2.1. Modules

A $\mathcal {C}$-module $M$ over $k$ means a covariant $k$-functor $M \colon \mathcal {C} \to \operatorname {Mod} k$. A morphism $f \colon M \to N$ of $\mathcal {C}$-modules means a natural transformation. In other words, it consists of a collection of maps $f_a \colon M(a) \to N(a)$ of $k$-modules for any $a \in \operatorname {Ob} \mathcal {C}$, such that $N(\alpha ) \circ f_a = f_b \circ M(\alpha )$ for any $\alpha \in \mathcal {C}(a,\, b)$.

Denote by $\operatorname {Mod} \mathcal {C}$ the category of $\mathcal {C}$-modules. It is well known that $\operatorname {Mod} \mathcal {C}$ is an abelian $k$-category. Given any $\mathcal {C}$-modules $M$ and $N$, we denote by $\operatorname {Hom}_\mathcal {C}(M,\,N)$ the set of morphisms of $\mathcal {C}$-modules. We have the faithful exact contravariant functor $D \colon \operatorname {Mod} \mathcal {C} \to \operatorname {Mod} \mathcal {C}^{\mathrm {op}}$ induced by $\operatorname {Hom}_k(-,k) \colon \operatorname {Mod} k \to \operatorname {Mod} k$.

We mention the following fact; see [Reference Gabriel and Roiter6, Section 3.7]. It implies that $\mathcal {C}(a,\,-)$ is projective and $D \mathcal {C}(-,\,a)$ is injective for any $a \in \operatorname {Ob} \mathcal {C}$.

Given a collection $\mathcal {A}$ of $\mathcal {C}$-modules, denote by $\operatorname {add} \mathcal {A}$ the full subcategory of $\operatorname {Mod} \mathcal {C}$ formed by direct summands of finite direct sums of objects in $\mathcal {A}$. Set $\operatorname {proj} \mathcal {C} = \operatorname {add} \left \{\mathcal {C}(a,-) \middle | a \in \operatorname {Ob} \mathcal {C}\right \}$ and $\operatorname {inj} \mathcal {C} = \operatorname {add} \left \{ D \mathcal {C}(-,a) \middle | a \in \operatorname {Ob} \mathcal {C}\right \}$. We observe that the restriction of $D$ gives a duality $D \colon \operatorname {proj} \mathcal {C} \to \operatorname {inj} \mathcal {C}^{\mathrm {op}}$.

A morphism $f \colon M \to N$ of $\mathcal {C}$-modules is called right minimal if any endomorphism $g \in \operatorname {End}_\mathcal {C}(M)$ with $f \circ g = f$ is an isomorphism. Dually, $f$ is called left minimal if any endomorphism $h \in \operatorname {End}_\mathcal {C}(N)$ with $h \circ f = f$ is an isomorphism.

Let $M$ be a $\mathcal {C}$-module. A right minimal epimorphism $P \to M$ with projective $P$ is called a projective cover of $M$. A left minimal monomorphism $M \to I$ with injective $I$ is called an injective envelope of $M$. It is well known that each $\mathcal {C}$-module admits an injective envelope; see [Reference Popescu17, Theorem 3.10.10]. Note that projective covers or injective envelopes may lie outside of $\operatorname {proj} \mathcal {C}$ or $\operatorname {inj} \mathcal {C}$.

We call $M$ finitely generated if there exists an epimorphism $f \colon P \to M$ with $P \in \operatorname {proj} \mathcal {C}$; call $M$ finitely presented if moreover $\operatorname {Ker} f$ is finitely generated. We denote by $\operatorname {fg} \mathcal {C}$ the category of finitely generated $\mathcal {C}$-modules, and by $\operatorname {fp} \mathcal {C}$ the one of finitely presented $\mathcal {C}$-modules.

Dually, we call $M$ finitely cogenerated if there exists a monomorphism $g \colon M \to I$ with $I \in \operatorname {inj} \mathcal {C}$; call $M$ finitely copresented if moreover $\operatorname {Cok} g$ is finitely cogenerated. We denote by $\operatorname {fcg} \mathcal {C}$ the category of finitely cogenerated $\mathcal {C}$-modules, and by $\operatorname {fcp} \mathcal {C}$ the one of finitely copresented $\mathcal {C}$-modules.

We observe that the restrictions of $D$ give dualities

\[ D \colon \operatorname{fg} \mathcal{C} \longrightarrow \operatorname{fcg} \mathcal{C}^{\mathrm{op}} \quad\text{and}\quad D \colon \operatorname{fp} \mathcal{C} \longrightarrow \operatorname{fcp} \mathcal{C}^{\mathrm{op}}. \]

It follows that each finitely generated $\mathcal {C}$-module $M$ admits a projective cover. Indeed, since $D M$ is finitely cogenerated, we can assume $f \colon D M \to I$ is an injective envelope in $\operatorname {Mod} \mathcal {C}^{\mathrm {op}}$ with $I \in \operatorname {inj} \mathcal {C}^{\mathrm {op}}$. Observe that both $D M(a)$ and $I(a)$ are finite dimensional for all $a \in \operatorname {Ob} \mathcal {C}$. Then $D f \colon D I \to M$ is a projective cover.

