, we provide a bijection between the equivalence classes of all support $\tau $ -tilting subcategories of $\mathrm{Mod}\mbox {-}R$ and the collection of all equivalence classes of certain R-modules, the so-called finendo quasitilting R-modules (see Theorem 7.1.10). It is known that all silting modules are finendo quasitilting. Dually, we also characterize all support $\tau ^{-}$ -tilting subcategories of $\mathrm{Mod}\mbox {-}R$ by providing a bijection between the equivalence classes of all support $\tau ^-$ -tilting subcategories of $\mathrm{Mod}\mbox {-}R$ and the collection of all equivalent classes of the quasicotilting R-modules.

Toward the end of the paper, we provide applications of this theory to the category of the representations of a finite and acyclic quiver in an abelian category $\mathscr {A}$ with enough projective objects. Let Q be such a quiver. It is known that the category $\mathrm {Rep}(Q, \mathscr {A})$ of representations of Q over $\mathscr {A}$ is again an abelian category with enough projective objects. See Section 8 for more details. We give a recipe to construct support $\tau $ -tilting subcategories in $\mathrm {Rep}(Q, \mathscr {A})$ from certain support $\tau $ -tilting subcategories of $\mathscr {A}$ . More explicitly, we have the following result.

Theorem 1.5 (Theorem 8.2.2)

Let $\mathscr {A}$ be an abelian category with enough projective objects, and let Q be a finite acyclic quiver. Let $\mathscr {T}$ be a support $\tau $ -tilting subcategory of $\mathscr {A}$ such that the induced torsion pair $({\mathrm {Fac}}(\mathscr {T}), \mathscr {T}^{\perp _0})$ is hereditary. Then

$$\begin{align*}\mathbb{T} ={\mathrm{add}}\lbrace e_i^{\rho}(T) \vert ~ i\in Q_0, T\in\mathscr{T}\rbrace\end{align*}$$

is a support $\tau $ -tilting subcategory of $\mathrm {Rep}(Q, \mathscr {A})$ .

Note that

$$\begin{align*}e_i^\rho: \mathscr{A}\longrightarrow \mathrm{Rep}(Q, \mathscr{A}) \end{align*}$$

is the right adjoint of the evaluation functor $e_i: \mathrm {Rep}(Q, \mathscr {A})\longrightarrow \mathscr {A}$ (see Section 8.1 for more details).

Let ${\mathrm {Prj}}(\mathscr {A})$ denote the category of all projective objects in $\mathscr {A}$ . In [7, Proposition 3.9], the authors have shown that

$$\begin{align*}\mathbb{T} ={\mathrm{add}}\lbrace e_i^{\rho}(P) \vert ~ i\in Q_0, P \in{\mathrm{Prj}}(\mathscr{A}) \rbrace \end{align*}$$

is a tilting subcategory of $\mathscr {A}$ . Then the previous result can be seen as a generalization of this result since $\mathscr {A} = {\mathrm {Fac}}({\mathrm {Prj}}(\mathscr {A}))$ is closed under kernels of epimorphisms.

Throughout the paper, several examples are provided using the theory of persistent modules, which are central objects of study in TDA. We refer the reader to the survey [17] by Carlsson, or the book [29] by Oudot, for an introduction to TDA and connections to quiver representations.

The paper is structured as follows. In Section 2, we fix the notations and give the necessary background for the rest of the paper. In Section 3, we define support $\tau $ -tilting subcategories in abelian categories with enough projective objects and we give some of its basic properties. Then, in Section 4, we introduce and study the notion of $\tau $ -cotorsion torsion triples. We also compare $\tau $ -cotorsion torsion triples with the left weak cotorsion torsion triples of [16]. In Section 5, we show the bijection between support $\tau $ -tilting subcategories and $\tau $ -cotorsion torsion triples. Section 6 is a compilation of dual definitions and results. In Section 7, we study support $\tau $ - and $\tau ^-$ - tilting subcategories in $\mathrm{Mod}\mbox {-}R$ , where R is a unitary associative ring, and show how they are connected to the silting and cosilting theory. We end the paper in Section 8 where we study support $\tau $ -tilting subcategories and (co)silting objects in the category $\mathrm {Rep}(Q, \mathscr {A})$ of quiver representations over $\mathscr {A}$ .

2 Preliminaries

Let $\mathscr {A}$ be an abelian category. In this paper, by subcategory, we always mean a full subcategory. Let $\mathscr {X}$ be a subcategory of $\mathscr {A}$ . A morphism $\varphi : X \longrightarrow A$ , where A is an object of $\mathscr {A}$ , is called a right $\mathscr {X}$ -approximation of A if, $X \in \mathscr {X}$ and for every $X' \in \mathscr {X}$ , the induced morphism $\mathscr {A}(X', X) \longrightarrow \mathscr {A}(X', A) \longrightarrow 0$ of abelian groups is exact. We say that $\mathscr {X}$ is a contravariantly finite subcategory of $\mathscr {A}$ if every object A of $\mathscr {A}$ admits a right $\mathscr {X}$ -approximation. The notions of left $\mathscr {X}$ -approximations and covariantly finite subcategories are defined dually. Moreover, we say that $\mathscr {X}$ is a functorially finite subcategory of $\mathscr {A}$ if it is both a contravariantly finite and a covariantly finite subcategory of $\mathscr {A}$ . If $\mathscr {X}$ is closed under taking finite direct sums and direct summands, we say that it is additively closed.

Let $\mathscr {A}$ be an abelian category with enough projective objects. For a nonnegative integer n, we set

$$\begin{align*}\mathscr{X}^{\perp_n}:=\lbrace A\in\mathscr{A} ~ \vert ~{\mathrm{Ext}}^n_{\mathscr{A}}(\mathscr{X}, A)=0\rbrace,\end{align*}$$

$$\begin{align*}{}^{\perp_n}\mathscr{X}:=\lbrace A\in\mathscr{A} ~ \vert ~{\mathrm{Ext}}^n_{\mathscr{A}}( A, \mathscr{X})=0 \rbrace.\end{align*}$$

Note that ${\mathrm {Ext}}^0$ is just the usual Hom-functor.

2.1 (Pointwise finite-dimensional representations)

Let k be a field, and let $(\mathcal {X} , \leq )$ be a poset. Consider $\mathcal {X} $ as a category. A persistence module is a (covariant and additive) functor $V: \mathcal {X} \to \mathrm{Mod}\mbox {-}k$ from $\mathcal {X} $ to $\mathrm{Mod}\mbox {-}k$ , the category of k-vector spaces. A persistence module V is called a pointwise finite-dimensional representation of $\mathcal {X} $ if it is a functor from $\mathcal {X} $ to ${\mathrm {{mod\mbox {-}}}} k$ , the category of finite-dimensional vector spaces.

The abelian category of pointwise finite-dimensional representations will be denoted by $\mathrm { Rep}_k^{\mathrm {pfd}}\mathcal {X} $ . A subset $\mathcal {C} $ of $\mathcal {X} $ is called a convex subset if for every $x \leq y\in \mathcal {C} $ , $x\leq z\leq y$ implies $z\in \mathcal {C} $ . Let $\mathcal {C} $ be a convex subset of $\mathcal {X} $ . The constant representation $k_{\mathcal {C}} $ is defined by $k_{\mathcal {C}} (x)=k$ , for $x\in \mathcal {C} $ , $k_{\mathcal {C}} (x)=0$ , for $x\not \in \mathcal {C} $ and $k_{\mathcal {C}} (x\leq y)=\mathrm {id}_k$ , for $x\leq y$ in $\mathcal {C} $ . Set $P_x=k_{\lbrace y\in \mathcal {X} \vert \ y\geq x\rbrace }.$ As an application of Yoneda’s lemma, one can see that, for every $x \in \mathcal {X} $ , $P_x$ is a projective object of $\mathrm {Rep}_k^{\mathrm {pfd}}\mathcal {X} $ and, vice versa, all projective modules are of this form (see, e.g., [7, Section 2]). A representation $V\in \mathrm {Rep}_k^{\mathrm {pfd}} \mathcal {X} $ is called finitely generated if there exists an epimorphism of functors $\bigoplus _{i\in J} P_{x_i}\longrightarrow V$ , where J is a finite indexing set. Moreover, if the kernel of this epimorphism is again finitely generated, it is called finitely presented. We denote the full subcategory of $\mathrm {Rep}_k^{\mathrm {pfd}} \mathcal {X} $ consisting of finitely presented representations by $\mathrm {Rep}_k^{\mathrm {fp}}\mathcal {X} $ .

One of the most important cases is when we consider the poset of nonnegative real numbers, denoted by $\mathbb {R}_{\geq 0}.$ It is proved in [18, Theorem 1.1] that the indecomposable objects in $\mathrm {Rep}^{\mathrm {fp}}_k \mathbb {R}_{\geq 0}$ are classified by the constant representations $k_{ [x, y)}$ , where $x< y\leq \infty $ . Moreover, for each $x\geq 0$ , the constant representations $k_{[ x, \infty )}$ and $k_{[ 0, x)}$ are projective and injective objects, respectively.

2.2 (Cotorsion torsion triple)

Here, we recall the notion of torsion pairs, cotorsion pairs, and cotorsion torsion triples. Let us begin by the following classical definition of Dickson [19].

Definition 2.3 Let $\mathscr {A}$ be an abelian category. A pair $(\mathscr {T},\mathscr {F})$ of full subcategories of $\mathscr {A}$ is called a torsion pair if ${\mathrm {Hom}}_{\mathscr {A}}(\mathscr {T},\mathscr {F})=0$ and for every $A \in \mathscr {A}$ there is a short exact sequence

$$ \begin{align*}0 \longrightarrow tA \longrightarrow A \longrightarrow fA \longrightarrow 0\end{align*} $$

such that $tA \in \mathscr {T}$ and $fA \in \mathscr {F}$ .

In this case, $\mathscr {T}$ is called the torsion class and $\mathscr {F}$ is called the torsion-free class. A torsion pair $(\mathscr {T}, \mathscr {F})$ is called hereditary if $\mathscr {T}$ is closed under subobjects. It is known that $\mathscr {T}$ and $\mathscr {F}$ determine each other and the short exact sequence is functorial. See, for instance, [11, Section I.1].

Let $\mathscr {A} = {\mathrm {{mod\mbox {-}}}} \Lambda $ . By [32, Theorem 1], if $(\mathscr {T}, \mathscr {F})$ is a torsion pair in $\mathscr {A}$ , then $\mathscr {T}$ is a functorially finite subcategory of $\mathscr {A}$ if and only if $\mathscr {F}$ is so. This is not true in an arbitrary abelian category with enough projective objects, as is shown by the following example.

Example 2.4 Let $\mathscr {A}=\mathrm {Rep}^{\mathrm {fp}}_k \mathbb {R}_{\geq 0}$ . Set

$$ \begin{align*} \mathscr{T}=&{\mathrm{add}}(\lbrace k_{[0, y)} \vert \ 0 < y\leq \infty\rbrace \cup \lbrace k_{[x, y)} \vert \ 1 \leq x< y\leq \infty\rbrace),\\ \mathscr{F}=&{\mathrm{add}}(\lbrace k_{[x, y)} \vert \ 0< x <y \leq 1\rbrace). \end{align*} $$

Then, by [7, Example 2.5], $(\mathscr {T}, \mathscr {F})$ is a torsion pair. Obviously, $\mathscr {T}$ is a functorially finite subcategory of $\mathscr {A}$ . We show that $\mathscr {F}$ is not a contravariantly finite subcategory of $\mathscr {A}$ . To see this, note that objects such as $K_{[0, b)}$ , where $b<1$ , do not have a right $\mathscr {F}$ -approximation. In fact, a right $\mathscr {F}$ -approximation of $K_{[0, b)}$ should be of the form $\theta : k_{[a, b)}\longrightarrow k_{[0, b)}$ , for some $0< a<b< 1$ . But nonzero morphisms such as $\eta : k_{[x, y)}\longrightarrow k_{[0, b)}$ , with $0\leq x<a$ and $b\leq y$ , cannot factor through $\theta $ .

Definition 2.5 (See [31])

Let $\mathscr {A}$ be an abelian category with enough projective objects. A pair $(\mathscr {C}, \mathscr {D})$ of full subcategories of $\mathscr {A}$ is called a cotorsion pair if $\mathscr {C} = {}^{\perp _1}\mathscr {D}$ , $\mathscr {D} = \mathscr {C}^{\perp _1}$ and for every object $A \in \mathscr {A}$ , there are short exact sequences

$$\begin{align*}0 \longrightarrow D \longrightarrow C \longrightarrow A \longrightarrow 0 \end{align*}$$

$$\begin{align*}0 \longrightarrow A \longrightarrow D' \longrightarrow C' \longrightarrow 0,\end{align*}$$

where C and $C'$ are in $\mathscr {C}$ and D and $D'$ are in $\mathscr {D}$ .

This notion is sometimes called complete cotorsion pair in the literature. See [7, Remark 2.7] for more details.

Remark 2.6 It is immediate from the definition that in a cotorsion pair $(\mathscr {C}, \mathscr {D})$ both $\mathscr {C}$ and $\mathscr {D}$ are closed under extensions.

Let $\mathscr {A}$ be an abelian category with enough projective and enough injective objects. In view of [11, Lemma V.3.3], a pair $(\mathscr {C}, \mathscr {D})$ of full subcategories of $\mathscr {A}$ is a cotorsion pair if and only if the following conditions are satisfied:

1. 1. $\mathscr {C} = {}^{\perp _1}\mathscr {D}$ ,

2. 2. for every object $A \in \mathscr {A}$ , there exists a short exact sequence

   $$\begin{align*}0 \longrightarrow A \stackrel{f}\longrightarrow D \longrightarrow C \longrightarrow 0,\end{align*}$$
   where $D \in \mathscr {D}$ and $C \in \mathscr {C}$ .

The following proposition shows that when $\mathscr {D}$ is closed under factors it is enough, in the above definition, to check Condition $2$ only for projective objects.

Proposition 2.7 Let $\mathscr {A}$ be an abelian category with enough projective and enough injective objects. Let $(\mathscr {C}, \mathscr {D})$ be a pair of full subcategories of $\mathscr {A}$ such that $\mathscr {D}$ is closed under factors. Then $(\mathscr {C}, \mathscr {D})$ is a cotorsion pair if and only if:

1. 1. $\mathscr {C} = {}^{\perp _1}\mathscr {D}$ ,

2. 2. for every projective object $P \in \mathscr {A}$ , there exists a short exact sequence

   $$\begin{align*}0 \longrightarrow P \stackrel{f}\longrightarrow D \longrightarrow C \longrightarrow 0,\end{align*}$$
   where $D \in \mathscr {D}\cap \mathscr {C}$ and $C \in \mathscr {C}$ .

Proof. We start by showing the necessity. Let A be an arbitrary object of $\mathscr {A}$ . Since $\mathscr {A}$ has enough projective objects, we have an epimorphism $P \longrightarrow A \longrightarrow 0$ . By assumption, P fits into the short exact sequence

$$\begin{align*}0 \longrightarrow P \stackrel{f}\longrightarrow D \longrightarrow C \longrightarrow 0.\end{align*}$$

The pushout diagram

induces the short exact sequence $0 \longrightarrow A \longrightarrow U \longrightarrow C \longrightarrow 0$ , where $U \in {\mathrm {Fac}}(\mathscr {D})$ . Since $\mathscr {D}$ is closed under factors, we deduce that $U \in \mathscr {D}$ , and hence this is the desired short exact sequence for A.

For the sufficiency, let P be a projective object in $\mathscr {A}$ . Then there exists a short exact sequence

$$\begin{align*}0 \longrightarrow P \stackrel{f}\longrightarrow D \longrightarrow C \longrightarrow 0,\end{align*}$$

where $D \in \mathscr {D}$ and $C \in \mathscr {C}$ . Since P is projective, $P \in {}^{\perp _1}\mathscr {D} = \mathscr {C}$ . Hence, $D \in \mathscr {C}$ because $\mathscr {C}$ is closed under extensions and the result follows.

The proof of the following lemma is included in the proof of [7, Proposition 2.28], but is restated here for convenience.

Lemma 2.8 Let $(\mathscr {C}, \mathscr {D})$ be a cotorsion pair in $\mathscr {A}$ such that $\mathscr {D}$ is a torsion class. Then $\mathscr {C} \cap \mathscr {D}$ is a contravariantly finite subcategory of $\mathscr {A}$ .

Proof. Let $A \in \mathscr {A}$ be an arbitrary object. Since $\mathscr {D}$ is a torsion class, there exists a monomorphism $tA \longrightarrow A$ with $tA \in \mathscr {D}$ . Since $(\mathscr {C}, \mathscr {D})$ is a cotorsion pair, there exists a short exact sequence

$$\begin{align*}0\longrightarrow D \longrightarrow C\stackrel{\imath}\longrightarrow tA\longrightarrow 0\end{align*}$$

such that $D \in \mathscr {D}$ and $C\in \mathscr {C}$ . Since $D, tA\in \mathscr {D}$ , and $\mathscr {D}$ is closed under extensions, $C \in \mathscr {C} \cap \mathscr {D}$ . We claim that the composition $C \stackrel {\imath }\longrightarrow tA\stackrel {\jmath }\longrightarrow A $ is a right $\mathscr {C} \cap \mathscr {D}$ -approximation of A. To prove the claim, let $X \stackrel {\ell }\longrightarrow A$ be a morphism with $X \in \mathscr {C}\cap \mathscr {D}$ . Since $X \in \mathscr {D}$ and every map from $\mathscr {D}$ to A factors through $tA$ , the morphism $\ell $ factors through $\jmath $ . Now since $X \in \mathscr {C}$ and ${\mathrm {Ext}}^1_A(X, D)=0$ , the morphism $\ell $ factors through $\jmath \imath $ . Hence, the claim is proved.

