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.

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 [Reference Dickson19].

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.

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.

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.

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.

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

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.

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.

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.

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 [Reference Buan and Zhou16, 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 [Reference Auslander and Smalø6] 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 [Reference Buan and Zhou16, Theorem 0.4].

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 [Reference Beligiannis8, Corollaries 3.9 and 3.10]). Now the result follows by the part $(ii)$ of the proof of [Reference Iyama, Jørgensen and Yang26, Proposition 5.3].

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 $\mathscr {A}$ .

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 ${\mathrm {Coker}}\varphi \in \mathscr {T}^{\perp _0}$ .

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.

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.

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, $T \in {\mathrm {Fac}}(\mathscr {C}\cap \mathscr {T})$ .

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.

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 [Reference Zhang36], 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, $\mathscr {T}={\mathrm {add}}(T)$ .

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