For each $\mathcal {C}$-module $M$, we denote by $M^{*}$ the $\mathcal {C}^{\mathrm {op}}$-module given by

\[ \begin{array}{r@{\;}l@{\;}c@{\;}l} \mathcal{C}^{\mathrm{op}} & \longrightarrow \operatorname{Mod} \mathcal{C} & \xrightarrow{\operatorname{Hom}_\mathcal{C}(M,-)} & \operatorname{Mod} k, \\ a & \longmapsto \mathcal{C}(a,-) & \longmapsto & \operatorname{Hom}_\mathcal{C}(M, \mathcal{C}(a,-)). \end{array} \]

Here, the left arrow is the Yoneda embedding. For each morphism $f \colon M \to N$ of $\mathcal {C}$-modules, we let $f^{*} \colon N^{*} \to M^{*}$ be the morphism of $\mathcal {C}^{\mathrm {op}}$-modules given by

\[ f^{*}_a := \operatorname{Hom}_\mathcal{C} (f, \mathcal{C}(a,-)) \colon N^{*} (a) \longrightarrow M^{*} (a), \]

for any $a \in \operatorname {Ob} \mathcal {C}$. Then we obtain a contravariant functor

\[ (-)^{*} \colon \operatorname{Mod} \mathcal{C} \longrightarrow \operatorname{Mod} \mathcal{C}^{\mathrm{op}}. \]

We mention that $(-)^{*}$ is left exact, since $\operatorname {Hom}_\mathcal {C}(-, \mathcal {C}(a,-)) \colon \operatorname {Mod} \mathcal {C} \to \operatorname {Mod} k$ is left exact for any $a \in \operatorname {Ob} \mathcal {C}$. We observe by Yoneda's lemma the duality

\[ (-)^{*} \colon \operatorname{proj} \mathcal{C} \longrightarrow \operatorname{proj} \mathcal{C}^{\mathrm{op}}. \]

2.2. Stable categories

Let $\mathcal {A}$ be an abelian $k$-category. Recall that a morphism $f \colon X \to Y$ in $\mathcal {A}$ is called projectively trivial if for any $Z \in \operatorname {Ob} \mathcal {A}$, the induced map $\operatorname {Ext}_\mathcal {A}^{1}(f, Z) \colon \operatorname {Ext}_\mathcal {A}^{1}(Y, Z) \to \operatorname {Ext}_\mathcal {A}^{1}(X, Z)$ is the zero map; see [Reference Lenzing and Zuazua14, Section 2]. We mention that $f$ is projectively trivial if and only if it factors through every epimorphism $f' \colon X' \to Y$. Dually, $f$ is called injectively trivial if for any $Z \in \operatorname {Ob} \mathcal {A}$, the induced map $\operatorname {Ext}_\mathcal {A}^{1}(Z, f) \colon \operatorname {Ext}_\mathcal {A}^{1}(Z, X) \to \operatorname {Ext}_\mathcal {A}^{1}(Z, Y)$ is the zero map. The morphism $f$ is injectively trivial if and only if it factors through every monomorphism $f' \colon X \to Y'$.

We mention the following observation; see [Reference Lenzing and Zuazua14, Lemma 2.2] and its dual.

Let $X,\, Y \in \operatorname {Ob} \mathcal {A}$. We denote by $\mathcal {P}(X,\, Y)$ the $k$-submodule of $\mathcal {A}(X,\, Y)$ formed by projectively trivial morphisms. Then $\mathcal {P}$ forms an ideal of $\mathcal {A}$. The projectively stable category $\underline {\mathcal {A}}$ attached to $\mathcal {A}$ is the factor category $\mathcal {A} / \mathcal {P}$. Given a morphism $f \in \mathcal {A}(X,\, Y)$, we denote by $\underline {f}$ its image in $\underline {\mathcal {A}}$.

Dually, we denote by $\mathcal {I}(X,\, Y)$ the $k$-submodule of $\mathcal {A}(X,\, Y)$ formed by injectively trivial morphisms. The injectively stable category $\overline {\mathcal {A}}$ attached to $\mathcal {A}$ is the factor category $\mathcal {A} / \mathcal {I}$. Given a morphism $f \in \mathcal {A}(X,\, Y)$, we denote by $\overline {f}$ its image in $\overline {\mathcal {A}}$.

We mention that $\operatorname {Ext}_\mathcal {A}^{1}(-,\,X)$ induces a functor $\operatorname {Ext}_\mathcal {A}^{1}(-,X) \colon \underline {\mathcal {A}} \to \operatorname {Mod} k$, and $\operatorname {Ext}_\mathcal {A}^{1}(X,\,-)$ induces a functor $\operatorname {Ext}_\mathcal {A}^{1}(X,-) \colon \overline {\mathcal {A}} \to \operatorname {Mod} k$, for any $X \in \operatorname {Ob} \mathcal {A}$.

Specially, we can consider the stable categories of $\operatorname {Mod} \mathcal {C}$. Since $\operatorname {Mod} \mathcal {C}$ contains enough projective modules, a morphism is projectively trivial if and only if it factors through some projective module by Lemma 2.3. Similarly, a morphism is injectively trivial if and only if it factors through some injective module.