Definition 2.9 Let $\mathscr {A}$ be an abelian category with enough projective objects. A triple $(\mathscr {C}, \mathscr {T}, \mathscr {F})$ of full subcategories in $\mathscr {A}$ is called a cotorsion torsion triple if $(\mathscr {C}, \mathscr {T})$ is a cotorsion pair and $(\mathscr {T}, \mathscr {F})$ is a torsion pair.

It has been recently shown in [7] that cotorsion torsion triples and tilting subcategories are closely related. Before stating the explicit relation between these two concepts, let us recall the definition of a tilting subcategory.

Definition 2.10 Let $\mathscr {A}$ be an abelian category with enough projective objects. An additively closed subcategory $\mathscr {T}$ of $\mathscr {A}$ is called a tilting subcategory if:

1. (i) $\mathscr {T}$ is a contravariantly finite subcategory of $\mathscr {A}$ .

2. (ii) ${\mathrm {Ext}}^1_{\mathscr {A}}(T_1, T_2)=0$ , for all $T_1, T_2\in \mathscr {T}$ .

3. (iii) Every object $T\in \mathscr {T}$ has projective dimension at most $1$ .

4. (iv) For every projective object P in $\mathscr {A}$ , there exists a short exact sequence

   $$\begin{align*}0\longrightarrow P\stackrel{f}{\longrightarrow}T^0\longrightarrow T^1\longrightarrow 0\end{align*}$$
   with $T^i\in \mathscr {T}$ .

If $\mathscr {T}$ satisfies only Conditions $(ii)$ – $(iv)$ , it is called a weak tilting subcategory of $\mathscr {A}$ .

Theorem 2.11 (See [7, Theorem 2.29])

Let $\mathscr {A}$ be an abelian category with enough projective objects. Then there is a bijection

$$\begin{align*}\lbrace \mbox{tilting subcategories}\rbrace &~~~~\longleftrightarrow~~~~ \lbrace \mbox{cotorsion torsion triples}\rbrace\\\mathscr {T}&~~~~\longrightarrow ~~~~~({}^{\perp _1}{(\mathrm {Fac}(\mathscr {T}))}, \mathrm {Fac}(\mathscr {T}), \mathscr {T}^{\perp _0})\\\mathscr{C}\cap\mathscr{T}&~~~~\longleftarrow ~~~~(\mathscr{C}, \mathscr{T}, \mathscr{F}),\end{align*}$$

where $\mathrm {Fac}(\mathscr {T})$ is the full subcategory of $\mathscr {A}$ consisting of factors of objects in $\mathscr {T}$ .

Remark 2.12 Note, as it is also mentioned in [7, Remark 1.9], that there is an extensive manuscript by Beligiannis [10], started in 2004, in which he developed a fully general tilting theory in any abelian category without any assumption on enough projective or injective objects. Although the paper is not yet publicly available, the results of this work have been announced at several conferences in the subsequent years. Specifically, he has proved and announced a result similar to the theorem above in the more general setting of an arbitrary abelian category.

3 $\tau $ -tilting subcategories

In this section, we recall the definition of $\tau $ -tilting subcategories and study some of their properties. The following definition is motivated by [26, Definition 1.5] and [28, Definition 2.1]. Note that for $X\in \mathscr {A}$ , we let ${\mathrm {add}}(X)$ be the category of all direct summands of finite direct sums of copies of X.

Definition 3.1 Let $\mathscr {A}$ be an abelian category with enough projective objects. Let $\mathscr {T}$ be an additive full subcategory of $\mathscr {A}$ . Then $\mathscr {T}$ is called a weak support $\tau $ -tilting subcategory of $\mathscr {A}$ if:

1. (i) ${\mathrm {Ext}}^1_{\mathscr {A}}(T_1,{\mathrm {Fac}}(T_2))=0,$ for all $T_1, T_2 \in \mathscr {T}$ .

2. (ii) For any projective P in $\mathscr {A}$ , there exists an exact sequence

   $$\begin{align*}P \stackrel{f}{\longrightarrow} T^0 \longrightarrow T^1 \longrightarrow 0\end{align*}$$
   such that $T^0$ and $T^1$ are in $\mathscr {T}$ and f is a left $\mathscr {T}$ -approximation of P.

If, furthermore, $\mathscr {T}$ is a contravariantly finite subcategory of $\mathscr {A}$ , it is called a support $\tau $ -tilting subcategory of $\mathscr {A}$ . A support $\tau $ -tilting subcategory is simply called $\tau $ -tilting if the approximation $f: P \longrightarrow T^0$ is nonzero for every projective object P. An object $T\in \mathscr {A}$ is called a support $\tau $ -tilting object if ${\mathrm {add}}(T)$ is a support $\tau $ -tilting subcategory of $\mathscr {A}$ .

Remark 3.2 We note that the previous definition is not the classical definition of support $\tau $ -tilting modules in ${\mathrm {{mod\mbox {-}}}}\Lambda $ introduced in [1]. However, it follows from [27, Proposition 2.14] that these two definitions coincide if $\mathscr {A}={\mathrm {{mod\mbox {-}}}}\Lambda $ .

A key difference between tilting and $\tau $ -tilting subcategories is that the approximations $f: P \longrightarrow T^0$ are always a monomorphism when $\mathscr {T}$ is a tilting subcategory. It is immediate from the definition that every (weak) tilting subcategory is a (weak) $\tau $ -tilting subcategory. In the following example, we present a support $\tau $ -tilting subcategory which is not a tilting subcategory.

Example 3.3 Let $\mathscr {A}=\mathrm {Rep}^{\mathrm {fp}}_k \mathbb {R}_{\geq 0}$ , and set

$$\begin{align*}\mathscr{T}={\mathrm{add}}( \lbrace k_{[x, \infty)} \vert \ x\geq 1\rbrace \cup \lbrace k_{[0, x)}\vert \ x\leq 1\rbrace).\end{align*}$$

Then $\mathscr {T}$ is a support $\tau $ -tilting subcategory. To see this, first note that

$$\begin{align*}{\mathrm{Fac}}(\mathscr{T})={\mathrm{add}}( \lbrace k_{[x, y)} \vert \ 1\leq x< y\leq\infty\rbrace \cup \lbrace k_{[0, x)}\vert \ x\leq 1\rbrace),\end{align*}$$

and obviously ${\mathrm {Ext}}^1_{\mathscr {A}}(\mathscr {T}, {\mathrm {Fac}}(\mathscr {T}))=0$ .

Now let $k_{[a, \infty )}$ be an indecomposable projective in $\mathscr {A}$ . If $a\geq 1$ , the exact sequence

$$\begin{align*}0\longrightarrow k_{[a, \infty)}\longrightarrow k_{[a, \infty)}\longrightarrow 0,\end{align*}$$

and if $0\leq a< 1$ , the exact sequence

$$\begin{align*}k_{[a, \infty)}\longrightarrow k_{[0, 1)} \longrightarrow k_{[0, a)}\longrightarrow 0,\end{align*}$$

are the desired exact sequences.

Moreover, $\mathscr {T}$ is a contravariantly finite subcategory of $\mathscr {A}$ . Indeed, depending on a and b, for an indecomposable representation $k_{[ a, b)}$ , we have the following right $\mathscr {T}$ -approximations:

$$\begin{align*}\left\{ \begin{array}{r@{\ \,}cl} k_{[0, b)}&\longrightarrow k_{[0, b)}, & 0= a< b \leq 1,\\ k_{[1, \infty)}&\longrightarrow k_{[a, b)}, & 0\leq a< 1 < b \leq \infty,\\ 0&\longrightarrow k_{[a, b)}, & 0< a< b \leq 1,\\ k_{[a, \infty)}&\longrightarrow k_{[a, b)}, & 1\leq a < b\leq \infty. \end{array} \right. \end{align*}$$

It is not a tilting subcategory of $\mathscr {A}$ , because $k_{[0, \infty )}$ is a projective object, for which we do not have a short exact sequence like Condition $(iv)$ of Definition 2.10.

One of the main reasons behind the success of $\tau $ -tilting theory [1] is the formal inclusion of the notion of support from the start (see Definition 5.8). Note that in this definition we are defining objects in ${\mathrm {{mod\mbox {-}}}} \Lambda $ by using a property of the object in a different category, namely ${\mathrm {{mod\mbox {-}}}} (\Lambda /\langle e\rangle )$ . So one needs to verify that the good properties of a support $\tau $ -tilting object M in ${\mathrm {{mod\mbox {-}}}} (\Lambda /\langle e\rangle )$ can be transported to ${\mathrm {{mod\mbox {-}}}} \Lambda $ . This was done in [1, Lemma 2.1]. Note that ${\mathrm {{mod\mbox {-}}}} (\Lambda /\langle e\rangle )$ is a functorially finite wide subcategory of ${\mathrm {{mod\mbox {-}}}} \Lambda $ which is at the same time a torsion and a torsion-free class. Recall that a subcategory $\mathscr {X}$ of an abelian category $\mathscr {A}$ is called a wide subcategory if it is closed under kernels, cokernels, and extensions. In particular, this implies that $\mathscr {X}$ itself is an abelian category. It is known that a subcategory $\mathscr {X}$ of an abelian category $\mathscr {A}$ is a torsion class if it is a contravariantly finite subcategory of $\mathscr {A}$ , closed with respect to quotients and extensions. Similarly, $\mathscr {X}$ is a torsion-free class if it is a covariantly finite subcategory of $\mathscr {A}$ which is furthermore closed under subobjects and extensions (see [9, Proposition 2.1] and [19, Theorem 2.3]).

Lemma 3.4 Let $\mathscr {A}$ be an abelian category with enough projective objects. Let $\mathscr {X}$ be a wide and functorially finite full subcategory of $\mathscr {A}$ . Then $\mathscr {X}$ is a torsion class of $\mathscr {A}$ if and only if $\mathscr {X}$ is a torsion-free class of $\mathscr {A}$ .

Proof. Let $\mathscr {X}$ be a torsion class of $\mathscr {A}$ . To show that it is a torsion-free class, we need to show that it is covariantly finite, closed under subobjects, and closed under extensions. By the assumption, $\mathscr {X}$ is a covariantly finite subcategory. It is closed under extensions, since $\mathscr {X}$ is a torsion class. Moreover, it is closed under subobjects, since it is a wide subcategory. The other implication follows similarly.

Our next result is inspired by [1, Lemma 2.1], and it justifies the name of support $\tau $ -tilting subcategories.

Proposition 3.5 Let $\mathscr {A}$ be an abelian category with enough projective objects. Let $\mathscr {X}$ be a wide and functorially finite torsion class of $\mathscr {A}$ . Then every support $\tau $ -tilting subcategory of $\mathscr {X}$ is a support $\tau $ -tilting subcategory of $\mathscr {A}$ .

Proof. Let $\mathscr {T}$ be a support $\tau $ -tilting subcategory of $\mathscr {X}$ . First, we note that, since $\mathscr {X}$ is a wide subcategory, then ${\mathrm {Fac}}(\mathscr {T})\subseteq {\mathrm {Fac}}(\mathscr {X})=\mathscr {X}$ . Therefore, since $\mathscr {T}$ is a support $\tau $ -tilting subcategory of $\mathscr {X}$ , we have ${\mathrm {Ext}}^1_{\mathscr {X}}(\mathscr {T}, {\mathrm {Fac}}(\mathscr {T}))=0$ . Hence, ${\mathrm {Ext}}^1_{\mathscr {A}}(\mathscr {T}, {\mathrm {Fac}}(\mathscr {T}))=0$ .

Now let P be a projective object in $\mathscr {A}$ . By Lemma 3.4, there is a subcategory $\mathscr {Y}$ of $\mathscr {A}$ such that $(\mathscr {Y}, \mathscr {X})$ is a torsion pair of $\mathscr {A}$ . Consider the canonical short exact sequence

$$\begin{align*}0\longrightarrow Y\longrightarrow P\stackrel{f}\longrightarrow X\longrightarrow 0\end{align*}$$

of P with respect to this torsion pair. So $Y \in \mathscr {Y}$ and $X \in \mathscr {X}$ . Let $X'$ be an arbitrary object of $\mathscr {X}$ . By applying the functor ${\mathrm {Hom}}_{\mathscr {A}}(-, X')$ to the above short exact sequence, we get ${\mathrm {Ext}}^1_{\mathscr {A}}(X, X')=0$ and therefore ${\mathrm {Ext}}^1_{\mathscr {X}}(X, X')=0$ . Hence, X is an Ext-projective object in $\mathscr {X}$ . Since $\mathscr {T}$ is a support $\tau $ -tilting subcategory of $\mathscr {X}$ , there exists an exact sequence

$$\begin{align*}X\stackrel{g}\longrightarrow T^0\longrightarrow T^1\longrightarrow 0,\end{align*}$$

where $T^0, T^1\in \mathscr {T}$ and g is a left $\mathscr {T}$ -approximation. Now the exact sequence

$$\begin{align*}P\stackrel{gf}\longrightarrow T^0\longrightarrow T^1\longrightarrow 0\end{align*}$$

is the desired one. It is easy to see that $gf$ is a left $\mathscr {T}$ -approximation of P.

Finally, we show that $\mathscr {T}$ is a contravariantly finite subcategory of $\mathscr {A}$ . Let $A\in \mathscr {A}$ . Since $\mathscr {X}$ is a functorially finite subcategory of $\mathscr {A}$ , there exists a right $\mathscr {X}$ -approximation $X\stackrel {f}\longrightarrow A$ of A. Consider a right $\mathscr {T}$ -approximation $T\stackrel {g}\longrightarrow X$ of X, which exists because $\mathscr {T}$ is a support $\tau $ -tilting subcategory of $\mathscr {A}$ . Now it is easy to see that $T\stackrel {fg}\longrightarrow A$ is a right $\mathscr {T}$ -approximation of A.

We illustrate our previous result with the following example.

Example 3.6 Let $\mathscr {A}=\mathrm {Rep}^{\mathrm {fp}}_k \mathbb {R}_{\geq 0}$ and $\mathscr {B}=\mathrm { Rep}^{\mathrm {fp}}_k\mathbb {R}_{\geq 1}$ . On one hand, since $\mathbb {R}_{\geq 0}$ and $\mathbb {R}_{\geq 1}$ are isomorphic as posets, it is clear that $\mathscr {A}$ and $\mathscr {B}$ are equivalent as categories. On the other hand, the natural inclusion of $\mathbb {R}_{\geq 1}$ into $\mathbb {R}_{\geq 0}$ induces an embedding of $\mathscr {B}$ into $\mathscr {A}$ . In fact, it is easy to see that $\mathscr {B}$ is a functorially finite wide subcategory of $\mathscr {A}$ which is both a torsion and a torsion-free class. Set

$$\begin{align*}\mathscr{T}={\mathrm{add}}(\lbrace k_{[ 1, y)} \vert \ 1<y\leq \infty\rbrace).\end{align*}$$

It is easy to see that $\mathscr {T}$ is a tilting subcategory of $\mathscr {B}$ and hence a support $\tau $ -tilting subcategory of $\mathscr {B}$ . So, Proposition 3.5 implies that $\mathscr {T}$ is a support $\tau $ -tilting subcategory of $\mathscr {A}$ . Note that $\mathscr {T}$ is not a tilting subcategory of $\mathscr {A}$ since the $\mathscr {T}$ -approximation $f: k_{[0,\infty )} \to k_{[1,\infty )}$ is not a monomorphism; in fact, it is a zero morphism.

By [30, Proposition 3.42], if $\mathscr {A}$ is an abelian category with enough projective objects such that ${\mathrm {Prj}}(\mathscr {A})={\mathrm {add}}(P)$ , then for every tilting subcategory $\mathscr {T}$ of $\mathscr {A}$ , there exists an object $T \in \mathscr {T}$ such that $\mathscr {T} = {\mathrm {add}}(T)$ . In fact, T is a tilting object. Note that an object $T\in \mathscr {A}$ is called a tilting object if:

1. (i) ${\mathrm {Ext}}^1_{\mathscr {A}}(T, T)=0$ .

2. (ii) Projective dimension of T is at most 1.

3. (iii) For every projective object $P\in \mathscr {A}$ , there exists a short exact sequence

   $$\begin{align*}0\longrightarrow P\longrightarrow T^0\longrightarrow T^1\longrightarrow 0\end{align*}$$
   such that $T^0, T^1\in {\mathrm {add}}(T)$ .

We prove a version of this fact for $\tau $ -tilting subcategories.

Let $\mathscr {A}$ be an abelian category with enough projective objects. Let T be a support $\tau $ -tilting object in $\mathscr {A}$ . It follows directly from the definition that ${\mathrm {add}}(T)$ is a weak support $\tau $ -tilting subcategory of $\mathscr {A}$ . The next proposition provides a partial converse.