We denote by $\operatorname {\underline {Mod}} \mathcal {C}$ the projectively stable category and by $\operatorname {\overline {Mod}} \mathcal {C}$ the injectively stable category. For any $\mathcal {C}$-modules $M$ and $N$, we denote $\operatorname {\underline {Hom}}_\mathcal {C}(M, N) = \operatorname {Hom}_\mathcal {C}(M, N) / \mathcal {P}(M, N)$ and $\operatorname {\overline {Hom}}_\mathcal {C}(M, N) = \operatorname {Hom}_\mathcal {C}(M, N) / \mathcal {I}(M, N)$.

2.3. Auslander–Reiten formula

Let $\delta \colon 0 \to X \to Y \to Z \to 0$ be an exact sequence of $\mathcal {C}$-modules. The covariant defect $\delta _*$ and the contravariant defect $\delta ^{*}$ are given by the following exact sequence of functors

\begin{align*} 0 \to \operatorname{Hom}_\mathcal{C}(Z, -) \to \operatorname{Hom}_\mathcal{C}(Y, -) \to \operatorname{Hom}_\mathcal{C}(X, -) \to \delta_* \to 0, \\ 0 \to \operatorname{Hom}_\mathcal{C}(-, X) \to \operatorname{Hom}_\mathcal{C}(-, Y) \to \operatorname{Hom}_\mathcal{C}(-, Z) \to \delta^{*} \to 0. \end{align*}

We mention that $\delta _*$ vanishes on injectively trivial morphisms, and $\delta ^{*}$ vanishes on projectively trivial morphisms. Therefore, they induce the functors

\[ \delta_* \colon \operatorname{\overline{Mod}} \mathcal{C} \longrightarrow \operatorname{Mod} k \quad \mbox{and} \quad \delta^{*} \colon \operatorname{\underline{Mod}} \mathcal{C} \longrightarrow \operatorname{Mod} k. \]

For each finitely presented $\mathcal {C}$-module $M$, we fix some exact sequence

\[ P_1(M) \overset{f_1}{\longrightarrow} P_0(M) \overset{f_0}{\longrightarrow} M \longrightarrow 0. \]

Here, $f_0$ and $P_0(M) \to \operatorname {Im} f_1$ are projective covers. We call $\operatorname {Cok} f_1^{*}$ the transpose of $M$, and denote by $\operatorname {Tr} M$; see [Reference Auslander and Reiten3, Section 2]. Moreover, we have a duality

\[ \operatorname{Tr} \colon \operatorname{\underline{fp}} \mathcal{C} \longrightarrow \operatorname{\underline{fp}} \mathcal{C}^{\mathrm{op}}, \]

Here, $\operatorname {\underline {fp}} \mathcal {C}$ is the full subcategory of $\operatorname {\underline {Mod}} \mathcal {C}$ formed by finitely presented $\mathcal {C}$-modules.

We mention that if $M$ is an indecomposable non-projective finitely presented $\mathcal {C}$-module, then $\operatorname {Tr} M$ is an indecomposable non-projective $\mathcal {C}^{\mathrm {op}}$-module, and $\operatorname {Tr} \operatorname {Tr} M \cong M$; see [Reference Auslander, Reiten and Smalø4, Proposition IV.1.7].

We have the Auslander's defect formula; see [Reference Krause12, Theorem].

As a consequence, the Auslander–Reiten formula follows; compare [Reference Auslander and Reiten3, Proposition 3.1] and [Reference Krause12, Corollaries].

Proposition 2.5 Let $N$ be a $\mathcal {C}$-module and $M$ be a finitely presented $\mathcal {C}$-module. Then there exist natural isomorphisms

\[ \operatorname{Ext}^{1}_\mathcal{C} ( N, D \operatorname{Tr} M ) \cong D \operatorname{\underline{Hom}}_\mathcal{C} (M, N) \]

and

\[ \operatorname{\overline{Hom}}_\mathcal{C} ( N, D \operatorname{Tr} M ) \cong D \operatorname{Ext}^{1}_\mathcal{C} (M, N). \]

The following result is useful in characterizing whether a morphism is projectively trivial or injectively trivial.

Proof. We only prove (1). It is sufficient to show the sufficiency. Proposition 2.5 implies the commutative diagram

Then $\operatorname {Ext}^{1}_\mathcal {C} (f, D \operatorname {Tr} M) = 0$ implies $D \operatorname {\underline {Hom}}_\mathcal {C} (M, \underline {f}) = 0$. Moreover, $\operatorname {\underline {Hom}}_\mathcal {C} (M, \underline {f}) = 0$ since $D$ is faithful. In particular, $\underline {f} = \operatorname {\underline {Hom}}_\mathcal {C} (M, \underline {f}) (\underline {{\mathbb 1}_M}) = 0$ in $\operatorname {\underline {Hom}}_\mathcal {C} (M,\, N)$. In other words, $f$ is projectively trivial in $\operatorname {Mod} \mathcal {C}$.

2.4. Almost split sequences

Recall that a morphism $f \colon M \to N$ is called right almost split if it is a non-retraction and each non-retraction $g \colon M' \to N$ factors through $f$. Dually, $f$ is called left almost split if it is a non-section and each non-section $g' \colon M \to N'$ factors through $f$. An exact sequence $0 \to X \xrightarrow {g} Y \xrightarrow {f} Z \to 0$ is called almost split if $f$ is right almost split and $g$ is left almost split.

We deduce the existence of almost split sequences; compare [Reference Auslander, Reiten and Smalø4, Theorem V.1.15].

Proof. We only prove (1). One can choose some non-zero $\theta \in D \operatorname {\underline {End}}_\mathcal {C}(M)$ vanishing on $\operatorname {rad} \operatorname {\underline {End}}_\mathcal {C}(M)$. Proposition 2.5 implies that $\operatorname {Ext}^{1}_\mathcal {C} ( M, D \operatorname {Tr} M ) \cong D \operatorname {\underline {End}}_\mathcal {C}(M)$. Assume the pre-image of $\theta$ under the isomorphism is the non-split exact sequence

\[ \delta \colon 0 \longrightarrow D \operatorname{Tr} M \longrightarrow E \overset{f}{\longrightarrow} M \longrightarrow 0. \]

Claim: $f$ is right almost split. Indeed, it is a non-retraction since $\delta$ is non-split. Assume $h \colon X \to M$ is a non-retraction. Consider the induced map

\[ D \operatorname{\underline{Hom}}_\mathcal{C} (M, \underline{h}) \colon D \operatorname{\underline{End}}_\mathcal{C}(M) \longrightarrow D \operatorname{\underline{Hom}}_\mathcal{C}(M,X). \]

Observe that $\operatorname {End}_\mathcal {C}(M)$ is local. Then $h \circ h' \in \operatorname {rad} \operatorname {End}_\mathcal {C}(M)$ for any $h' \colon M \to X$. Since $\theta$ vanishing on $\operatorname {rad} \operatorname {\underline {End}}_\mathcal {C}(M)$, it follows that

\[ D \operatorname{\underline{Hom}}_\mathcal{C} (M, \underline{h}) (\theta) (\underline{h'}) = (\theta \circ \operatorname{\underline{Hom}}_\mathcal{C}(M, \underline{h})) (\underline{h'}) = \theta (\underline{h} \circ \underline{h'}) = 0. \]

Hence, $D \operatorname {\underline {Hom}}_\mathcal {C} (M, \underline {h}) (\theta ) = 0$. Consider the commutative diagram

We have that $\operatorname {Ext}^{1}_\mathcal {C} ( h, D \operatorname {Tr} M ) (\delta ) = 0$. That is to say, the pullback of $\delta$ along $h$ splits. In other words, $h$ factors through $f$. It follows that $f$ is right almost split.

Observe that $D \operatorname {Tr} M$ is indecomposable, and hence $\operatorname {End}_\mathcal {C}(D \operatorname {Tr} M)$ is local. It follows that $\delta$ is an almost split sequence; see [Reference Auslander2, Proposition I.4.4].

3.1. Finitely presented modules

We begin with the following well-known fact.

Recall that a $\mathcal {C}$-module $M$ is called finite dimensional if there exist only finitely many $i \in \operatorname {Ob} \mathcal {C}$ with $M(i) \neq 0$ and these $M(i)$ are both finite dimensional. We denote by $\operatorname {fd} \mathcal {C}$ the category of finite-dimensional $\mathcal {C}$-modules.

We mention the following observation.

As a consequence, we obtain the following fact.

Observe that $\operatorname {fp} \mathcal {C}$ is a Hom-finite Krull–Schmidt abelian category; see Lemmas 2.2 and 3.1. We consider its stable categories $\underline {\operatorname {fp} \mathcal {C}}$ and $\overline {\operatorname {fp} \mathcal {C}}$. The first step is to study the projectively trivial morphisms and injectively trivial morphisms in $\operatorname {fp} \mathcal {C}$.

As a consequence of Lemma 3.4, we have that $\underline {\operatorname {fp} \mathcal {C}} = \operatorname {\underline {fp}} \mathcal {C}$ and $\operatorname {\overline {fd}} \mathcal {C}$ is a full subcategory of $\overline {\operatorname {fp} \mathcal {C}}$. Here, $\underline {\operatorname {fp} \mathcal {C}}$ is the projectively stable category of $\operatorname {fp} \mathcal {C}$, and $\operatorname {\underline {fp}} \mathcal {C}$ is the full subcategory of $\operatorname {\underline {Mod}} \mathcal {C}$ formed by finitely presented $\mathcal {C}$-modules.

Assume $f \colon M \to N$ is an injectively trivial morphism in $\operatorname {fp} \mathcal {C}$ such that $N \in \operatorname {fd} \mathcal {C}$ but $M \notin \operatorname {fd} \mathcal {C}$. We mention that $f$ needs not factor through some injective object in $\operatorname {fp} \mathcal {C}$. But Lemma 3.4(2) implies that $f$ is injectively trivial in $\operatorname {Mod} \mathcal {C}$. It follows from Lemma 2.3 that $f$ factors through some injective $\mathcal {C}$-module $I$. Here, $I$ needs not lie in $\operatorname {fp} \mathcal {C}$.

There may exist some injectively trivial morphisms $f \colon M \to N$ in $\operatorname {fp} \mathcal {C}$, such that $M,\, N \notin \operatorname {fd} \mathcal {C}$. In this case, we have no idea about the properties of these $f$, including whether $f$ factors through some injective object in $\operatorname {fp} \mathcal {C}$ or $\operatorname {Mod} \mathcal {C}$.