Proposition 3.7 Let $\mathscr {A}$ be an abelian category with enough projective objects such that ${\mathrm {Prj}}(\mathscr {A})={\mathrm {add}}(P)$ , for some object $P\in \mathscr {A}$ . If $\mathscr {T}$ is a support $\tau $ -tilting subcategory of $\mathscr {A}$ , then there exists a support $\tau $ -tilting object $T\in \mathscr {T}$ such that ${\mathrm {Fac}}(\mathscr {T})={\mathrm {Fac}}(T)$ .

Proof. Consider the exact sequence

$$\begin{align*}P\longrightarrow T^0\longrightarrow T^1\longrightarrow 0\end{align*}$$

which exists, because $\mathscr {T}$ is a support $\tau $ -tilting subcategory. We claim that ${\mathrm {Fac}}(T^0\oplus T^1)={\mathrm {Fac}}(\mathscr {T})$ . First, we prove that $T^0\oplus T^1$ is a support $\tau $ -tilting object. To this end, we just need to show that every projective object $P'\in {\mathrm {Prj}} (\mathscr {A})={\mathrm {add}}(P)$ admits an exact sequence

$$\begin{align*}P'\stackrel{f'}\longrightarrow T^0_{P'}\longrightarrow T^1_{P'}\longrightarrow 0,\end{align*}$$

where $f'$ is a left ${\mathrm {add}}(T^0\oplus T^1)$ -approximation and $T^0_{P'} , T^1_{P'}\in {\mathrm {add}}(T^0\oplus T^1)$ . Since ${\mathrm {Prj}}(\mathscr {A})={\mathrm {add}}(P)$ , there exists a nonnegative integer n and projective object $Q\in {\mathrm {add}}(P)$ such that $P'\oplus Q=P^n$ . So we get the exact sequence

$$\begin{align*}P'\oplus Q \stackrel{(g \ h)}\longrightarrow (T^0)^n\longrightarrow (T^1)^n\longrightarrow 0.\end{align*}$$

Based on this sequence, we can construct the following commutative diagram:

Since the second vertical short exact sequence splits, both ${\mathrm {Coker}} g $ and ${\mathrm {Coker}} h $ are in ${\mathrm {add}}(T_0\oplus T_1)$ . Therefore, the exact sequence

$$\begin{align*}P'\longrightarrow (T^0)^n\longrightarrow {\mathrm{Coker}} g \longrightarrow 0\end{align*}$$

is the desired one.

Now we show that ${\mathrm {Fac}}(\mathscr {T}) = {\mathrm {Fac}}(T^0\oplus T^1)$ . It is clear that ${\mathrm {Fac}}(T^0\oplus T^1)\subseteq {\mathrm {Fac}}(\mathscr {T})$ . So it is enough to show the reverse inclusion. To do this, we show that every object $\bar {T}\in \mathscr {T}$ lies in ${\mathrm {Fac}}(T^0\oplus T^1)$ . Let

$$\begin{align*}P_1\stackrel{h_1}\longrightarrow P_0\stackrel{h_0}\longrightarrow \bar{T}\longrightarrow 0\end{align*}$$

be a projective presentation of $\bar {T}$ . Since $T^0\oplus T^1$ is a support $\tau $ -tilting object, for $i=0, 1$ , there are exact sequences

$$\begin{align*}P_i\stackrel{f_i}\longrightarrow T^0_i\stackrel{g_i}\longrightarrow T^1_i\longrightarrow 0\end{align*}$$

with $T^0_i , T^1_i\in {\mathrm {add}}(T_0\oplus T_1)$ . Therefore, since $f_i$ is a left ${\mathrm {add}}(T^0\oplus T^1)$ -approximation of $P_i$ , we have the commutative diagram

where the first row and the first column are exact. Since $g_0k_1f_1=0$ and $k_0k_1f_1=0$ , by the cokernel property, we have morphisms $l_1: T^1_1\longrightarrow T^1_0$ and $l_0: T^1_1\longrightarrow \bar {T}$ such that $g_0k_1=l_1g_1$ and $l_0g_1=k_0k_1$ . To finish the proof, we show that the sequence

is exact. Indeed, let $x\oplus y\in T^1_0\oplus \bar {T}$ . Since $g_0$ is an epimorphism, there exists $t^{\prime }_0\in T^0_0$ such that $g_0(t^{\prime }_0)=x$ . Now since $h_0$ is an epimorphism, there exists $p_0\in P_0$ such that $h_0(p_0)=k_0(t^{\prime }_0)-y$ . Consider $ (t^{\prime }_0-f_0(p_0))\oplus 0\in T^0_0\oplus T^1_1$ . Therefore, it is easy to see that

$$\begin{align*}{\begin{bmatrix} g_0\ l_0\\ k_0 \ l_1 \end{bmatrix}} \begin{bmatrix} t^{\prime}_0-f_0(p_0) \\ 0 \end{bmatrix} =\begin{bmatrix} x\\y \end{bmatrix}.\end{align*}$$

Hence, $\bar {T}\in {\mathrm {Fac}}(T^0_0\oplus T^1_1)\subseteq {\mathrm {Fac}}(T_0\oplus T_1)$ .

When $\mathscr {T}$ is a weak tilting subcategory of $\mathscr {A}$ , then ${\mathrm {Fac}}(\mathscr {T}) = \mathscr {T}^{\perp _1}$ (see [7, Proposition 2.22]). The following example shows that this equality does not hold in general for $\tau $ -tilting subcategories.

Example 3.8 Let A be the path algebra of the quiver

modulo the ideal generated by all paths of length 2. The Auslander–Reiten quiver of A is

Let $\mathscr {T}={\mathrm {add}}({\tiny \begin {matrix}2\\ 3\end {matrix}}\oplus {{\tiny \begin {matrix}2\end {matrix}}} \oplus {{\tiny \begin {matrix}1\\ 2\end {matrix}}})$ . Then $\mathscr {T}$ is a $\tau $ -tilting subcategory of $\mathrm {mod}\mbox {-}A$ . It is easy to see that

$$\begin{align*}{\mathrm{Fac}}(\mathscr{T})= {\mathrm{add}}({\tiny\begin{matrix}2\\ 3\end{matrix}} \oplus {\tiny\begin{matrix}2\end{matrix}} \oplus {\tiny\begin{matrix}1\\ 2\end{matrix}} \oplus {\tiny\begin{matrix}1\end{matrix}})\neq {\mathrm{add}}({\tiny\begin{matrix}2\\ 3\end{matrix}} \oplus {\tiny\begin{matrix}2\end{matrix}} \oplus {\tiny\begin{matrix}1\\ 2\end{matrix}} \oplus{\tiny\begin{matrix}1\end{matrix}}\oplus {\tiny\begin{matrix}3\\ 1\end{matrix}})=\mathscr{T}^{\perp_1}. \end{align*}$$

4 $\tau $ -cotorsion torsion triples

In this section, we introduce a triple of full additive subcategories of $\mathscr {A}$ , that will be called a $\tau $ -cotorsion torsion triple, or simply a $\tau $ -triple, and show that there is a bijection between the collection of all $\tau $ -tilting subcategories of $\mathscr {A}$ and the collection of all $\tau $ -triples. Proposition 2.7 motivates the following definition.

Definition 4.1 Let $\mathscr {A}$ be an abelian category with enough projective objects. A pair of full subcategories $(\mathscr {C}, \mathscr {D})$ of $\mathscr {A}$ is called a $\tau $ -cotorsion pair if:

1. 1. $\mathscr {C} = {}^{\perp _1}\mathscr {D}$ .

2. 2. For every projective object $P \in \mathscr {A}$ , there exists an exact sequence

   $$\begin{align*}P \stackrel{f}\longrightarrow {D}\longrightarrow C \longrightarrow 0,\end{align*}$$
   where $D \in \mathscr {C}\cap \mathscr {D}$ , $C \in \mathscr {C}$ , and f is a left $\mathscr {D}$ -approximation.

3. 3. $\mathscr {C}\cap \mathscr {D}$ is a contravariantly finite subcategory of $\mathscr {A}$ .

Example 4.2 Let $\mathscr {A}=\mathrm {Rep}^{\mathrm {fp}}_k \mathbb {R}_{\geq 0}$ and set

$$ \begin{align*} \mathscr{C}=&{\mathrm{add}}(\lbrace k_{[x, \infty)} \vert \ 0\leq x < \infty\rbrace \cup \lbrace k_{[x, y)} \vert \ 0 \leq x < y< 1\rbrace)\\ \mathscr{D}=&{\mathrm{add}}(\lbrace k_{[x, y)} \vert \ 1\leq x< y \leq \infty \rbrace). \end{align*} $$

Then $(\mathscr {C}, \mathscr {D})$ is a $\tau $ -cotorsion pair. Note that it is not a cotorsion pair, because $\mathscr {D}$ does not contain the injective objects. To see it is a $\tau $ -cotorsion pair, first note that

$$\begin{align*}\mathscr{C}\cap \mathscr{D}={\mathrm{add}}(\lbrace k_{[x, \infty)} \vert x\geq 1\rbrace)\end{align*}$$

is a contravariantly finite subcategory of $\mathscr {A}$ . Indeed, every indecomposable representation $k_{[ a, b)}$ admits a right $\mathscr {C}\cap \mathscr {D}$ -approximation as follows:

$$\begin{align*}\left\{ \begin{array}{r@{\ \,}cl} k_{[a, \infty)}&\longrightarrow k_{[a, b)}, & 1\leq a < b\leq \infty,\\ k_{[1, \infty)}&\longrightarrow k_{[a, b)}, & 0< a< 1 < b \leq \infty,\\ 0&\longrightarrow k_{[a, b)}, & 0\leq a< b \leq 1. \end{array} \right. \end{align*}$$

Now let $k_{[a, \infty )}$ be an indecomposable projective in $\mathscr {A}$ . If $a\geq 1$ , the exact sequence

$$\begin{align*}0\longrightarrow k_{[a, \infty)}\longrightarrow k_{[a, \infty)}\longrightarrow 0,\end{align*}$$

and if $a< 1$ , the exact sequence

$$\begin{align*}k_{[a, \infty)}\stackrel{0}\longrightarrow k_{[b, \infty)} \longrightarrow k_{[b, \infty)}\longrightarrow 0\end{align*}$$

with $b\geq 1$ , are the desired exact sequences. Finally, it is straightforward to see that $\mathscr {C}={}^{\perp _1}\mathscr {D}$ .

It is shown in [7, Lemma 2.12] that if $\mathscr {C}$ and $\mathscr {D}$ are two subcategories of an abelian category $\mathscr {A}$ such that

1. (1) ${\mathrm {Ext}}^1_{\mathscr {A}}(\mathscr {C}, \mathscr {D})=0$ ,

2. (2) $\mathscr {D}$ is closed under factor objects,

3. (3) for every object $A\in \mathscr {A}$ , there exists a short exact sequence $0 \longrightarrow A \stackrel {\varphi }\longrightarrow D \longrightarrow C \longrightarrow 0$ where $D \in \mathscr {D}$ and $C \in \mathscr {C}$ ,

then every object in $\mathscr {C}$ is of projective dimension at most 1. In particular, a cotorsion pair $(\mathscr {C}, \mathscr {T})$ satisfies all the above conditions when it is embedded in a cotorsion torsion triple $(\mathscr {C}, \mathscr {T}, \mathscr {F})$ . So the projective dimension of every object in $\mathscr {C}$ is at most 1.

The following example shows that the injectivity of the morphism $\varphi $ in Condition $(3)$ is essential. We note that this is implicit in the proof of [7, Lemma 2.12]. Also, note that it follows automatically from the previous definition that the map $\varphi $ is a left $\mathscr {D}$ -approximation.

Example 4.3 Let A be the path algebra of the quiver

modulo the ideal generated by the composition of the two arrows. The Auslander–Reiten quiver of A is the following.

Let $\mathscr {T}={\mathrm {add}}({\tiny \begin {matrix}1\\ 2\end {matrix}}\oplus {\tiny \begin {matrix}1\end {matrix}}\oplus {\tiny \begin {matrix}{3}\end {matrix}})$ . It is easy to see that $\mathscr {T}$ is a $\tau $ -tilting subcategory of $\mathrm {mod}\mbox {-}A$ . Then we have

$$ \begin{align*} \mathscr{D} & = {\mathrm{Fac}}(\mathscr{T})={\mathrm{add}}({\tiny\begin{matrix}1\\ 2\end{matrix}}\oplus {\tiny\begin{matrix}{1}\end{matrix}}\oplus {\tiny\begin{matrix}{3}\end{matrix}}), \\ \mathscr{C} & ={}^{\perp_1}{\mathrm{Fac}}(\mathscr{T})={\mathrm{add}}({\tiny\begin{matrix}1\\ 2\end{matrix}}\oplus {\tiny\begin{matrix}{1}\end{matrix}}\oplus {\tiny\begin{matrix}{3}\end{matrix}}\oplus {\tiny\begin{matrix}2 \\ 3\end{matrix}}). \end{align*} $$

It is clear that ${\mathrm {Fac}}(\mathscr {T})$ is closed under factor modules. However, the object ${\tiny \begin {matrix}{1}\end {matrix}} \in \mathscr {C}$ has projective dimension $2$ .

Definition 4.4 A triple $(\mathscr {C}, \mathscr {T}, \mathscr {F})$ of full subcategories in $\mathscr {A}$ is called a $\tau $ -cotorsion torsion triple, or simply a $\tau $ -triple, if $(\mathscr {C}, \mathscr {T})$ is a $\tau $ -cotorsion pair and $(\mathscr {T}, \mathscr {F})$ is a torsion pair.

Example 4.5 Let $\mathscr {A}=\mathrm {Rep}^{\mathrm {fp}}_k \mathbb {R}_{\geq 0}$ and $(\mathscr {C}, \mathscr {D})$ be the $\tau $ -cotorsion pair as in Example 4.2. Since $\mathscr {D}$ is a contravariantly finite subcategory of $\mathscr {A}$ which is closed under factors and extensions, it is a torsion class. Hence, $(\mathscr {C}, \mathscr {D}, \mathscr {D}^{\perp _0})$ is a $\tau $ -triple.

Proposition 4.6 Let $\mathscr {A}$ be an abelian category with enough projective objects. Then every cotorsion torsion triple $(\mathscr {C}, \mathscr {T}, \mathscr {F})$ in $\mathscr {A}$ is a $\tau $ -cotorsion torsion triple.

Proof. Since $(\mathscr {T}, \mathscr {F})$ is a torsion pair, it is enough to show that $(\mathscr {C}, \mathscr {T})$ is a $\tau $ -cotorsion pair. Condition $1$ of Definition 4.1 holds trivially. Now let P be a projective object in $\mathscr {A}$ and consider the short exact sequence $0\longrightarrow P \stackrel {f}\longrightarrow T \longrightarrow C\longrightarrow 0,$ in which $T\in \mathscr {T}, C\in \mathscr {C}$ , and f is a left $\mathscr {T}$ -approximation. It exists by definition of a cotorsion pair. To show the validity of Condition $2$ of Definition 4.1, we just need to show that $T\in \mathscr {C}\cap \mathscr {T}$ . This follows using the facts that $P, C \in \mathscr {C}$ and $\mathscr {C}$ is closed under extensions. The contravariantly finiteness of $\mathscr {C} \cap \mathscr {T}$ follows from Lemma 2.8.

Proposition 4.7 Let $\mathscr {A}$ be an abelian category with enough projective objects. Let $(\mathscr {C}, \mathscr {T}, \mathscr {F})$ be a $\tau $ -triple in $\mathscr {A}$ . Then, for every object $A\in \mathscr {A}$ , there exists an exact sequence

$$\begin{align*}A\stackrel{g}\longrightarrow T\longrightarrow C\longrightarrow 0,\end{align*}$$

where $T\in \mathscr {T}$ , $C\in \mathscr {C}$ , and g is a left $\mathscr {T}$ -approximation.

Proof. Let A be an arbitrary object of $\mathscr {A}$ . Let $P\stackrel {\pi }\longrightarrow A\longrightarrow 0$ be an epimorphism from a projective object in $\mathscr {A}$ . By $(2)$ of Definition 4.1, we can construct the following pushout diagram, where $C,\tilde {C}\in \mathscr {C}\cap \mathscr {T}$ and f is a left $\mathscr {T}$ -approximation:

We show that the second row in the above diagram is the desired exact sequence. First, we note that $T\in \mathscr {T}$ , because $T\in {\mathrm {Fac}}(\mathscr {T})$ and $\mathscr {T}$ is closed under quotients. To complete the proof, it remains to show that $g: A\longrightarrow T$ is a left $\mathscr {T}$ -approximation. Let $h: A\longrightarrow T'$ be a morphism in $\mathscr {A}$ with $T'\in \mathscr {T}$ . Since f is a left $\mathscr {T}$ -approximation, there exists a morphism $l: C\longrightarrow T'$ such that $lf=h\pi $ . Thus, the pushout property implies the existence of the morphism $t: T\longrightarrow T'$ such that the diagram

is commutative. In other words, we have that $tg\pi = h\pi $ . Since $\pi : P \longrightarrow A$ is an epimorphism, we obtain that $tg=h$ . Hence, g is a left $\mathscr {T}$ -approximation.

Corollary 4.8 Let $\mathscr {A}$ be an abelian category with enough projective objects. Let $(\mathscr {C}, \mathscr {T}, \mathscr {F})$ be a $\tau $ -triple in $\mathscr {A}$ . Then $\mathscr {T}$ is a functorially finite torsion class of $\mathscr {A}$ .