3.2. Generalized Auslander–Reiten duality

Recall from [Reference Jiao9, Section 2] that the generalized Auslander–Reiten duality on $\operatorname {fp} \mathcal {C}$ consists of a pair of full categories $(\operatorname {fp} \mathcal {C})_r$ and $(\operatorname {fp} \mathcal {C})_l$, and a pair of functors

\[ \tau \colon \underline{(\operatorname{fp} \mathcal{C})_r} \longrightarrow \overline{(\operatorname{fp} \mathcal{C})_l} \quad \mbox{and} \quad \tau \colon \overline{(\operatorname{fp} \mathcal{C})_l} \longrightarrow \underline{(\operatorname{fp} \mathcal{C})_r}. \]

Here, $\underline {(\operatorname {fp} \mathcal {C})_r}$ is the image of $(\operatorname {fp} \mathcal {C})_r$ under the factor functor $\operatorname {fp} \mathcal {C} \to \operatorname {\underline {fp}} \mathcal {C}$, and $\overline {(\operatorname {fp} \mathcal {C})_l}$ is the image of $(\operatorname {fp} \mathcal {C})_l$ under the factor functor $\operatorname {fp} \mathcal {C} \to \overline {\operatorname {fp} \mathcal {C}}$.

The subcategories $(\operatorname {fp} \mathcal {C})_r$ and $(\operatorname {fp} \mathcal {C})_l$ are given as follows

\[ (\operatorname{fp} \mathcal{C})_r = \left\{M \in \operatorname{fp} \mathcal{C} \middle| D \operatorname{Ext}^{1}_\mathcal{C}(M,-) \colon \overline{\operatorname{fp} \mathcal{C}} \to \operatorname{Mod} k \text{ is representable}\right\} \]

and

\[ (\operatorname{fp} \mathcal{C})_l = \left\{M \in \operatorname{fp} \mathcal{C} \middle| D \operatorname{Ext}^{1}_\mathcal{C}(-,M) \colon \underline{\operatorname{fp} \mathcal{C}} \to \operatorname{Mod} k \text{ is representable}\right\}. \]

We mention that $(\operatorname {fp} \mathcal {C})_r$ and $(\operatorname {fp} \mathcal {C})_l$ are both additive.

For any $M \in (\operatorname {fp} \mathcal {C})_l$ and $N \in \operatorname {fp} \mathcal {C}$, there exists a natural isomorphism

\[ \operatorname{\underline{Hom}}_\mathcal{C} (\tau^{-} M, N) \cong D \operatorname{Ext}^{1}_\mathcal{C} (N, M). \]

For any $N \in (\operatorname {fp} \mathcal {C})_r$ and $M \in \operatorname {fp} \mathcal {C}$, there exists a natural isomorphism

\[ \operatorname{\overline{Hom}}_\mathcal{C} (M, \tau N) \cong D \operatorname{Ext}^{1}_\mathcal{C} (N, M). \]

Moreover, the functors $\tau$ and $\tau ^{-}$ are mutually quasi-inverse equivalences. They are called the generalized Auslander–Reiten translation functors.

We mention the following characterizations for objects in $(\operatorname {fp} \mathcal {C})_r$ and $(\operatorname {fp} \mathcal {C})_l$; see [Reference Jiao9, Proposition 2.4].

Considering the above lemma, it is necessary to study the almost split sequences in $\operatorname {fp} \mathcal {C}$.

Proof. The sufficiency is immediate. For the necessary, we assume

\[ \delta \colon 0 \longrightarrow M \longrightarrow E \longrightarrow N \longrightarrow 0 \]

is an almost split sequence in $\operatorname {fp} \mathcal {C}$. We observe that $N$ is a finitely presented non-projective $\mathcal {C}$-module. Then there exists an almost split sequence

\[ \epsilon \colon 0 \longrightarrow D \operatorname{Tr} N \longrightarrow E' \longrightarrow N \longrightarrow 0 \]

in $\operatorname {Mod} \mathcal {C}$ by Proposition 2.7(1). We observe that $D \operatorname {Tr} N$ is finitely copresented. Proposition 3.3 implies that $D \operatorname {Tr} N$ is finitely presented, and hence $\epsilon$ lies in $\operatorname {fp} \mathcal {C}$. Then $\epsilon$ is an almost split sequence in $\operatorname {fp} \mathcal {C}$, and hence is isomorphic to $\delta$. It follows that $\delta$ is an almost split sequence in $\operatorname {Mod} \mathcal {C}$.

The following result gives the generalized Auslander–Reiten duality on $\operatorname {fp} \mathcal {C}$. It is analogous to [Reference Jiao9, Proposition 4.4].

Theorem 3.7 Let $\mathcal {C}$ be a Hom-finite category of type $A_\infty$ such that $\operatorname {Mod} \mathcal {C}$ is locally Noetherian. Then

\[ ( \operatorname{fp} \mathcal{C} )_r = \operatorname{fp} \mathcal{C} \]

and

\[ ( \operatorname{fp} \mathcal{C} )_l = \operatorname{add} \left( \operatorname{fd} \mathcal{C} \cup \left\{\mbox{injective objects in} \operatorname{fp} \mathcal{C}\right\} \right). \]

Moreover, the functors $D \operatorname {Tr}$ and $\operatorname {Tr} D$ induce the generalized Auslander–Reiten translation functors.

Proof. We observe that projective objects lie in $(\operatorname {fp} \mathcal {C})_r$. Let $M$ be an indecomposable non-projective object in $\operatorname {fp} \mathcal {C}$. Proposition 2.7(1) gives an almost split sequence

\[ \delta \colon 0 \longrightarrow D \operatorname{Tr} M \longrightarrow E \longrightarrow M \longrightarrow 0. \]

We observe by Proposition 3.3 that $D \operatorname {Tr} M$ is finitely presented. Then $\delta$ is an almost split sequence in $\operatorname {fp} \mathcal {C}$. Lemma 3.5(1) implies that $M$ lies in $(\operatorname {fp} \mathcal {C})_r$. Then the first equality follows.

Observe that injective objects lie in $(\operatorname {fp} \mathcal {C})_l$. Let $N$ be a finite-dimensional indecomposable non-injective object in $\operatorname {fp} \mathcal {C}$. We observe by Proposition 3.3 that $N$ is finitely copresented. Proposition 2.7(2) gives an almost split sequence starting at $N$, which lies in $\operatorname {fp} \mathcal {C}$. Lemma 3.5(2) implies that $N$ lies in $(\operatorname {fp} \mathcal {C})_l$.

On the other hand, let $N$ be an indecomposable non-injective object lying in $(\operatorname {fp} \mathcal {C})_l$. Lemma 3.5(2) implies that there exists an almost split sequence

\[ \delta \colon 0 \longrightarrow N \longrightarrow E \longrightarrow M \longrightarrow 0 \]

in $\operatorname {fp} \mathcal {C}$. Lemma 3.6 implies that $\delta$ is an almost split sequence in $\operatorname {Mod} \mathcal {C}$. Since $M$ is non-projective, we observe by Proposition 2.7(1) that $N \cong D \operatorname {Tr} M$ and is finitely copresented. Proposition 3.3 implies that $N$ is finite dimensional. Then the second equality follows.

We observe that $\operatorname {\overline {fd}} \mathcal {C}$ is a dense full subcategory of $\overline {(\operatorname {fp} \mathcal {C})_l}$, since any injective object becomes zero in $\overline {\operatorname {fp} \mathcal {C}}$. Then $D \operatorname {Tr}$ and $\operatorname {Tr} D$ induce functors

\[ \tau \colon \operatorname{\underline{fp}} \mathcal{C} \longrightarrow \overline{(\operatorname{fp} \mathcal{C})_l} \quad \mbox{and} \quad \tau^{-} \colon \overline{(\operatorname{fp} \mathcal{C})_l} \longrightarrow \operatorname{\underline{fp}} \mathcal{C}, \]

which are mutually quasi-inverse equivalences.

Proposition 2.5 gives natural isomorphisms

\[ \phi \colon \operatorname{\overline{Hom}}_\mathcal{C}(M, \tau N) \overset\cong\longrightarrow D \operatorname{Ext}^{1}_\mathcal{C}(N, M) \]

for any $M,\, N \in \operatorname {fp} \mathcal {C}$, and

\[ \psi \colon \operatorname{\underline{Hom}}_\mathcal{C}(\tau^{-} M, N) \overset\cong\longrightarrow D \operatorname{Ext}^{1}_\mathcal{C}(N, M) \]

for any $M \in (\operatorname {fd} \mathcal {C})_l$ and $N \in \operatorname {fp} \mathcal {C}$. Here, we mention that $\operatorname {\overline {Hom}}_\mathcal {C}(M,\, \tau N)$ is the Hom-set in $\operatorname {\overline {Mod}} \mathcal {C}$ by Proposition 2.6(2). Then the result follows.

4.1. Quivers

Let $Q = (Q_0,\, Q_1)$ be a quiver, where $Q_0$ is the set of vertices and $Q_1$ is the set of arrows. For any arrow $\alpha \colon a \to b$, we denote by $s(\alpha ) = a$ its source and by $t(\alpha ) = b$ its target.

Every vertex $a$ is associated with a trivial path (of length 0) $e_a$ with $s(e_a) = a = t(e_a)$. A path $p$ of length $l \geq 1$ is a sequence of arrows $\alpha _l \cdots \alpha _2 \alpha _1$ that $s(\alpha _{i+1}) = t(\alpha _i)$ for any $1 \leq i < l$. We set $s(p) = s(\alpha _1)$ and $t(p) = t(\alpha _l)$. For any path $p$, we have $e_{t(p)} p = p = p e_{s(p)}$. For any vertices $a$ and $b$, we denote by $Q(a,\, b)$ the set of paths $p$ with $s(p) = a$ and $t(p) = b$.

In this subsection, we assume $Q_0 = \mathbb {N}$ and $0< \left \lvert {Q(i,\,j)}\right \rvert < \infty$ and $\left \lvert {Q(j,\,i)}\right \rvert = 0$ for any $0 \leq i < j$. In particular, $Q$ has a subquiver of the form

View $Q$ as a small category, and let $\mathcal {C}$ be its $k$-linearization; see [Reference Gabriel and Roiter6, Section2.1]. Then $\mathcal {C}$ is a Hom-finite $k$-category of type $A_\infty$. The category of representations of $Q$ is isomorphic to $\operatorname {Mod} \mathcal {C}$. Denote $P_a = \mathcal {C}(a,\, -)$ and $I_a = D \mathcal {C}(-,\, a)$ for any $a \in \operatorname {Ob} \mathcal {C}$. It is well known that $\operatorname {Mod} \mathcal {C}$ is hereditary; see [Reference Gabriel and Roiter6, Section 8.2].