Proof. The contravariantly finiteness of $\mathscr {T}$ follows from the fact that it is a torsion class and its covariantly finiteness follows from the previous proposition.

Remark 4.9 In case $\mathscr {A}={\mathrm {{mod\mbox {-}}}} \Lambda $ , where $\Lambda $ is an Artin algebra, a generalization of the notion of a cotorsion pair, called a left weak cotorsion pair, is introduced and studied in [16]. Based on Definition 0.2 of [16], a pair $(\mathscr {C}, \mathscr {D})$ of subcategories of ${\mathrm {{mod\mbox {-}}}} \Lambda $ is a left weak cotorsion pair if:

1. (1) ${\mathrm {Ext}}^1_{\Lambda }(\mathscr {C}, \mathscr {D})=0$ .

2. (2) For every $M \in {\mathrm {{mod\mbox {-}}}} \Lambda $ , there are exact sequences

   $$\begin{align*}M \stackrel{f}{\longrightarrow} D \longrightarrow C \longrightarrow 0\end{align*}$$
   and
   $$\begin{align*}0 \longrightarrow D' \longrightarrow C' \stackrel{g}{\longrightarrow} M \longrightarrow 0\end{align*}$$
   such that $C, C' \in \mathscr {C}$ , $D, D' \in \mathscr {D}$ , f is a left $\mathscr {D}$ -approximation of M, and g is a right $\mathscr {C}$ -approximation of M.

A triple $(\mathscr {C}, \mathscr {T}, \mathscr {F})$ of full subcategories in ${\mathrm {{mod\mbox {-}}}}\Lambda $ is called a left weak cotorsion torsion triple if $(\mathscr {C}, \mathscr {T})$ is a left weak cotorsion pair and $(\mathscr {T}, \mathscr {F})$ is a torsion pair.

In the following theorem, we show that in the module category of an Artin algebra, $\tau $ -triples are exactly left weak cotorsion torsion triples.

Theorem 4.10 Let $\mathscr {A}={\mathrm {{mod\mbox {-}}}}\Lambda $ , where $\Lambda $ is an Artin algebra. Then the triple $(\mathscr {C}, \mathscr {T}, \mathscr {F})$ of full subcategories in $\mathscr {A}$ is a $\tau $ -triple if and only if it is a left weak cotorsion torsion triple.

Proof. First, let $(\mathscr {C}, \mathscr {T}, \mathscr {F})$ be a $\tau $ -triple. Since $(\mathscr {T}, \mathscr {F})$ is already a torsion pair, we just need to show that $(\mathscr {C}, \mathscr {T})$ is a left weak cotorsion pair. Condition $(1)$ of the definition of a left weak cotorsion pair is followed by the first condition of the definition of $\tau $ -cotorsion pair. Also, for $M\in {\mathrm {{mod\mbox {-}}}}\Lambda $ , by Proposition 4.7, there is an exact sequence

$$\begin{align*}M \stackrel{f}{\longrightarrow} T \longrightarrow C \longrightarrow 0,\end{align*}$$

where $T\in \mathscr {T}$ , $C\in \mathscr {C}$ , and f is a left $\mathscr {T}$ -approximation. Now, for $M\in {\mathrm {{mod\mbox {-}}}}\Lambda $ , we construct a short exact sequence as in Condition $(2)$ of the definition of left weak cotorsion pairs. Let $P_1\stackrel {\beta }\longrightarrow P_0\longrightarrow M\longrightarrow 0$ be a projective presentation of M. By Condition $(2)$ of the definition of a $\tau $ -cotorsion pair, for projective module $P_1$ , there exists an exact sequence $P_1\stackrel {\alpha }\longrightarrow T\longrightarrow C\longrightarrow 0$ . Consider the pushout diagram:

In view of the construction of pushouts in ${\mathrm {{mod\mbox {-}}}}\Lambda $ , we have the exact sequence

$$\begin{align*}P_1\stackrel{\phi=\begin{bmatrix} \alpha\\ \beta \end{bmatrix}}\longrightarrow T\oplus P_0\longrightarrow C'\longrightarrow 0,\end{align*}$$

and therefore the short exact sequence

$$\begin{align*}0\longrightarrow {\mathrm{Im}}\phi\stackrel{\varphi: \begin{bmatrix} \iota\\ \kappa \end{bmatrix}}\longrightarrow T\oplus P_0\longrightarrow C'\longrightarrow 0.\end{align*}$$

Now let $\pi : P_1\longrightarrow {\mathrm {Im}}\varphi \longrightarrow 0$ and let $\psi :{\mathrm {Im}}\varphi \longrightarrow T'$ be a morphism with $T'\in \mathscr {T}$ . We show that $\psi $ factors through $\varphi $ . Since $\alpha $ is a left $\mathscr {T}$ -approximation, then there exists a morphism $\gamma : T\longrightarrow T'$ such that $\psi \pi =\gamma \alpha =\gamma \iota \pi $ . Thus, $\psi =\gamma \iota $ . By applying the functor ${\mathrm {Hom}}_\Lambda (-,\mathscr {T})$ on the above short exact sequence and using the fact that $T\oplus P_0\in \mathscr {C}$ , we get $C'\in \mathscr {C}$ . Now since $\mathscr {T}$ is a torsion class, it is closed under quotients, and so we have ${\mathrm {Ker}} g\in \mathscr {T}$ . Therefore, the short exact sequence

$$\begin{align*}0\longrightarrow {\mathrm{Ker}} g\longrightarrow C'\stackrel{g}\longrightarrow M\longrightarrow 0\end{align*}$$

is the desired one.

Now we show the converse. Let $(\mathscr {C}, \mathscr {T}, \mathscr {F})$ be a left weak cotorsion torsion triple. Again since $(\mathscr {T}, \mathscr {F})$ is a torsion pair, we just need to show that $(\mathscr {C}, \mathscr {T})$ is a $\tau $ -cotorsion pair. By [16, Lemma 4.1], we have Condition $(1)$ of the definition of $\tau $ -cotorsion pair. To conclude Condition $(2)$ of the definition of $\tau $ -cotorsion pair, we note that $\mathscr {T}$ is a functorially finite torsion class and by [6] there exists an exact sequence

$$\begin{align*}P \stackrel{f}{\longrightarrow} T \longrightarrow C \longrightarrow 0,\end{align*}$$

where f is a left $\mathscr {T}$ -approximation, $T\in \mathscr {T}\cap \mathscr {C}$ , and $C\in \mathscr {C}$ . Finally, Condition $(3)$ of the definition of $\tau $ -cotorsion pairs follows by [16, Theorem 0.4].

5 $\tau $ -tilting subcategories and $\tau $ -triples

In this section, we show that there is a bijection between the collection of all support $\tau $ -tilting subcategories of $\mathscr {A}$ and the collection of all $\tau $ -cotorsion torsion triples in $\mathscr {A}$ . In case we start with a tilting subcategory, this bijection specializes to the one introduced in [7, Theorem 2.29]. We prepare the ground with some preliminary results.

Lemma 5.1 Let $\mathscr {A}$ be an abelian category with enough projective objects. Let $\mathscr {T}$ be a support $\tau $ -tilting subcategory of $\mathscr {A}$ . Then ${}^{\perp _1} {\mathrm {Fac}}(\mathscr {T}) \cap {\mathrm {Fac}}(\mathscr {T})=\mathscr {T}$ .

Proof. Since $\mathscr {A}$ is an abelian category with enough projectives ${\mathrm {Prj}}(\mathscr {A})$ , we have $\mathscr {A}\simeq {\mathrm {{mod\mbox {-}}}}{\mathrm {Prj}}(\mathscr {A})$ , where ${\mathrm {{mod\mbox {-}}}}{\mathrm {Prj}}(\mathscr {A})$ is the category of finitely presented functors on ${\mathrm {Prj}}(\mathscr {A})$ (see [8, Corollaries 3.9 and 3.10]). Now the result follows by the part $(ii)$ of the proof of [26, Proposition 5.3].

Proposition 5.2 Let $\mathscr {A}$ be an abelian category with enough projective objects. Let $\mathscr {T}$ be a support $\tau $ -tilting subcategory of $\mathscr {A}$ . Then

$$\begin{align*}({}^{\perp_1} {\mathrm{Fac}}(\mathscr{T}), {\mathrm{Fac}}(\mathscr{T}))\end{align*}$$

is a $\tau $ -cotorsion pair.

Proof. The first condition of Definition 4.1 holds trivially. For the second condition, consider the exact sequence

$$\begin{align*}P\stackrel{f}\longrightarrow T^0\longrightarrow T^1\longrightarrow 0,\end{align*}$$

where f is a left $\mathscr {T}$ -approximation of P and $T^0, T^1\in \mathscr {T}$ , which exists for every projective object P. Now, by Lemma 5.1, we observe that $T^0, T^1\in {}^{\perp _1} {\mathrm {Fac}}(\mathscr {T}) \cap {\mathrm {Fac}}(\mathscr {T})$ . So it remains to show that f is a left ${\mathrm {Fac}}(\mathscr {T})$ -approximation of P. To show this, let $X\in {\mathrm {Fac}}(\mathscr {T})$ and let $g: P\longrightarrow X$ be a morphism. Consider an epimorphism $\pi :T\longrightarrow X$ with $T\in \mathscr {T}$ . Since P is a projective object, there is a morphism $h: P\longrightarrow T$ such that $\pi h=g$ . Now, because f is a left $\mathscr {T}$ -approximation, there is a morphism $t: T^0\longrightarrow T$ such that $tf=h$ . Therefore, a morphism $\pi t: T^0\longrightarrow X$ exists such that $\pi t f=\pi h=g$ . Finally, Lemma 5.1 implies that ${}^{\perp _1} {\mathrm {Fac}}(\mathscr {T}) \cap {\mathrm {Fac}}(\mathscr {T})=\mathscr {T}$ is a contravariantly finite subcategory of $\mathscr {A}$ .

By [7, Proposition 2.22], if $\mathscr {T}$ is a tilting subcategory of $\mathscr {A}$ , then the pair $({\mathrm {Fac}}(\mathscr {T}), \mathscr {T}^{\perp _0})$ is a torsion pair. We use the same technique to prove the same result for $\tau $ -tilting subcategories.

Proposition 5.3 Let $\mathscr {A}$ be an abelian category with enough projective objects. If ${\mathscr {T}\subseteq \mathscr {A}}$ is a $\tau $ -tilting subcategory, then $({\mathrm {Fac}}(\mathscr {T}), \mathscr {T}^{\perp _0})$ is a torsion pair.

Proof. Let $X\in {\mathrm {Fac}}(\mathscr {T})$ . Then there exists an epimorphism $\mathscr {T}\longrightarrow X\longrightarrow 0$ with $T\in \mathscr {T}$ . The exact sequence $0\longrightarrow {\mathrm {Hom}}_{\mathscr {A}}(X, Y)\longrightarrow {\mathrm {Hom}}_{\mathscr {A}}(T, Y)$ shows that ${\mathrm {Hom}}_{\mathscr {A}}(X, Y)=0$ whenever $Y\in \mathscr {T}^{\perp _0}$ .

Now let $A\in \mathscr {A}$ be an arbitrary object. Since $\mathscr {T}$ is contravariantly finite subcategory, there exists a right $\mathscr {T}$ -approximation $\varphi : T\longrightarrow A$ . Consider the short exact sequence

$$\begin{align*}0\longrightarrow {\mathrm{Im}} \varphi \stackrel{f}\longrightarrow A \longrightarrow {\mathrm{Coker}}\varphi\longrightarrow 0,\end{align*}$$

where ${\mathrm {Im}} \varphi \in {\mathrm {Fac}}(\mathscr {T})$ . By applying the functor ${\mathrm {Hom}}_{\mathscr {A}}(\mathscr {T}, -)$ , we have a long exact sequence

$$\begin{align*}0&\longrightarrow {\mathrm{Hom}}_{\mathscr{A}}(-, {\mathrm{Im}}\varphi)\vert_{\mathscr{T}}\stackrel{f_*} \longrightarrow {\mathrm{Hom}}_{\mathscr{A}}(-, A)\vert_{\mathscr{T}} \longrightarrow {\mathrm{Hom}}_{\mathscr{A}}(-, {\mathrm{Coker}} \varphi)\vert_{\mathscr{T}}\\&\quad \longrightarrow {\mathrm{Ext}}^1_{\mathscr{A}}(-, {\mathrm{Im}} \varphi)\vert_{\mathscr{T}}.\end{align*}$$

Now, since f is a right $\mathscr {T}$ -approximation, $f_*$ is an epimorphism. Also, since $\mathscr {T}$ is $\tau $ -tilting and ${\mathrm {Im}}\varphi \in {\mathrm {Fac}}(\mathscr {T})$ , ${\mathrm {Ext}}^1_{\mathscr {A}}(-, {\mathrm {Im}} \varphi )\vert _{\mathscr {T}}=0$ . So ${\mathrm {Hom}}_{\mathscr {A}}(-, {\mathrm {Coker}} \varphi )\vert _{\mathscr {T}}=0$ and hence ${\mathrm {Coker}}\varphi \in \mathscr {T}^{\perp _0}$ .

As a consequence of this proposition, we obtain the following corollary. This is a generalization of [1, Theorem 2.7], which proves the same result for the case when $\mathscr {A}={\mathrm {{mod\mbox {-}}}}\Lambda $ , the category of finitely generated modules over an Artin algebra $\Lambda $ .

Corollary 5.4 Let $\mathscr {A}$ be an abelian category with enough projective objects. Let $\mathscr {T}$ be a support $\tau $ -tilting subcategory of $\mathscr {A}$ . Then ${\mathrm {Fac}}(\mathscr {T})$ is a functorially finite torsion class of $\mathscr {A}$ .

Proof. By Propositions 5.2 and 5.3, $({}^{\perp _1} {\mathrm {Fac}}(\mathscr {T}), {\mathrm {Fac}}(\mathscr {T}), \mathscr {T}^{\perp _0})$ is a $\tau $ -cotorsion torsion triple. So the result follows from Corollary 4.8.

Lemma 5.5 Let $\mathscr {A}$ be an abelian category with enough projective objects. Let $(\mathscr {C}, \mathscr {T}, \mathscr {F})$ be a $\tau $ -cotorsion torsion triple in $\mathscr {A}$ . Then $\mathscr {C}\cap \mathscr {T}$ is a support $\tau $ -tilting subcategory of $\mathscr {A}$ .

Proof. First, we observe that, by the definition of $\tau $ -cotorsion torsion triples, $\mathscr {C}\cap \mathscr {T}$ is a contravariantly finite subcategory of $\mathscr {A}$ and

$$\begin{align*}{\mathrm{Ext}}^1_{\mathscr{A}}(\mathscr{C}\cap\mathscr{T}, {\mathrm{Fac}}(\mathscr{C}\cap\mathscr{T}))\subseteq {\mathrm{Ext}}^1_{\mathscr{A}}(\mathscr{C}, {\mathrm{Fac}}(\mathscr{T}))={\mathrm{Ext}}^1_{\mathscr{A}}(\mathscr{C}, \mathscr{T})=0.\end{align*}$$

Moreover, for every projective object P, there is an exact sequence

$$\begin{align*}P\stackrel{f}\longrightarrow T \longrightarrow T' \longrightarrow 0,\end{align*}$$

where $T, T' \in \mathscr {C} \cap \mathscr {T}$ and f is a left $\mathscr {T}$ -approximation. To verify the last condition of support $\mathscr {T}$ -tilting subcategories, it is enough to note that the left $\mathscr {T}$ -approximation f is also a left $\mathscr {C}\cap \mathscr {T}$ -approximation.

Lemma 5.6 Let $\mathscr {A}$ be an abelian category with enough projective objects. Let $(\mathscr {C}, \mathscr {T}, \mathscr {F})$ be a $\tau $ -cotorsion torsion triple in $\mathscr {A}$ . Then $\mathscr {T}={\mathrm {Fac}}(\mathscr {C}\cap \mathscr {T})$ .

Proof. Since $\mathscr {T}$ is closed under factors, we observe that ${\mathrm {Fac}}(\mathscr {C}\cap \mathscr {T})\subseteq \mathscr {T}$ . Now let $T\in \mathscr {T}$ and $P\stackrel {\pi }\longrightarrow T\longrightarrow 0$ be an epimorphism with projective object P. By Condition $(2)$ of Definition 4.1, there is a left $\mathscr {T}$ -approximation $f: P\longrightarrow T'$ , where $T' \in \mathscr {C}\cap \mathscr {T}$ . So there exists an epimorphism $g: T' \longrightarrow T$ such that $gf=\pi $ . Hence, $T \in {\mathrm {Fac}}(\mathscr {C}\cap \mathscr {T})$ .