We mention the following fact.

Proof. Let $f \colon P \to P'$ be a morphism in $\operatorname {proj} \mathcal {C}$. Since $\operatorname {Mod} \mathcal {C}$ is hereditary, then $\operatorname {Im} f$ is projective. Therefore, the induced exact sequence

\[ 0 \longrightarrow \operatorname{Ker} f \longrightarrow P \longrightarrow \operatorname{Im} f \longrightarrow 0 \]

splits, and hence $\operatorname {Ker} f \in \operatorname {proj} \mathcal {C}$. Then the result follows from [Reference Auslander1, Proposition 2.1] and the horseshoe lemma.

We mention that $\operatorname {Mod} \mathcal {C}$ needs not be locally Noetherian in general, even though $\operatorname {fp} \mathcal {C}$ is abelian by Lemma 4.1. See the following example.

Example 4.2 Assume $Q$ is the following quiver.

We have the injection

\[ (f_1, f_2, \dots, f_i, \dots) \colon \bigoplus_{i \geq 1} P_i \longrightarrow P_0. \]

Here, $f_i$ is induced by $\alpha _i$. It follows that $\operatorname {Mod} \mathcal {C}$ is not locally Noetherian.

Recall that $Q$ is called uniformly interval finite if there exists some integer $N$ such that $\left \lvert {Q(a,\, b)}\right \rvert \leq N$ for any $a,\, b \in Q_0$; see [Reference Jiao11, Definition 2.3]. We have the following characterization.

Proof. We observe that $\left \lvert {Q(i, j)}\right \rvert \leq \left \lvert {Q(i', j')}\right \rvert$ for any $i' \leq i$ and $j' \geq j$, since $Q(i',\,i)$ and $Q(j,\,j')$ are non-empty. Then $Q$ is uniformly interval finite if and only if $\left \{{\left \lvert {Q(0, j)}\right \rvert | j \in Q_0}\right \}$ is bounded.

If $\left \{{\left \lvert {Q(0, j)}\right \rvert | j \in Q_0}\right \}$ is bounded, there exists some $n \in \mathbb {N}$ such that $\left \lvert {Q(0, n)}\right \rvert = \left \lvert {Q(0, j)}\right \rvert$ for any $j \geq n$. Then for any $i \geq 0$, we have that $\dim P_i (j)$ coincide for all $j \geq \max \left \{{i,\, n}\right \}$.

For any submodule $M$ of $P_i$, there exists some $m \in \mathbb {N}$ such that $\dim M(i) = \dim M(m)$ for any $i \geq m$. Consider the submodule $M'$ of $M$ such that

\[ M'(i) = \left\{ \begin{array}{ll} M(i), & \mbox{if } i \geq m, \\ 0, & \mbox{if } i < m. \end{array} \right. \]

We observe that $M' \cong P_m^{\oplus \dim M(m)}$ and $M / M'$ is finite dimensional. It follows that $M$ is finitely generated.

We observe that Noetherian property is closed under finite direct sums and factor modules. Then $\operatorname {Mod} \mathcal {C}$ is locally Noetherian.

If $\left \{{\left \lvert {Q(0, j)}\right \rvert | j \in Q_0}\right \}$ is unbounded, we consider $P_0$. There exists some $i_1>1$ such that $\left \lvert {Q(0, i_1)}\right \rvert > \left \lvert {Q(0, i_1-1)}\right \rvert \geq 1$. Moreover, there exists some $i_2$ such that $\left \lvert {Q(0, i_2)}\right \rvert > \left \lvert {Q(0, i_1)}\right \rvert$. Then at least two paths in $Q(0,\, i_2)$ are not the form $u p_1$ for any $u \in Q(i_1,\, i_2)$. We denote one of them by $p_2$.

Inductively, for any $j \geq 2$, there exists some $i_j$ such that $\left \lvert {Q(0, i_j)}\right \rvert > \left \lvert {Q(0, i_{j-1})}\right \rvert$. Then at least two paths in $Q(0,\, i_j)$ are not the form $u p_r$ for any $1 \leq r < j$ and $u \in Q(i_r,\, i_j)$. Denote one of them by $p_j$.

We then obtain the monomorphism

\[ (f_1, f_2, \dots, f_j, \dots) \colon \bigoplus_{j \geq 1} P_{i_j} \longrightarrow P_0, \]

where $f_j$ is induced by $p_j$. It follows that $\operatorname {Mod} \mathcal {C}$ is not locally Noetherian.

We study the generalized Auslander–Reiten duality on $\operatorname {fp} \mathcal {C}$ when $Q$ is uniformly interval finite.

For each $a \in Q_0$, we denote by $Q(a,\,\infty )$ the set of infinite sequences of arrows $\cdots \alpha _i\cdots \alpha _2\alpha _1$, such that $s(\alpha _1) = a$ and $s(\alpha _{i+1}) = t(\alpha _i)$ for any $i \geq 1$.

We introduce the representation $Y$ as follows. For each vertex $a$, let $Y(a) = \operatorname {Hom}_k (\bigoplus _{p \in Q(a,\infty )} k p, k)$. For each arrow $\alpha \colon a \to b$, let $Y(\alpha ) \colon Y(a) \to Y(b)$ be given by $Y(\alpha )(f)(q) = f(q\alpha )$, for any $f \in Y(a)$ and $q \in Q(b,\,\infty )$.

We mention that $Y$ is an indecomposable injective object in $\operatorname {fp} \mathcal {C}$. Moreover, we have the following characterization of indecomposable injective objects in $\operatorname {fp} \mathcal {C}$; see [Reference Jiao10, Theorem 3.11].

Lemma 4.4 If $Q$ is uniformly interval finite, then

\[ \left\{{Y}\right\} \cup \left\{{I_a | a \in Q_0}\right\} \]

is a complete set of indecomposable injective objects in $\operatorname {fp} \mathcal {C}$.

Then we can make the subcategory $(\operatorname {fp} \mathcal {C})_l$ more explicit.

Proposition 4.5 Assume $Q$ is uniformly interval finite. Then

\[ (\operatorname{fp} \mathcal{C})_l = \operatorname{add} \left( \operatorname{fd} \mathcal{C} \cup \left\{{Y}\right\} \right). \]

Example 4.6 Assume $Q$ is the following quiver.

We observe that $Q$ is uniformly interval finite. Then $\operatorname {Mod} \mathcal {C}$ is locally Noetherian by Proposition 4.3.

For any $j \geq i \geq 0$, we denote the indecomposable $\mathcal {C}$-module

We observe that

\[ \left\{X_{ij} \middle| j \geq i \geq 0\right\} \cup \left\{P_i \middle| i \geq 0\right\} \]

is a complete set of indecomposable $\mathcal {C}$-modules. Here, $P_0 \cong Y$ and $X_{0j} \cong I_j$ for any $j \geq 0$. It follows from Theorem 3.7 and Proposition 4.5 that

\[ (\operatorname{fp} \mathcal{C})_r = \operatorname{fp} \mathcal{C} = \operatorname{add} \left( \left\{X_{ij} \middle| j \geq i \geq 0\right\} \cup \left\{P_i \middle| i \geq 0\right\} \right) \]

and

\[ ( \operatorname{fp} \mathcal{C} )_l = \operatorname{add} \left( \left\{X_{ij} \middle| j \geq i \geq 0\right\} \cup \left\{{P_0}\right\} \right). \]

4.2. FI and VI

Assume the field $k$ is of characteristic 0. Recall that $\mathrm {FI}$ is the category whose objects are finite sets and morphisms are injections, and $\mathrm {VI}$ is the one whose objects are finite-dimensional vector spaces over a finite field $\mathbb {F}_q$ and morphisms are $\mathbb {F}_q$-linear injections.

Let $G$ be a finite group. Recall from [Reference Gan and Li7, Definition 1.1] that $\mathrm {FI}_G$ is the category whose objects are finite sets, and $\mathrm {FI}_G (S,\, T)$ is the set of pairs $(f,\, g)$ where $f \colon S \to T$ is an injection and $g \colon S \to G$ is an arbitrary map. The composition of $(f,\, g) \in \mathrm {FI}_G (S,\, T)$ and $(f',\, g') \in \mathrm {FI}_G (T,\, T')$ is given by

\[ (f', g') \circ (f, g) = (f' \circ f, g''), \]

where $g''(x) = g'(f(x)) \cdot g(x)$ for any $x \in S$. We observe that $\mathrm {FI}_G$ is isomorphic to $\mathrm {FI}$ if $G$ is the trivial group.

Given a skeleton of $\mathrm {FI}_G$ (or $\mathrm {VI}$), we will denote every object by its cardinal (or its $\mathbb {F}_q$-dimension) $n \in \mathbb {N}$. Let $\mathcal {C}$ be the $k$-linearization of the skeleton. Then $\mathcal {C}$ is a Hom-finite $k$-category of type $A_\infty$. The category of $\mathrm {FI}_G$-modules (or $\mathrm {VI}$-modules) over $k$ is isomorphic to $\operatorname {Mod} \mathcal {C}$.

The following result follows from [Reference Gan and Li8, Theorem 3.7].

We will study the generalized Auslander–Reiten duality on $\operatorname {fp} \mathcal {C}$.

The following characterization of injective objects in $\operatorname {fp} \mathcal {C}$ is counter-intuitive; see [Reference Gan and Li7, Theorems 1.5 and 1.7] and [Reference Nagpal16, Theorems 1.9 and 5.23].

The above fact implies that any projectively trivial morphism in $\operatorname {fp} \mathcal {C}$ is also an injectively trivial morphism in $\operatorname {fp} \mathcal {C}$. Therefore, $\overline {\operatorname {fp} \mathcal {C}}$ is a factor category of $\operatorname {\underline {fp}} \mathcal {C}$. But, Theorem 3.7 implies that $\operatorname {\underline {fp}} \mathcal {C}$ is equivalent to the full subcategory $\operatorname {\overline {fd}} \mathcal {C}$ of $\overline {\operatorname {fp} \mathcal {C}}$. It is somehow surprising.

We can make the subcategory $(\operatorname {fp} \mathcal {C})_l$ more explicit.

Proposition 4.9 Let $\mathcal {C}$ be the $k$-linearization of a skeleton of $\mathrm {FI}_G$ or $\mathrm {VI}$. Then

\[ (\operatorname{fp} \mathcal{C})_l = \operatorname{add} \left( \operatorname{fd} \mathcal{C} \cup \operatorname{proj} \mathcal{C} \right). \]