Theorem 5.7 Let $\mathscr {A}$ be an abelian category with enough projective objects. Then there exists a bijection between the collection of all support $\tau $ -tilting subcategories of $\mathscr {A}$ and the collections of all $\tau $ -cotorsion torsion triples in $\mathscr {A}$ . This bijection is induced by the following maps:

$$\begin{align*}\lbrace \mbox{support } \tau\mbox{-tilting subcategories}\rbrace ~~~~&\longleftrightarrow~~~~ \lbrace \tau\mbox{-cotorsion torsion triples}\rbrace\\\mathscr {T}~~~~&\stackrel {\Phi }\longrightarrow ~~~~~({}^{\perp _1}{(\mathrm {Fac}(\mathscr {T}))}, \mathrm {Fac}(\mathscr {T}), \mathscr {T}^{\perp _0})\\\mathscr{C}\cap\mathscr{T}~~&\stackrel{\Psi}\longleftarrow ~~~(\mathscr{C}, \mathscr{T}, \mathscr{F})\qquad\qquad\qquad\qquad\end{align*}$$

that are well-defined and mutually inverse. Moreover, this bijection restricts to the bijection between the collection of all tilting subcategories of $\mathscr {A}$ and the collections of all cotorsion torsion triples in $\mathscr {A}$ .

Proof. It follows from Propositions 5.2 and 5.3 and Lemma 5.5 that $\Phi $ and $\Psi $ are well defined. We show that they are mutually inverse. Let $\mathscr {T}$ be a support $\tau $ -tilting subcategory. We have that $\mathscr {T} = {}^{\perp _1}({\mathrm {Fac}}(\mathscr {T}))\cap {\mathrm {Fac}}(\mathscr {T})$ by Lemma 5.1. Thus, $\Psi \Phi \cong 1$ .

Now let $(\mathscr {C}, \mathscr {T}, \mathscr {F})$ be a $\tau $ -cotorsion torsion triple. First, we note that by Lemma 5.6, $\mathscr {T}={\mathrm {Fac}}(\mathscr {C}\cap \mathscr {T})$ and so $\mathscr {C}={}^{\perp _1}{\mathscr {T}}={}^{\perp _1} ({\mathrm {Fac}} (\mathscr {C}\cap \mathscr {T}))$ . Also, we have $\mathscr {F}=(\mathscr {C}\cap \mathscr {T})^{\perp _0}$ . Hence, $\Phi \Psi \cong 1$ and so the first statement holds.

We now prove that this bijection restricts to the bijection between cotorsion torsion triples in $\mathscr {A}$ and tilting subcategories in $\mathscr {A}$ . Let $\mathscr {T}$ be a tilting subcategory. We show that the $\tau $ -triple

$$\begin{align*}({}^{\perp_1}{(\mathrm{Fac}(\mathscr{T}))}, \mathrm{Fac}(\mathscr{T}), \mathscr{T}^{\perp_0})\end{align*}$$

is a cotorsion torsion triple. To see this, it is enough to show that $({}^{\perp _1}{(\mathrm {Fac}(\mathscr {T}))}, \mathrm {Fac}(\mathscr {T}))$ is a cotorsion pair. First, we note that, since $\mathscr {T}$ is a tilting subcategory of $\mathscr {A}$ , for every object $A\in \mathscr {A}$ , there is a short exact sequence

$$\begin{align*}0\longrightarrow A\longrightarrow X_A\longrightarrow Y_A\longrightarrow 0,\end{align*}$$

where $X_A\in {\mathrm {Fac}}(\mathscr {T})$ and $Y_A\in {}^{\perp _1}{(\mathrm {Fac}(\mathscr {T}))}$ . Next, we construct the second short exact sequence in the definition of the cotorsion pair. Let $A\in \mathscr {A}$ and let $0\longrightarrow K\longrightarrow P\longrightarrow A\longrightarrow 0$ be a short exact sequence in $\mathscr {A}$ , where P is projective. Let $0\longrightarrow K\longrightarrow X_K\longrightarrow Y_K\longrightarrow 0$ be a short exact sequence such that $X_K\in {\mathrm {Fac}}(\mathscr {T})$ and $Y_K\in {}^{\perp _1}{(\mathrm {Fac}(\mathscr {T}))}$ . Consider the following pushout diagram:

Since $P, Y_K\in {}^{\perp _1}{(\mathrm {Fac}(\mathscr {T}))}$ and ${}^{\perp _1}{(\mathrm {Fac}(\mathscr {T}))}$ is closed under extensions, then $U\in {}^{\perp _1}{(\mathrm {Fac}(\mathscr {T}))}$ . Hence, the second row in the diagram is the desired short exact sequence. Finally, it is clear that ${\mathrm {Ext}}^1_{\mathscr {A}}({}^{\perp _1}{(\mathrm {Fac}(\mathscr {T}))}, {\mathrm {Fac}}(\mathscr {T}))=0$ . So we have verified all of the conditions of a cotorsion pair.

As a corollary of the above theorem, we can recover one of the main results of [16]. For this, we need some preparations. Let us begin by recalling the original definition of a support $\tau $ -tilting module [1]. This definition is based on $\tau : {\mathrm {{mod\mbox {-}}}} \Lambda \to {\mathrm {{mod\mbox {-}}}} \Lambda $ , the Auslander–Reiten translation in ${\mathrm {{mod\mbox {-}}}} \Lambda $ (see [4, Chapter IV]).

Definition 5.8 (See [1, Definition 0.1])

Let $\Lambda $ be an Artin algebra. A module T in ${\mathrm {{mod\mbox {-}}}}\Lambda $ is called $\tau $ -rigid if ${\mathrm {Hom}}_{\Lambda }(T, \tau T)=0$ . It is called $\tau $ -tilting if it is $\tau $ -rigid and $|T|=|\Lambda |$ . A support $\tau $ -tilting module T in ${\mathrm {{mod\mbox {-}}}} \Lambda $ is a module T that is a $\tau $ -tilting module in ${\mathrm {{mod\mbox {-}}}}(\Lambda /\langle e\rangle )$ , where $\langle e \rangle $ is the ideal generated by some idempotent $e \in \Lambda $ .

As it is mentioned in Remark 3.2, by [27, Proposition 2.14], T is a support $\tau $ -tilting module in ${\mathrm {{mod\mbox {-}}}}\Lambda $ if and only if ${\mathrm {add}}(T)$ is a support $\tau $ -tilting subcategory of ${\mathrm {{mod\mbox {-}}}}\Lambda $ . We now show that every support $\tau $ -tilting subcategory in ${\mathrm {{mod\mbox {-}}}}\Lambda $ is of the form ${\mathrm {add}}(T)$ for some $\tau $ -tilting module T in ${\mathrm {{mod\mbox {-}}}}\Lambda $ .

Proposition 5.9 Let $\Lambda $ be an Artin algebra. Then every support $\tau $ -tilting subcategory $\mathscr {T}$ of ${\mathrm {{mod\mbox {-}}}} \Lambda $ is of the form ${\mathrm {add}}(T)$ , where T is a support $\tau $ -tilting module in ${\mathrm {{mod\mbox {-}}}} \Lambda $ .

Proof. Let $\mathscr {T}$ be a support $\tau $ -tilting subcategory of ${\mathrm {{mod\mbox {-}}}} \Lambda $ . By Proposition 3.7, there exists a support $\tau $ -tilting module T such that ${\mathrm {Fac}}(T)={\mathrm {Fac}}(\mathscr {T})$ . We show that ${\mathscr {T}={\mathrm {add}}(T)}$ . First, we note that by the construction of T in Proposition 3.7, we have $T\in \mathscr {T}$ . Therefore, ${\mathrm {add}}(T)\subseteq \mathscr {T}$ , since T is additively closed subcategory of ${\mathrm {{mod\mbox {-}}}}\Lambda $ . Next, let $X\in \mathscr {T}$ , then $X\in {\mathrm {Fac}}(T)$ . By Proposition 2.5 of [36], there exists a short exact sequence

$$\begin{align*}0\longrightarrow K'\longrightarrow T'\longrightarrow X\longrightarrow 0,\end{align*}$$

where $T'\in {\mathrm {add}}(T)$ and $K' \in {\mathrm {Fac}}(T)$ . Since T is a support $\tau $ -tilting module, the above short exact sequence splits and hence $X\in {\mathrm {add}}(T)$ . Thus, $\mathscr {T}={\mathrm {add}}(T)$ .

Corollary 5.10 (See [16, Theorem 4.6])

Let $\mathscr {A}={\mathrm {{mod\mbox {-}}}}\Lambda $ , where $\Lambda $ is an Artin algebra. Then there is a bijection between the collection of all support $\tau $ -tilting modules and the collection of all left weak cotorsion torsion triples.

Proof. This is a direct consequence of Theorems 4.10 and 5.7 and Proposition 5.9.

6 Summary of dual results

In this section, we collect the dual of our results in the previous sections. Given the similarity in their arguments, we list the results without their corresponding proofs. Throughout this section, we assume that $\mathscr {A}$ is an abelian category with enough injective objects. For a subcategory $\mathscr {U}$ of $\mathscr {A}$ , let ${\mathrm {Sub}} \mathscr {U}$ be the full subcategory of $\mathscr {A}$ consisting of all subobjects of finite direct sums of objects in $\mathscr {U}$ .

We start with the definition of a $\tau ^{-}$ -tilting subcategory of $\mathscr {A}$ . Recall [1, Section. 2.2] that a $\Lambda $ -module M, where $\Lambda $ is an Artin algebra, is called $\tau ^{-}$ -tilting if it is $\tau ^{-}$ -rigid, i.e., ${\mathrm {Hom}}_{\Lambda }(\tau ^{-}M, M)=0$ , and $|M|=|\Lambda |$ . It follows from [6, Proposition 5.6] that M is $\tau ^{-}$ -rigid if and only if ${\mathrm {Ext}}^1_{\mathscr {A}}({\mathrm {Sub}} M,M)=0.$ Here, ${\mathrm {Sub}} M$ means the subcategory of ${\mathrm {{mod\mbox {-}}}} \Lambda $ consisting of all subobjects of ${\mathrm {add}}(M)$ . This motivates the following definition.

Definition 6.1 Let $\mathscr {A}$ be an abelian category with enough injective objects. Let $\mathscr {U}$ be an additive full subcategory of $\mathscr {A}$ . Then $\mathscr {U}$ is called a weak support $\tau ^{-}$ -tilting subcategory if:

1. 1. ${\mathrm {Ext}}^1_{\mathscr {A}}({\mathrm {Sub}}\mathscr {U},\mathscr {U})=0.$

2. 2. For every injective object I in $\mathscr {A}$ , there exists an exact sequence

   $$\begin{align*}0\longrightarrow U^0\longrightarrow U^1\stackrel{g}\longrightarrow I\end{align*}$$
   such that $U^0$ and $U^1$ are in $\mathscr {U}$ and g is a right $\mathscr {U}$ -approximation of I.

If, furthermore, $\mathscr {U}$ is a covariantly finite subcategory of $\mathscr {A}$ , it is called a support $\tau ^-$ -tilting subcategory of $\mathscr {A}$ . A support $\tau ^-$ -tilting subcategory $\mathscr {U}$ of $\mathscr {A}$ is called a $\tau ^-$ -tilting subcategory if the approximation $g: U^1 \longrightarrow I$ is nonzero for every injective object I.

Definition 6.2 Let $\mathscr {A}$ be an abelian category with enough injective objects. A pair of full subcategories $(\mathscr {C}, \mathscr {D})$ of $\mathscr {A}$ is called a $\tau ^-$ -cotorsion pair if:

1. 1. $\mathscr {D} = \mathscr {C}^{\perp _1}$ .

2. 2. For every injective object $I \in \mathscr {A}$ , there is an exact sequence

   $$\begin{align*}0\longrightarrow D\longrightarrow C\stackrel{g}\longrightarrow I,\end{align*}$$
   where $C \in \mathscr {C}\cap \mathscr {D}$ , $D \in \mathscr {D}$ , and g is a right $\mathscr {C}$ -approximation.

3. 3. $\mathscr {C}\cap \mathscr {D}$ is a covariantly finite subcategory of $\mathscr {A}$ .

Definition 6.3 A triple $(\mathscr {T}, \mathscr {F}, \mathscr {D})$ of full subcategories in $\mathscr {A}$ is called a $\tau ^-$ -torsion cotorsion triple, or simply a $\tau ^-$ -triple, if $(\mathscr {T}, \mathscr {F})$ is a torsion pair and $(\mathscr {F}, \mathscr {D})$ is a $\tau ^-$ -cotorsion pair.

A triple $(\mathscr {T}, \mathscr {F}, \mathscr {D})$ of subcategories of $\mathscr {A}$ is called a torsion cotorsion triple if $(\mathscr {T}, \mathscr {F})$ is a torsion pair and $(\mathscr {F}, \mathscr {D})$ is a cotorsion pair.

Proposition 6.4 (Dual of Proposition 4.6)

Let $\mathscr {A}$ be an abelian category with enough injective objects. Then every torsion cotorsion pair $(\mathscr {T}, \mathscr {F}, \mathscr {D})$ in $\mathscr {A}$ is a $\tau ^-$ -triple.

Proposition 6.5 (Dual of Proposition 4.7)

Let $\mathscr {A}$ be an abelian category with enough injective objects. Let $(\mathscr {T}, \mathscr {F}, \mathscr {D})$ be a $\tau ^-$ -triple in $\mathscr {A}$ . Then, for every object $A\in \mathscr {A}$ , there exists an exact sequence

$$\begin{align*}0\longrightarrow D\longrightarrow F\stackrel{g}\longrightarrow A,\end{align*}$$

where $D\in \mathscr {D}$ , $F\in \mathscr {F}$ , and g is a right $\mathscr {F}$ -approximation.

Lemma 6.6 (Dual of Lemma 5.6)

Let $\mathscr {A}$ be an abelian category with enough injective objects. Let $(\mathscr {T}, \mathscr {F}, \mathscr {D})$ be a $\tau ^-$ -triple in $\mathscr {A}$ . Then $\mathscr {F}={\mathrm {Sub}}(\mathscr {F}\cap \mathscr {D})$ .

Proposition 6.7 (Dual of Propositions 5.2 and 5.3 and Lemma 5.1)

Let $\mathscr {A}$ be an abelian category with enough injective objects. Let $\mathscr {U}$ be a support $\tau ^-$ -tilting subcategory of $\mathscr {A}$ . Then

$$\begin{align*}({\mathrm{Sub}}\mathscr{U}, ({\mathrm{Sub}}\mathscr{U})^{\perp_1})\end{align*}$$

is a $\tau ^-$ -cotorsion pair and

$$\begin{align*}({}^{\perp_0}{\mathscr{U}}, {\mathrm{Sub}}\mathscr{U})\end{align*}$$

is a torsion pair in $\mathscr {A}$ . Moreover,

$$\begin{align*}({\mathrm{Sub}}\mathscr{U}){}^{\perp_1}\cap {\mathrm{Sub}}\mathscr{U}=\mathscr{U}.\end{align*}$$

Corollary 6.8 (Dual of Corollary 5.4)

Let $\mathscr {A}$ be an abelian category with enough injectives. Let $\mathscr {U}$ be a support $\tau ^-$ -tilting subcategory of $\mathscr {A}$ . Then ${\mathrm {Sub}}\mathscr {U}$ is a functorially finite torsion-free class of $\mathscr {A}$ .

Lemma 6.9 (Dual of Lemma 5.5)

Let $\mathscr {A}$ be an abelian category with enough injective objects. Let $(\mathscr {T}, \mathscr {F}, \mathscr {D})$ be a $\tau ^-$ -triple in $\mathscr {A}$ . Then $\mathscr {F}\cap \mathscr {D}$ is a support $\tau ^-$ -tilting subcategory of $\mathscr {A}$ .

Let $\mathscr {U}$ be an additively closed full subcategory $\mathscr {A}$ . By [7, Section 2.3], $\mathscr {U}$ is called a cotilting subcategory if it satisfies the following conditions.

1. (i) $\mathscr {U}$ is a covariantly finite subcategory of $\mathscr {A}$ .

2. (ii) ${\mathrm {Ext}}^1_{\mathscr {A}}(U_1, U_2)=0$ , for all $U_1, U_2 \in \mathscr {U}$ .

3. (iii) Every object $U \in \mathscr {U}$ has injective dimension at most $1$ .

4. (iv) For every injective object I in $\mathscr {A}$ , there exists a short exact sequence

   $$\begin{align*}0 \longrightarrow U_1 {\longrightarrow} U_0 \longrightarrow I \longrightarrow 0,\end{align*}$$
   with $U^i \in \mathscr {U}$ .

If $\mathscr {U}$ only satisfies the conditions $(ii)$ – $(iv)$ , it is called a weak cotilting subcategory of $\mathscr {A}$ .

Theorem 6.10 (Dual of Theorem 5.7)

Let $\mathscr {A}$ be an abelian category with enough injective objects. Then there is a bijection

$$\begin{align*}\lbrace \mbox{support } \tau^{-}\mbox{-tilting subcategories}\rbrace &~~~~\longleftrightarrow~~~~ \lbrace \tau^{-}\mbox{-torsion cotorsion triples}\rbrace\\\mathscr {U}&~~~~\longrightarrow ~~~~~({}^{\perp _0}{\mathscr {U}}, {\mathrm {Sub}}\mathscr {U}, ({\mathrm {Sub}}\mathscr {U})^{\perp _1})\\\mathscr{F}\cap\mathscr{D}&~~\longleftarrow ~~~(\mathscr{T}, \mathscr{F}, \mathscr{D}).\end{align*}$$

This bijection restricts to a bijection between the collection of all cotilting subcategories of $\mathscr {A}$ and the collections of all torsion cotorsion triples in $\mathscr {A}$ .

Proposition 6.11 (Dual of Proposition 5.9)

Let $\Lambda $ be an Artin algebra. Then every support $\tau ^{-}$ -tilting subcategory $\mathscr {T}$ of ${\mathrm {{mod\mbox {-}}}} \Lambda $ is of the form ${\mathrm {add}}(T)$ , where T is a support $\tau ^{-}$ -tilting module in ${\mathrm {{mod\mbox {-}}}} \Lambda $ .

7 Connection to silting and cosilting modules

This section is divided into two subsections and is devoted to the study of the connections between support $\tau $ -tilting and support $\tau ^{-}$ -tilting subcategories with silting and cosilting theories in $\mathrm{Mod}\mbox {-}R$ , respectively, where R is an associative unitary ring. In particular, we characterize all support $\tau $ - and support $\tau ^-$ -tilting subcategories of $\mathrm{Mod}\mbox {-}R$ .

7.1 Silting modules and support $\tau $ -tilting subcategories

Our aim in this subsection is to characterize all support $\tau $ -tilting subcategories of $\mathrm{Mod}\mbox {-} R$ . We do this by providing a bijection between the equivalence classes of all support $\tau $ -tilting subcategories of $\mathrm{Mod}\mbox {-}R$ and the collection of all equivalent classes of the certain R-modules, the so-called finendo quasitilting R-modules. It is known that all silting modules are finendo quasitilting. As a result, it will be shown that ${\mathrm {Add}}(S)$ is a $\tau $ -tilting subcategory of $\mathrm{Mod}\mbox {-}R$ , where S is a silting R-module.

For a module M in $\mathrm{Mod}\mbox {-}R$ , let ${\mathrm {Add}}(M)$ denote the class of all modules isomorphic to a direct summand of an arbitrary direct sum of copies of M. We also let ${\mathrm {Gen}}(M)$ be the subcategory of $\mathrm{Mod}\mbox {-}R$ consisting of all M-generated modules, i.e., all modules isomorphic to epimorphic images of modules in ${\mathrm {Add}}(M)$ . Moreover, we let $\mathrm {Pres}(M)$ be the subcategory of $\mathrm{Mod}\mbox {-}R$ consisting of all M-presented modules, i.e., all modules that admit an ${\mathrm {Add}}(M)$ -presentation. Recall that an ${\mathrm {Add}}(M)$ -presentation of an R-module X is an exact sequence

$$\begin{align*}M_1 \longrightarrow M_0 \longrightarrow X \longrightarrow 0,\end{align*}$$

with $M_1$ and $M_0$ in ${\mathrm {Add}}(M)$ .

Let $\sigma $ be a morphism in ${\mathrm {Prj}}(R) $ . Let $\mathscr {D}_{\sigma }$ be the class of all modules M in $\mathrm{Mod}\mbox {-}R$ such that the induced homomorphism ${\mathrm {Hom}}_R(\sigma , M)$ is surjective.

Definition 7.1.1 (See[2, Definition 3.7])

An R-module S is called a partial silting module if there exists a projective presentation $\sigma $ of S such that $\mathscr {D}_{\sigma }$ contains S and is a torsion class in $\mathrm{Mod}\mbox {-}R$ . S is called a silting module if there is a projective presentation $\sigma $ of S such that ${\mathrm {Gen}}(S)=\mathscr {D}_{\sigma }.$

Remark 7.1.2 By [2, Remark 3.8], every silting module is a partial silting module; hence, ${\mathrm {Gen}}(S)$ is a torsion class. Support $\tau $ -tilting modules over a finite-dimensional k-algebra are examples of silting modules (see [2, Proposition 3.15]).

Definition 7.1.3 (See [2, Lemdef 3.1])

An R-module T is called quasitilting if $\mathrm {Pres}(T)={\mathrm {Gen}}(T)$ and T is Ext-projective in ${\mathrm {Gen}}(T)$ .

Recall that an R-module T is Ext-projective with respect to a subcategory $\mathscr {X}$ of $\mathrm{Mod}\mbox {-}R$ , if $T \in \mathscr {X}$ and ${\mathrm {Ext}}^1_R(T, \mathscr {X})=0$ .

The following proposition collects some basic properties of quasitilting modules. Their proofs can be found in [2, Lemdef 3.1, Lemma 3.3, and Proposition 3.2]. Recall that an R-modules T is called finendo if it is finitely generated over its endomorphism ring.

Proposition 7.1.4 Let T be a quasitilting R-module. Then the following statements hold.

1. 1. If $X\in {\mathrm {Gen}}(T)$ then, there exist a set J and a short exact sequence

   $$\begin{align*}0\longrightarrow{\mathrm{Ker}}\pi\longrightarrow T^{(J)}\stackrel{\pi}\longrightarrow X\longrightarrow 0\end{align*}$$
   such that ${\mathrm {Ker}}\pi \in {\mathrm {Gen}}(T)$ and $T^{(J)}$ is the coproduct of copies of T indexed by J. That is, ${\mathrm {Gen}}(T)$ is closed with respect to the kernels of epimorphisms.

2. 2. ${\mathrm {Add}}(T)$ is the class of ${\mathrm {Ext}}$ -projective modules in ${\mathrm {Gen}}(T)$ .

3. 3. The following are equivalent.

   1. (i) T is a finendo quasitilting R-module.

   2. (ii) ${\mathrm {Gen}}(T)$ is a torsion class and T is a tilting $R/{\mathrm {ann}}(R)$ -module.

   3. (iii) T is an ${\mathrm {Ext}}$ -projective module in ${\mathrm {Gen}}(T)$ and there exists an exact sequence

      $$\begin{align*}R\stackrel{f}\longrightarrow T_0\longrightarrow T_1\longrightarrow 0\end{align*}$$
      such that $T_0, T_1\in {\mathrm {Add}}(T)$ and f is a left ${\mathrm {Gen}}(T)$ -approximation.

Proposition 7.1.5 Let T be a finendo quasitilting module in $\mathrm{Mod}\mbox {-}R$ . Then ${\mathrm {Add}}(T)$ is a support $\tau $ -tilting subcategory of $\mathrm{Mod}\mbox {-}R$ .

Proof. We show the validity of conditions of Definition 3.1. Note that ${\mathrm {Fac}}({\mathrm {Add}}(T))={\mathrm {Gen}}(T)$ . Then ${\mathrm {Ext}}^1_R({\mathrm {Add}}(T), {\mathrm {Fac}}({\mathrm {Add}}(T)))=0$ follows from Proposition 7.1.4.2.

By Proposition 7.1.4.3, there exists an exact sequence

$$\begin{align*}R\stackrel{f}\longrightarrow T_0\longrightarrow T_1\longrightarrow 0,\end{align*}$$

where $T_0, T_1\in {\mathrm {Add}}(T)$ and f is a left ${\mathrm {Gen}}(T)$ -approximation. Since $T_0 \in {\mathrm {Add}}(T)$ , f is also a left ${\mathrm {Add}}(T)$ -approximation. Now let P be a projective R-module. There exists a set J and an epimorphism $R^{(J)}\longrightarrow P\longrightarrow 0$ which is split. Consider the pushout diagram

Since h is a split epimorphism, we get $\tilde {T}\in {\mathrm {Add}}(T)$ . Also, it follows easily from the universal property of the pushout diagrams that g is a left ${\mathrm {Add}}(T)$ -approximation of P.

To complete the proof, we just need to show that ${\mathrm {Add}}(T)$ is a contravariantly finite subcategory of $\mathrm{Mod}\mbox {-}R$ . Let $M\in \mathrm{Mod}\mbox {-}R$ . Since, by Proposition 7.1.4.3, ${\mathrm {Gen}}(T)$ is a torsion class, it is a contravariantly finite subcategory of $\mathrm{Mod}\mbox {-}R$ . Therefore, there is a monomorphism $0\longrightarrow X\stackrel {\imath }\longrightarrow M $ such that $X\in {\mathrm {Gen}}(T)$ . By Proposition 7.1.4.1, there exist a set J and a short exact sequence

$$\begin{align*}0\longrightarrow{\mathrm{Ker}}\pi \longrightarrow T^{(J)}\stackrel{\pi}\longrightarrow X\longrightarrow 0\end{align*}$$

such that ${\mathrm {Ker}} \pi \in {\mathrm {Gen}}(T)$ . We show that $T^{(J)}\stackrel {\imath \pi }\longrightarrow M$ is a right ${\mathrm {Add}}(T)$ -approximation. To do this, let $T'\in {\mathrm {Add}}(T)$ and $T'\stackrel {\ell }\longrightarrow M$ be a morphism. Since $\imath $ is a right ${\mathrm {Gen}}(T)$ -approximation, there is a morphism $T'\stackrel {\jmath }\longrightarrow X$ such that $\imath \jmath =\ell $ . By applying ${\mathrm {Hom}}_R(T', -)$ on the above short exact sequence and using the fact that ${\mathrm {Ker}}\pi \in {\mathrm {Gen}}(T)$ , we conclude that $\jmath $ factors through $\pi $ . Hence, $\ell $ factors through $\imath \pi $ and the result follows.

As a direct consequence of the above proposition, we have the following.

Corollary 7.1.6 Let S be a silting module in $\mathrm{Mod}\mbox {-}R$ . Then ${\mathrm {Add}}(S)$ is a support $\tau $ -tilting subcategory of $\mathrm{Mod}\mbox {-}R$ .

Proof. By [2, Proposition 3.10], every silting R-module is a finendo quasitilting R-module. Now, by the above proposition, the result follows.

We also have a kind of converse to the previous proposition.

Proposition 7.1.7 Let $\mathscr {T}$ be a support $\tau $ -tilting subcategory of $\mathrm{Mod}\mbox {-}R$ such that ${\mathscr {T}={\mathrm {Add}}(T)}$ , for some R-module T. Then T is a finendo quasitilting R-module.

Proof. Since $\mathscr {T}$ is a support $\tau $ -tilting subcategory, there exists a short exact sequence

$$\begin{align*}R\stackrel{f}\longrightarrow T^0\longrightarrow T^1\longrightarrow 0,\end{align*}$$

where $T^0, T^1\in \mathscr {T}$ and f is a left $\mathscr {T}$ -approximation. We note that f also is a left ${\mathrm {Fac}}(\mathscr {T})$ -approximation (see, for instance, the proof of Proposition 5.2). On the other hand, since ${\mathrm {Fac}}(\mathscr {T})={\mathrm {Fac}}({\mathrm {Add}}(T))={\mathrm {Gen}}(T)$ and $\mathscr {T}$ is a support $\tau $ -tilting subcategory, T is Ext-projective in ${\mathrm {Gen}}(T)$ . Now Proposition 7.1.4.3 implies that T is a finendo quasitilting module.

Let R be a right perfect ring. In this case, it is known that every functorially finite torsion class is generated by a silting module that is constructed by taking an approximation of the ring and then its cokernel [16, Theorem 2.4]. In the following, we use a similar idea to provide a version of Proposition 3.7 in $\mathrm{Mod}\mbox {-}R$ .

Proposition 7.1.8 Let $\mathscr {T}$ be a support $\tau $ -tilting subcategory of $\mathrm{Mod}\mbox {-}R$ . Then there exists a finendo quasitilting module T such that ${\mathrm {Fac}}(\mathscr {T})={\mathrm {Gen}}(T)$ .

Proof. By Theorem 5.7, $({}^{\perp _1}{\mathrm {Fac}}(\mathscr {T}), {\mathrm {Fac}}(\mathscr {T}), \mathscr {T}^{\perp _0})$ is a $\tau $ -triple. Hence, by Proposition 4.7, for every $M\in \mathrm{Mod}\mbox {-}R$ , there exists an exact sequence

$$\begin{align*}M\stackrel{\phi}\longrightarrow B\longrightarrow C\longrightarrow 0\end{align*}$$

such that $\phi $ is a left ${\mathrm {Fac}}(\mathscr {T})$ -approximation and $C\in {}^{\perp _1}{\mathrm {Fac}}(\mathscr {T})$ , that is, C is an Ext-projective in ${\mathrm {Fac}}(\mathscr {T})$ . Hence, by [2, Theorem 3.4], we deduce that there exists a finendo quasitilting R-module T such that ${\mathrm {Fac}}(\mathscr {T})={\mathrm {Gen}}(T)$ .

Definition 7.1.9 Let $\mathscr {A}$ be an abelian category with enough projective objects. Let $\mathscr {T}$ and $\mathscr {T}'$ be two support $\tau $ -titling subcategories of $\mathscr {A}$ . We say that $\mathscr {T}$ and $\mathscr {T}'$ are equivalent if ${\mathrm {Fac}}(\mathscr {T})={\mathrm {Fac}}(\mathscr {T}')$ .

We now have enough ingredients for the proof of our main theorem. Recall that, by [2, p. 12], two quasitilting modules $T_1$ and $T_2$ are equivalent if ${\mathrm {Add}}(T_1)={\mathrm {Add}}(T_2)$ .

Theorem 7.1.10 There is a bijection between the equivalence classes of support $\tau $ -tilting subcategories of $\mathrm{Mod}\mbox {-}R$ and the equivalence classes of finendo quasitilting R-modules.

Proof. The result is followed by Propositions 7.1.8 and 7.1.5.

7.2 Cosilting modules and support $\tau ^-$ -tilting subcategories

Our aim in this subsection is to characterize all support $\tau ^{-}$ -tilting subcategories of $\mathrm{Mod}\mbox {-}R$ in terms of cosilting R-modules. Note that cosilting modules coincide with quasicotilting modules coincide (see [34, Theorem 4.18]). We show that ${\mathrm {Prod}}(T)$ is a $\tau ^{-}$ -tilting subcategory of $\mathrm{Mod}\mbox {-}R$ , where T is a quasicotilting R-module, where ${\mathrm {Prod}}(M)$ denotes the class of all modules isomorphic to an arbitrary direct product of copies of M.

For a module M in $\mathrm{Mod}\mbox {-}R$ , let ${\mathrm {Cogen}}(M)$ be the subcategory of $\mathrm{Mod}\mbox {-} R$ consisting of all M-cogenerated modules, i.e., all modules isomorphic to a submodule of modules in ${\mathrm {Prod}}(M)$ and let $\mathrm {Copres}(M)$ be the subcategory of $\mathrm{Mod}\mbox {-}R$ consisting of all M-copresented modules, i.e., all modules that admit a ${\mathrm {Prod}}(M)$ -copresentation. Recall that a ${\mathrm {Prod}}(M)$ -copresentation of an R-module X is an exact sequence

$$\begin{align*}0 \longrightarrow X \longrightarrow M_0 \longrightarrow M_1,\end{align*}$$

with $M_0$ and $M_1$ in ${\mathrm {Prod}}(M)$ .

Let $\zeta $ be a morphism in ${\mathrm {Inj}}(R)$ . Let $\mathscr {B}_{\zeta }$ denote the class of all modules M in $\mathrm{Mod}\mbox {-}R$ such that the induced homomorphism ${\mathrm {Hom}}_R(M, \zeta )$ is surjective.

Definition 7.2.1 (See [14, Definition 3.1])

An R-module C is called a partial cosilting module if there is an injective copresentation $\zeta $ of C such that $\mathscr {B}_{\zeta }$ contains C and the class $\mathscr {B}_\zeta $ is closed under direct products. Moreover, C is called a cosilting module if there is an injective copresentation $\zeta $ of C such that ${\mathrm {Cogen}}(C)=\mathscr {B}_{\zeta }.$

By [14, Remark 3.2], every cosilting module is a partial cosilting module. In particular, for every cosilting module C, ${\mathrm {Cogen}}(C)$ is a torsion-free class.

Definition 7.2.2 (See [34, Definition 3.1])

An R-module T is called a quasicotilting module if ${\mathrm {Cogen}}(T)=\mathrm {Copres}(T)$ and T is ${\mathrm {Ext}}$ -injective in ${\mathrm {Cogen}}(T)$ .

Recall that an R-module T is Ext-injective with respect to a subcategory $\mathscr {X}$ of $\mathrm{Mod}\mbox {-}R$ , if $T \in \mathscr {X}$ and ${\mathrm {Ext}}^1_R(\mathscr {X}, T)=0$ .

By [34, Theorem 4.18], quasicotilting modules and cosilting modules coincide, so if T is a quasicotilting R-module, then ${\mathrm {Cogen}}(T)$ is a torsion-free class. Also, [34, Proposition 3.2.2] implies that all quasicotilting R-modules are cofinendo. Recall that R-module T is cofinendo if and only if there exists a right ${\mathrm {Prod}}(T)$ -approximation of an injective cogenerator $\mathrm{Mod}\mbox {-}R$ (see [3, Proposition 1.6]).

The following proposition collects some of the basic properties of the quasicotilting modules. The proofs can be found in [35, Lemma 3.1 and Theorem 3.2].

Proposition 7.2.3 Let E be an injective cogenerator of $\mathrm{Mod}\mbox {-}R$ . Let T be a quasicoilting R-module. Then the following statements hold.

1. 1. If $X\in {\mathrm {Cogen}}(T)$ , then there exist a set J and a short exact sequence

   $$\begin{align*}0\longrightarrow X\stackrel{\imath}\longrightarrow T^J\longrightarrow {\mathrm{Coker}}\imath\longrightarrow 0\end{align*}$$
   such that ${\mathrm {Coker}}\imath \in {\mathrm {Cogen}}(T)$ and $T^J$ is the product of copies of T indexed by J. That is, ${\mathrm {Cogen}}(T)$ is closed with respect to the cokernels of monomorphisms.

2. 2. ${\mathrm {Prod}}(T)$ is the class of ${\mathrm {Ext}}$ -injective modules in ${\mathrm {Cogen}}(T)$ .

3. 3. T is quasicotilting if and only if T is an $Ext$ -injective in ${\mathrm {Cogen}}(T)$ and there exists an exact sequence

   $$\begin{align*}0\longrightarrow T_0\longrightarrow T_1\stackrel{f}\longrightarrow E \end{align*}$$
   such that $T_0, T_1\in {\mathrm {Prod}}(T)$ and f is a right ${\mathrm {Cogen}}(T)$ -approximation.

In the following, we show that every quasicotilting module induces a support $\tau ^-$ -tilting subcategory of $\mathrm{Mod}\mbox {-}R$ . The proof is essentially dual to that of Proposition 7.1.5, but we include it for the sake of completeness.

Proposition 7.2.4 Let E be an injective cogenerator of $\mathrm{Mod}\mbox {-}R$ . Let T be a quasicotilting R-module. Then ${\mathrm {Prod}}(T)$ is a support $\tau ^-$ -tilting subcategory of $\mathrm{Mod}\mbox {-}R$ .

Proof. We show the validity of the conditions of Definition 6.1. For the first condition, note that ${\mathrm {Sub}}{\mathrm {Prod}}(T)={\mathrm {Cogen}}(T)$ . The fact that

$$\begin{align*}{\mathrm{Ext}}^1_R({\mathrm{Sub}}{\mathrm{Prod}}(T), {\mathrm{Prod}}(T))=0\end{align*}$$

follows from Proposition 7.2.3(2). For the second condition, consider the exact sequence

$$\begin{align*}0\longrightarrow T_0\longrightarrow T_1\stackrel{f}\longrightarrow E \end{align*}$$

of Proposition 7.2.3(3), in which $T_0, T_1\in {\mathrm {Prod}}(T)$ and f is a right ${\mathrm {Cogen}}(T)$ -approximation of E. Now let I be an injective R-module. There exists a set J and a monomorphism $0\longrightarrow I\longrightarrow E^{J}$ which is a split morphism. Consider the pullback diagram

Since h is a split monomorphism, we get $\tilde {T}\in {\mathrm {Prod}}(T)$ . Moreover, it follows easily from the universal property of the pullback diagrams that g is a right ${\mathrm {Prod}}(T)$ -approximation of I.

To complete the proof, it remains to show that ${\mathrm {Prod}}(T)$ is a covariantly finite subcategory of $\mathrm{Mod}\mbox {-}R$ . Let $M\in \mathrm{Mod}\mbox {-}R$ . Since ${\mathrm {Cogen}}(T)$ is a torsion-free class, it is a covariantly finite subcategory of $\mathrm{Mod}\mbox {-}R$ . Therefore, there is an epimorphism $ M\stackrel {\pi }\longrightarrow X\longrightarrow 0$ such that $X\in {\mathrm {Cogen}}(T)$ . By Proposition 7.2.3(1), there exist a set J and a short exact sequence

$$\begin{align*}0\longrightarrow X\stackrel{\imath}\longrightarrow T^{J}\longrightarrow {\mathrm{Coker}}\imath\longrightarrow 0\end{align*}$$

such that ${\mathrm {Coker}} \imath \in {\mathrm {Cogen}}(T)$ . We claim that $M \stackrel {\imath \pi }\longrightarrow T^{J}$ is a left ${\mathrm {Prod}}(T)$ -approximation. Indeed, let $f: M \longrightarrow \tilde {T}$ be a map with $\tilde {T} \in {\mathrm {Prod}}(T)$ . Then f factors through X because $\tilde {T} \in {\mathrm {Cogen}}(T)$ . Moreover, ${\mathrm {Cogen}}(T)$ is closed under cokernels of monomorphisms by Proposition 7.2.3(1). Hence, we can lift the factorization f through X using $\pi $ and conclude that f factors through $\imath \pi $ .

The following is a kind of converse to the previous proposition which is also the dual of Proposition 7.1.7.

Proposition 7.2.5 Let E be an injective cogenerator of $\mathrm{Mod}\mbox {-}R$ . Let $\mathscr {U}$ be a support $\tau ^-$ -tilting subcategory of $\mathrm{Mod}\mbox {-}R$ such that $\mathscr {U}={\mathrm {Prod}}(T)$ , for some R-module T. Then T is a quasicotilting R-module.

Proof. Since $\mathscr {U}$ is a support $\tau ^-$ -tilting subcategory, for E there exists an exact sequence

$$\begin{align*}0\longrightarrow U^0\longrightarrow U^1\stackrel{f}\longrightarrow E,\end{align*}$$

where $U^0, U^1\in \mathscr {T}$ and f is a right $\mathscr {U}$ -approximation. We note that f is also a right ${\mathrm {Sub}}\mathscr {U}$ -approximation. On the other hand, since ${\mathrm {Sub}}\mathscr {U} ={\mathrm {Sub}} {\mathrm {Prod}}(T) ={\mathrm {Cogen}}(T)$ and $\mathscr {U}$ is a support $\tau ^-$ -tilting subcategory, T is Ext-injective in ${\mathrm {Cogen}}(T)$ . Now Proposition 7.2.3(3) implies that T is a quasicotilting module.

The following result is a dual of Proposition 7.1.8.

Proposition 7.2.6 Let $\mathscr {U}$ be a support $\tau ^-$ -tilting subcategory of $\mathrm{Mod}\mbox {-}R$ . Then there exists a quasicotilting module T such that ${\mathrm {Sub}} \mathscr {U}={\mathrm {Cogen}}(T)$ .

Proof. By Theorem 6.10, $({}^{\perp _0}{\mathscr {U}}, {\mathrm {Sub}}\mathscr {U}, ({\mathrm {Sub}}\mathscr {U})^{\perp _1})$ is a $\tau ^-$ -triple. Hence, by Proposition 6.5, for every $M\in \mathrm{Mod}\mbox {-}R$ , there exists an exact sequence

$$\begin{align*}0\longrightarrow B\longrightarrow C\stackrel{\phi}\longrightarrow M\end{align*}$$

such that $\phi $ is a right ${\mathrm {Sub}} \mathscr {U}$ -approximation and $B\in ({\mathrm {Sub}}\mathscr {U})^{\perp _1}$ , that is, B is an ${\mathrm {Ext}}$ -injective module in ${\mathrm {Sub}}\mathscr {U}$ . Hence, by [35, Theorem 3.5], we deduce that there exists a quasicotilting R-module T such that ${\mathrm {Sub}} \mathscr {U}={\mathrm {Cogen}}(T)$ .

Definition 7.2.7 Let $\mathscr {A}$ be an abelian category with enough injective objects. Let $\mathscr {U}$ and $\mathscr {U}'$ be two support $\tau ^-$ -titling subcategories of $\mathscr {A}$ . We say that $\mathscr {U}$ and $\mathscr {U}'$ are equivalent if ${\mathrm {Sub}} \mathscr {U}={\mathrm {Sub}} \mathscr {U}'.$

Now we can state the main theorem of this subsection which is the dual of Theorem 7.1.10. Recall that, by [35, p. 12], two quasicotilting modules $T_1$ and $T_2$ are equivalent if ${\mathrm {Prod}}(T_1)={\mathrm {Prod}}(T_2)$ .

Theorem 7.2.8 There is a bijection between the equivalence classes of support $\tau ^-$ -tilting subcategories of $\mathrm{Mod}\mbox {-}R$ and the equivalence classes of cosilting R-modules.

Proof. The result follows by Propositions 7.2.6 and 7.2.4 together with [34, Theorem 4.18].

8 Applications to quiver representations

This section is devoted to producing support $\tau $ -tilting subcategories of the category of representation of quivers. We divide the section into three subsections. In the first subsection, we recall some known definitions and properties of the category $\mathrm {Rep}(Q, \mathscr {A})$ of representations of a finite acyclic quiver Q over an abelian category $\mathscr {A}$ with enough projective objects. In the second subsection, we produce support $\tau $ -tilting subcategories in $\mathrm {Rep}(Q, \mathscr {A})$ from certain $\tau $ -tilting subcategories of $\mathscr {A}$ . Finally, in the last subsection, we construct (co)silting modules in $\mathrm{Mod}\mbox {-}RQ$ from (co)silting modules in $\mathrm{Mod}\mbox {-}R$ .

8.1 Notions on quiver representations

Let $\mathscr {A}$ be an abelian category with enough projective objects ${\mathrm {Prj}}(\mathscr {A})$ and $Q=(Q_0, Q_1)$ be a finite acyclic quiver with vertex set $Q_0$ of cardinality n and arrow set $Q_1$ . An arrow $\alpha \in Q_1$ of source $i=s(\alpha )$ and target $j=t(\alpha )$ is usually denoted by $\alpha : i\rightarrow j$ . We denote by $\mathrm {Rep}(Q, \mathscr {A})$ the category of representations of Q in $\mathscr {A}$ . An object X in $\mathrm {Rep}(Q, \mathscr {A})$ is defined by the following data:

1. 1. To each vertex, $i\in Q_0$ is associated an object $X_i$ in $\mathscr {A}$ .

2. 2. To each arrow, $\alpha : i\rightarrow j$ in $Q_1$ is associated a morphism $X_\alpha : X_i\longrightarrow X_j$ .

A morphism $\varphi : X \longrightarrow Y$ in $\mathrm {Rep}(Q,\mathscr {A})$ is a family $\lbrace \varphi _i: X_i\longrightarrow Y_i \rbrace _{ i\in Q_0}$ of morphisms in $\mathscr {A}$ such that for each arrow $\alpha : i\longrightarrow j$ in $Q_1$ , the diagram

is commutative.

The category $\mathrm {Rep}(Q, \mathscr {A})$ is an abelian category. Kernels, cokernels, and images in $\mathrm { Rep}(Q,\mathscr {A})$ are computed vertexwise in $\mathscr {A}$ . A sequence $X \longrightarrow Y \longrightarrow Z$ in $\mathrm {Rep}(Q,\mathscr {A})$ is exact if and only if for every vertex $i \in Q_0$ , the sequence $X_i\longrightarrow Y_i\longrightarrow Z_i$ is exact in $\mathscr {A}$ .

For each vertex $i\in Q_0$ , there exists the evaluation functor

$$\begin{align*}e_i: \mathrm{Rep}(Q, \mathscr{A})&\longrightarrow \mathscr{A}\\X&\mapsto X_i\end{align*}$$

which sends each representation $X\in \mathrm {Rep}(Q, \mathscr {A})$ to the object $X_i\in \mathscr {A}$ at vertex i. It is clear that the evaluation functor $e_i$ is exact and it has an exact left and also an exact right adjoint, which will be denoted by $e_i^{\lambda }$ and $e_i^\rho $ , respectively. Let us recall the constructions of $e_i^{\lambda },e_i^\rho : \mathscr {A}\longrightarrow \mathrm {Rep}(Q, \mathscr {A})$ more explicitly (cf. [25]).

Let $A\in \mathscr {A}$ . Then $e_i^{\lambda }(A)_j= \bigoplus _{Q(i, j)} A$ , where $Q(i, j)$ denotes the set of paths starting in i and ending in j. The morphisms are natural inclusions, that is, for any arrow $\alpha : j\longrightarrow k$ , we set $e_i^{\lambda }(A)_\alpha : \bigoplus _{Q(i, j)} A\longrightarrow \bigoplus _{Q(i, k)} A$ .

The right adjoint $e_i^\rho $ is defined dually. Let $A\in \mathscr {A}$ . Then $e_i^\rho (A)_j=\bigoplus _{Q(j, i)} A$ . The morphisms are natural projections. Moreover, the functor $e_i^\rho $ has a right adjoint, which will be denoted by $Re_i^\rho $ .

One can show that the sets

$$\begin{align*}\lbrace e_i^{\lambda} (P): \ i\in Q_0, P\in{\mathrm{Prj}}(\mathscr{A})\rbrace \ \ \mathrm{and} \ \ \lbrace e_i^{\rho}(I): \ i\in Q_0, I\in{\mathrm{Inj}}(\mathscr{A}) \rbrace\end{align*}$$

are sets of projective generators and injective cogenerators for the category $\mathrm {Rep}(Q, \mathscr {A})$ , respectively. For details of the proofs, see, e.g., [20, 21].

8.2 Constructing $\tau $ -tilting subcategories of $\mathrm {Rep}(Q, \mathscr {A})$

Our aim in this subsection is to provide a systematic technique to construct, starting from a certain $\tau $ -tilting subcategory $\mathscr {T}$ of an abelian category $\mathscr {A}$ , a new $\tau $ -tilting subcategory in the category $\mathrm {Rep}(Q, \mathscr {A})$ of representations of a finite and acyclic quiver Q in $\mathscr {A}$ . For the proof of the main result of this subsection, we need the following lemma. Although it seems that it is known to the experts, we could not find a reference. So we provide proof for the sake of completeness.

Lemma 8.2.1 Let $\mathscr {A}$ be an abelian category with enough projective objects, and let $Q=(Q_0, Q_1)$ be a finite and acyclic quiver. Then, for every $X, Y \in \mathrm {Rep}(Q, \mathscr {A})$ , there exists the long exact sequence

$$ \begin{align*} 0 \longrightarrow & \ {\mathrm{Hom}}_{\mathscr{R}}(X, Y)\longrightarrow \bigoplus_{r\in Q_0} {\mathrm{Hom}}_{\mathscr{A}}(X_r, Y_r) \stackrel{\varphi}\longrightarrow \bigoplus_{\alpha: r\rightarrow l} {\mathrm{Hom}}_{\mathscr{A}}(X_r, Y_l) \\ \longrightarrow & \ {\mathrm{Ext}}^1_{\mathscr{R}}(X, Y) \ \longrightarrow \ \bigoplus_{r\in Q_0}{\mathrm{Ext}}^1_{\mathscr{A}}(X_r, Y_r) \ \longrightarrow \bigoplus_{\alpha: r\rightarrow l}{\mathrm{Ext}}^1_{\mathscr{A}}(X_r, Y_l) \\ \longrightarrow & \ {\mathrm{Ext}}^2_{\mathscr{R}}(X, Y) \ \longrightarrow \cdots, \end{align*} $$

where here and throughout we set $\mathscr {R}:=\mathrm {Rep}(Q, \mathscr {A})$ .

Proof. Let $X \in \mathscr {R}:=\mathrm {Rep}(Q, \mathscr {A})$ . By [7, Lemma 3.5], there exists a short exact sequence

$$\begin{align*}0\longrightarrow \bigoplus_{\alpha:r\rightarrow l} e_l^{\lambda}(X_r)\longrightarrow \bigoplus _{r\in Q_0} e_r^{\lambda}(X_r)\longrightarrow X\longrightarrow 0,\end{align*}$$

which is natural in X. By applying the functor ${\mathrm {Hom}}_{\mathscr {R}}(-,Y)$ to this sequence, we get the following long exact sequence:

$$ \begin{align*} 0 \longrightarrow & \ {\mathrm{Hom}}_{\mathscr{R}}(X, Y) \longrightarrow \bigoplus_{r\in Q_0}{\mathrm{Hom}}_{\mathscr{R}}( e_r^{\lambda}(X_r), Y)\longrightarrow \bigoplus_{\alpha:r\rightarrow l}{\mathrm{Hom}}_{\mathscr{R}}(e_l^{\lambda}(X_r), Y)\\ \longrightarrow & \ {\mathrm{Ext}}^1_{\mathscr{R}}(X, Y) \ \ \longrightarrow\bigoplus_{r\in Q_0}{\mathrm{Ext}}^1_{\mathscr{R}}( e_r^{\lambda}(X_r), Y) \ \longrightarrow \bigoplus_{\alpha:r\rightarrow l}{\mathrm{Ext}}^1_{\mathscr{R}}(e_l^{\lambda}(X_r), Y)\\ \longrightarrow & \ {\mathrm{Ext}}^2_{\mathscr{R}}(X, Y) \ \ \longrightarrow \cdots. \end{align*} $$

Now the result follows in view of the adjoint pair $(e_i^{\lambda }, e_i)$ and using the fact that the adjunction between $e_i^{\lambda }$ and $e_i$ extends to ${\mathrm {Ext}}^t$ , for all $t \geq i$ (see [25, Proposition 5.2]).

Now we can state and prove the main result of this part.

Theorem 8.2.2 Let $\mathscr {A}$ be an abelian category with enough projective objects, and let $Q=(Q_0, Q_1)$ be a finite and acyclic quiver. Let $\mathscr {T}$ be a support $\tau $ -tilting subcategory of $\mathscr {A}$ such that the induced torsion pair $({\mathrm {Fac}}(\mathscr {T}), \mathscr {T}^{\perp _0})$ is hereditary. Then

$$\begin{align*}\mathbb{T} ={\mathrm{add}}\lbrace e_i^{\rho}(T) \vert ~ i\in Q_0, T\in\mathscr{T}\rbrace\end{align*}$$

is a support $\tau $ -tilting subcategory of $\mathrm {Rep}(Q, \mathscr {A})$ .

Proof. We show the validity of conditions of Definition 3.1. For the first condition, let $i, j\in Q_0$ and $T\in \mathscr {T}$ . We show that

$$\begin{align*}{\mathrm{Ext}}^1_{\mathscr{R}}(e_i^{\rho}(T), {\mathrm{Fac}}(e_j^{\rho}(T))=0.\end{align*}$$

Set $X:=e_i^{\rho }(T)$ and pick $Y\in {\mathrm {Fac}}(e_j^{\rho }(T))$ . Then, for every $r\in Q_0$ , $X_r$ is the sum of some finite copies of T, maybe zero, and $Y_r$ is in ${\mathrm {Fac}}(T)$ . Moreover, for every $\alpha : r\rightarrow l\in Q_1$ , $Y_\alpha : Y_r\longrightarrow Y_l$ is an epimorphism. Hence, since by assumption ${\mathrm {Fac}}(\mathscr {T})$ is closed under subobjects, we deduce that ${\mathrm {Ker}} Y_\alpha \in {\mathrm {Fac}}(T)$ . This, in particular, implies that for every $T \in \mathscr {T}$ , the induced morphism

$$\begin{align*}{\mathrm{Hom}}_{\mathscr{A}}(T, Y_r) \longrightarrow {\mathrm{Hom}}_{\mathscr{A}}(T, Y_l)\end{align*}$$

is an epimorphism.

Hence, in the exact sequence

$$\begin{align*}\bigoplus_{r\in Q_0} {\mathrm{Hom}}_{\mathscr{A}}(X_r, Y_r) \stackrel{\varphi}\longrightarrow \bigoplus_{\alpha: r\rightarrow l} {\mathrm{Hom}}_{\mathscr{A}}(X_r, Y_l) \longrightarrow \\ {\mathrm{Ext}}^1_{\mathscr{R}}(X, Y)\longrightarrow \bigoplus_{r\in Q_0}{\mathrm{Ext}}^1_{\mathscr{A}}(X_r, Y_r),\end{align*}$$

of the above lemma, we deduce that $\varphi $ is an epimorphism. So, to show the result, it is enough to show that

$$\begin{align*}\bigoplus_{r\in Q_0}{\mathrm{Ext}}^1_{\mathscr{A}}(X_r, Y_r)=0.\end{align*}$$

This follows from the fact ${\mathrm {Ext}}^1_{\mathscr {A}}(\mathscr {T}, {\mathrm {Fac}}(\mathscr {T}))$ , because $\mathscr {T}$ is a support $\tau $ -tilting subcategory of $\mathscr {A}$ and the fact that

$$\begin{align*}\bigoplus_{r\in Q_0}{\mathrm{Ext}}^1_{\mathscr{A}}(X_r, Y_r)\subseteq{\mathrm{Ext}}^1_{\mathscr{A}}(\mathscr{T}, {\mathrm{Fac}}(\mathscr{T})).\end{align*}$$

Now we show the validity of the second condition. It is enough to show it only for the projective generators of $\mathrm {Rep}(Q, \mathscr {A})$ , i.e., for representations of the form $e_i^{\lambda }(P)$ , where P is a projective object in $\mathscr {A}$ . Let $\lbrace \rho _1, \ldots , \rho _k \rbrace $ be the set of all longest paths in Q starting from i. Since $\mathscr {T}$ is a support $\tau $ -tilting subcategory of $\mathscr {A}$ , for P there exists an exact sequence $P\stackrel {f}\longrightarrow T^0\stackrel {g}\longrightarrow T^1\longrightarrow 0$ , such that $T^0, T^1\in \mathscr {T}$ and f is a left $\mathscr {T}$ -approximation of P. Since ${\mathrm {Fac}}(\mathscr {T})$ is closed under subobjects, ${\mathrm {Ker}} g \in {\mathrm {Fac}}(\mathscr {T})$ and so the induced short exact sequence

$$\begin{align*}0\longrightarrow {\mathrm{Ker}} g\longrightarrow T^0\longrightarrow T^1\longrightarrow 0\end{align*}$$

splits and so ${\mathrm {Ker}} g\in \mathscr {T}$ . Take the exact sequence

$$\begin{align*}e^{\lambda}_{i}(P)\stackrel{\psi}\longrightarrow \bigoplus_{q=1}^ k e^\rho_{t(\rho_q)}(T^0)\longrightarrow \bigoplus_{q=1}^k e^\rho_{t(\rho_q)}(T^1)\oplus \bigoplus_{\alpha \in \mathfrak{I} }e^\rho_{s(\alpha)}({\mathrm{Ker}} g)\longrightarrow 0,\end{align*}$$

where $\mathfrak {I}\subset Q_1$ is the set of arrows $\alpha $ of Q such that $\alpha $ is not part of any path in the set $\lbrace \rho _1, \ldots , \rho _k \rbrace $ but there is a path in that set passing through $t(\alpha )$ . An easy verification shows that this is the desired sequence for $e^{\lambda }_i(P)$ . In particular, $\psi $ is a left $\mathbb {T}$ -approximation of $e^{\lambda }_{i}(P)$ .

To complete the proof, we have to show that $\mathbb {T}$ is a contravariantly finite subcategory of $\mathrm { Rep}(Q, \mathscr {A})$ . Let $X \in \mathrm {Rep}(Q, \mathscr {A})$ . For each i, consider a right $\mathscr {T}$ -approximation $\pi ^i: T^i \longrightarrow Re_i^\rho (X)$ , where $Re_i^\rho $ is the right adjoint of $e_i^\rho $ . Following the same argument as in [7, Proposition 3.9], one can show that

$$\begin{align*}\bigoplus_{i\in Q_0} e_i^\rho(T^i) \longrightarrow X\end{align*}$$

is a right $\mathbb {T}$ -approximation of X.

The examples below provide situations where the torsion pair $({\mathrm {Fac}}(\mathscr {T}), \mathscr {T}^{\perp _0})$ induced by the support $\tau $ -tilting subcategory $\mathscr {T}$ of $\mathscr {A}$ is hereditary.

Example 8.2.3 Let $\Lambda $ be an Artin algebra, and let S be a simple injective object in ${\mathrm {{mod\mbox {-}}}}\Lambda $ . Then ${\mathrm {add}}(S)$ is a support $\tau $ -tilting subcategory such that ${\mathrm {Fac}}(S)={\mathrm {add}}(S)$ is closed under subobjects.

Example 8.2.4 Let A be a finite-dimensional algebra, let e be an idempotent of A, and let $B=A/AeA$ . Let $Q=(Q_0, Q_1)$ be a finite and acyclic quiver. By [7, Proposition 3.9], we have $\mathscr {T}={\mathrm {add}} (\lbrace e_i^\rho (B) \vert i\in Q_0\rbrace )$ is a tilting subcategory in $\mathrm {Rep}(Q, {\mathrm {{mod\mbox {-}}}} B)$ . On the other hand, it is obvious that ${\mathrm {add}}(B)$ is a support $\tau $ -tilting subcategory of ${\mathrm {{mod\mbox {-}}}} A$ and ${\mathrm {Fac}}({\mathrm {add}}(B))={\mathrm {{mod\mbox {-}}}} B$ is closed under subobjects. So, by Theorem 8.2.2, $\mathscr {T}$ is a support $\tau $ -tilting of $\mathrm {Rep}(Q, {\mathrm {{mod\mbox {-}}}} A)$ .

We end this subsection with the following example which is also an application of Theorem 3.5.

Example 8.2.5 Let Q be a finite and acyclic quiver, and let $Q'$ be a full subquiver of Q. Then it is easy to see that $\mathrm {Rep}(Q', \mathscr {A})$ is a wide and functorially finite torsion class of $\mathrm {Rep}(Q, A)$ . By [7, Proposition 3.9], $\mathscr {T}={\mathrm {add}}(\lbrace e_i^\rho (P)\ \vert i\in Q^{\prime }_0, P\in {\mathrm {Prj}}(\mathscr {A})\rbrace )$ is a tilting subcategory of $\mathrm { Rep}(Q', \mathscr {A})$ . So, by Theorem 3.5, $\mathscr {T}$ is a support $\tau $ -tilting subcategory of $\mathrm {Rep}(Q, \mathscr {A})$ .

8.3 (Co)silting objects in $\mathrm {Rep}(Q, \mathscr {A})$

Let R be an associative ring with unity, and let Q be a finite and acyclic quiver. In this subsection, we construct silting (resp. cosilting) objects in the category of representations of Q in $\mathrm{Mod}\mbox {-} R$ , $\mathrm {Rep}(Q, \mathrm{Mod}\mbox {-}R)$ , from silting (resp. cosilting) modules in $\mathrm{Mod}\mbox {-} R$ . Note that $\mathrm {Rep}(Q, \mathrm{Mod}\mbox {-}R)$ is equivalent to the $\mathrm{Mod}\mbox {-}RQ$ , where $RQ$ denotes the path algebra of Q over R. So, by a silting (resp. cosilting) object in $\mathrm {Rep}(Q, \mathrm{Mod}\mbox {-}R)$ , we mean a silting (resp. cosilting) module in $\mathrm{Mod}\mbox {-}RQ$ .

Theorem 8.3.1 Let $Q=(Q_0, Q_1)$ be a finite and acyclic quiver.

1. (i) Let S be a silting module in $\mathrm{Mod}\mbox {-}R$ . Let $i\in Q_0$ be an arbitrary vertex of Q. Then $e_i^{\lambda }(S)$ is a silting object in $\mathrm {Rep}(Q, \mathrm{Mod}\mbox {-}R)$ .

2. (ii) Let C be a cosilting module in $\mathrm{Mod}\mbox {-}R$ . Let $i\in Q_0$ be an arbitrary vertex of Q. Then $e_i^\rho (C)$ is a cosilting object in $\mathrm {Rep}(Q, \mathrm{Mod}\mbox {-}R)$ .

Proof. $(i)$ Let S be a silting R-module. By definition, there exists a projective presentation

$$\begin{align*}P_1\stackrel{\sigma} \longrightarrow P_0\longrightarrow S\longrightarrow 0\end{align*}$$

of S such that $\mathscr {D}_\sigma ={\mathrm {Gen}}(S)$ . By applying the exact functor $e_i^{\lambda }$ on the projective presentation of S and using the fact that $e_i^{\lambda }$ is an exact functor that preserves projectives, we get the projective presentation

$$\begin{align*}e_i^{\lambda}(P_1)\stackrel{e_i^{\lambda}(\sigma)}\longrightarrow e_i^{\lambda}(P_0)\longrightarrow e_i^{\lambda}(S)\longrightarrow 0\end{align*}$$

of $e_i^{\lambda }(S)$ . To complete the proof, we show that $\mathscr {D}_{e_i^{\lambda }(\sigma )}={\mathrm {Gen}}(e_i^{\lambda }(S))$ .

Let $X \in \mathscr {D}_{e_i^{\lambda }(\sigma )}$ . So there exists an epimorphism

$$\begin{align*}{\mathrm{Hom}}_{\mathscr{R}}(e_i^{\lambda}(P_0), X)\longrightarrow {\mathrm{Hom}}_{\mathscr{R}}(e_i^{\lambda}(P_1), X)\longrightarrow 0,\end{align*}$$

where $\mathscr {R}$ means $\mathrm {Rep}(Q, \mathrm{Mod}\mbox {-}R)$ . The adjoint pair $(e_i^{\lambda }, e_i)$ induces the epimorphism

$$\begin{align*}{\mathrm{Hom}}_R(P_0, X_i)\longrightarrow {\mathrm{Hom}}_R(P_1, X_i)\longrightarrow 0.\end{align*}$$

This, in turn, implies that $X_i \in \mathscr {D}_\sigma ={\mathrm {Gen}}(S)$ . Therefore, $e_i^{\lambda }(X_i) \in e_i^{\lambda }({\mathrm {Gen}}(S))$ . On the other hand, by [7, Lemma 3.5], there exists an epimorphism $\bigoplus _{i\in Q_0} e_i^{\lambda }(X_i)\longrightarrow X\longrightarrow 0$ which shows that $X\in e_i^{\lambda }({\mathrm {Gen}}(S))$ . But it follows directly from the definition of $e_i^{\lambda }$ that $e_i^{\lambda }({\mathrm {Gen}}(S)) = {\mathrm {Gen}}(e_i^{\lambda }(S))$ . Thus, $\mathscr {D}_{e_i^{\lambda }(\sigma )}\subseteq {\mathrm {Gen}}(e_i^{\lambda }(S))$ .

To see the reverse inclusion, let $X\in e_i^{\lambda }({\mathrm {Gen}}(S))$ . So $X=e_i^{\lambda }(U)$ such that $U\in {\mathrm {Gen}}(S)$ . Since ${\mathrm {Gen}}(S)=\mathscr {D}_\sigma $ , we have $U\in \mathscr {D}_\sigma $ . Therefore, there exists an epimorphism

$$\begin{align*}{\mathrm{Hom}}_R(P_0, U)\longrightarrow {\mathrm{Hom}}_R(P_1, U)\longrightarrow 0.\end{align*}$$

By the using of adjoint properties of adjoint pair $(e_i^{\lambda }, e_i)$ , we have an epimorphism

$$\begin{align*}{\mathrm{Hom}}_{\mathscr{R}}(e_i^{\lambda}(P_0), X)\longrightarrow {\mathrm{Hom}}_{\mathscr{R}}(e_i^{\lambda}(P_1), X)\longrightarrow 0,\end{align*}$$

which shows that $X\in \mathscr {D}_{e_i^{\lambda }(\sigma )}$ . So we show that $e_i^{\lambda }({\mathrm {Gen}}(S))= {\mathrm {Gen}}(e_i^{\lambda }(S)) \subseteq \mathscr {D}_{e_i^{\lambda }(\sigma )}$ . Hence, the proof is complete.

$(ii)$ The proof is just dual of the proof of part $(i)$ , so we skip the proof.

The following result provides a partial converse to the above theorem. Recall that a vertex $i\in Q_0$ is called a source (resp. a sink) of Q if there are no arrows $\alpha \in Q_1$ such that $t(\alpha )=i$ (resp. $s(\alpha )=i$ ).

Theorem 8.3.2 Let $Q=(Q_0, Q_1)$ be a finite and acyclic quiver.

1. (i) Let X be a silting object in $\mathrm {Rep}(Q, \mathrm{Mod}\mbox {-}R)$ . Then $e_i(X)$ is a silting module in $\mathrm{Mod}\mbox {-}R$ , provided $i\in Q_0$ is a source of Q.

2. (ii) Let Y be a cosilting object in $\mathrm {Rep}(Q, \mathrm{Mod}\mbox {-}R)$ . Then $e_i(Y)$ is a cosilting module in $\mathrm{Mod}\mbox {-}R$ , provided $i\in Q_0$ is a sink of Q.

Proof. $(i)$ Since X is a silting object in $\mathrm {Rep}(Q, \mathrm{Mod}\mbox {-}R)$ , there exists a projective presentation

$$\begin{align*}P^1\stackrel{\sigma}\longrightarrow P^0\longrightarrow X\longrightarrow 0\end{align*}$$

of X such that $\mathscr {D}_\sigma ={\mathrm {Gen}}(X)$ . By applying the exact functor $e_i$ on this exact sequence, we get the exact sequence

$$\begin{align*}P^1_i\stackrel{\sigma_i}\longrightarrow P^0_i\longrightarrow X_i\longrightarrow 0,\end{align*}$$

which is a projective presentation of $e_i(X)=X_i$ .

To complete the proof, we have to show that $\mathscr {D}_{\sigma _i}={\mathrm {Gen}}(X_i)$ . First, let $M\in \mathscr {D}_{\sigma _i}$ . Then there exists an epimorphism

$$\begin{align*}{\mathrm{Hom}}_R( P^0_i, M)\longrightarrow {\mathrm{Hom}}_R( P^1_i, M)\longrightarrow 0.\end{align*}$$

By using the adjoint pair $(e_i, e_i^\rho )$ , we get the epimorphism

$$\begin{align*}{\mathrm{Hom}}_{\mathscr{R}}(P^0, e_i^\rho(M))\longrightarrow {\mathrm{Hom}}_{\mathscr{R}}(P^1, e_i^\rho(M)\longrightarrow 0,\end{align*}$$

where $\mathscr {R}$ means $\mathrm {Rep}(Q, \mathrm{Mod}\mbox {-}R)$ . So $e_i^\rho (M)\in \mathscr {D}_\sigma ={\mathrm {Gen}}(X)$ . Therefore, $M=e_i e_i^\rho (M)\in e_i({\mathrm {Gen}}(X))$ . But $e_i({\mathrm {Gen}}(X))={\mathrm {Gen}}(e_i(X))={\mathrm {Gen}}(X_i)$ . Hence, $\mathscr {D}_{\sigma _i}\subseteq {\mathrm {Gen}}(X_i)$ .

For the reverse inclusion, let $Y_i\in {\mathrm {Gen}}(X_i)$ . We define $Y\in \mathrm {Rep}(Q, \mathrm{Mod}\mbox {-}R)$ by setting $Y_i$ at source vertex i, and $0$ elsewhere. Since i is a source, it follows easily that $Y \in {\mathrm {Gen}}(X)$ . Therefore, $Y \in \mathscr {D}_\sigma $ , which implies that $Y_i \in \mathscr {D}_{\sigma _i}$ . Hence, we have the equality $\mathscr {D}_{\sigma _i}={\mathrm {Gen}}(X_i)$ and the proof is complete.

$(ii)$ The proof is just dual of the proof of part $(i)$ . So we skip the proof.