2. Preliminaries

Throughout this section, we fix a positive integer n. We first recall from [Reference Jasso30, definition 3.14] the definition of n-cluster tilting subcategories of abelian categories. Then we discuss the (Gabriel) functor rings T associated with n-cluster tilting subcategories $\mathcal{M}$ of Λ-mod and also the relationship between the functor category ${\rm Mod}(\mathcal{M})$ and the category TMod. We investigate the weak global dimension of (Gabriel) functor rings associated with n-cluster tilting subcategories of module categories, generalizing results from Fuller [Reference Fuller and Hullinger23, proposition 1.5] and Auslander [Reference Auslander4, proposition 4.2] (see theorem 2.3). By using this outcome, we prove some basic properties of the subcategory of filtered colimit ${\underrightarrow{\lim}}\mathcal{M}$ of n-cluster tilting subcategory $\mathcal{M}$ of Λ-mod.

Osamu Iyama introduced n-cluster tilting modules and higher Auslander algebras, known as n-Auslander algebras, in [Reference Iyama27, Reference Iyama28] to construct a higher Auslander correspondence (see [Reference Iyama27, theorem 0.2]), generalizing the classical correspondence between algebras of finite representation type and Auslander algebras as developed by Auslander in [Reference Auslander4, corollary 4.7].

Let $\mathcal{M}$ be a full subcategory of an abelian category $\mathcal{A}$. Given an object $A \in \mathcal{A}$, a morphism $\theta: M \rightarrow A$ with $M \in \mathcal{M}$ is called a right $\mathcal{M}$-approximation of A if any morphism from an object in $\mathcal{M}$ to A factors through θ. The subcategory $\mathcal{M}$ is called contravariantly finite if every object in $\mathcal{A}$ admits a right $\mathcal{M}$-approximation. Left $\mathcal{M}$-approximations and covariantly finite subcategories are defined dually. Also $\mathcal{M}$ is called functorially finite if it is both contravariantly and covariantly finite. We refer the reader to [Reference Auslander and Reiten5, Reference Auslander and Smalø6] for more details on functorially finite subcategories.

For a positive integer n, an abelian category $\mathcal{A}$ is called an n-Auslander category if $\mathcal{A}$ has enough projectives and satisfies ${\rm gl.dim}\mathcal{A} \leq n + 1 \leq {\rm dom.dim}\mathcal{A}$. Beligiannis in [Reference Beligiannis11] proved a categorified version of higher Auslander’s correspondence for arbitrary abelian categories. He proved that there is a bijection between equivalence classes of contravariantly finite subcategories $\mathcal{M}$ in abelian categories $\mathcal{A}$ with enough injectives that satisfy the following conditions:

1. (i) Any right $\mathcal{M}$-approximation is an epimorphism,

2. (ii) $\mathcal{M} = \mathcal{M}^{\bot_n}=\lbrace Y\in \mathcal{A}~|~ {\rm Ext}^i_{\mathcal{A}}(\mathcal{M},Y)=0 ~~~ {\rm for ~all}~ 0 \lt i \lt n \rbrace$.

and equivalence classes of n-Auslander categories (see [Reference Beligiannis11, theorem 8.23]). It is given by $\mathcal{M} \mapsto {\rm mod}(\mathcal{M})$, where ${\rm mod}(\mathcal{M})$ denotes the category of contravariant coherent functors from $\mathcal{M}$ to the category $\mathfrak{Ab}$ of all abelian groups. An additive functor $F: \mathcal{M} \rightarrow \mathfrak{Ab}$ is called coherent if there exists an exact sequence $\mathcal{M}(-, M_1) \rightarrow \mathcal{M}(-, M_0) \rightarrow F \rightarrow 0$, where $M_0, M_1 \in \mathcal{M}$ (see [Reference Auslander3]).

Before presenting the definition of the n-cluster tilting subcategories of arbitrary abelian categories, let us recall some notions.

Let $\mathcal{M}$ be a full subcategory of an abelian category $\mathcal{A}$. We recall that $\mathcal{M}$ is generating if any object in $\mathcal{A}$ is a quotient of an object in $\mathcal{M}$; that is, for every object $A \in \mathcal{A}$, there exists an exact sequence $M \rightarrow A \rightarrow 0$ with $M \in \mathcal{M}$. Cogenerating subcategories are defined dually. $\mathcal{M}$ is called a generating-cogenerating subcategory of $\mathcal{A}$ if it is both generating and cogenerating.

Definition 2.1. (See [Reference Jasso30] definition 3.14)

Let $\mathcal{A}$ be an abelian category and $\mathcal{M}$ be a generating-cogenerating full subcategory of $\mathcal{A}$. The subcategory $\mathcal{M}$ of $\mathcal{A}$ is called n-cluster tilting if it is functorially finite in $\mathcal{A}$ and $\mathcal{M}^{\bot_n}=\mathcal{M}={^{\bot_n}\mathcal{M}}$, where

\begin{align*} {^{\bot_n}\mathcal{M}}&=\lbrace X\in \mathcal{A}~|~ {\rm Ext}^i_{\mathcal{A}}(X,\mathcal{M})=0 ~~~ {\rm for ~all }~ 0 \lt i \lt n \rbrace\\ \mathcal{M}^{\bot_n}&=\lbrace Y\in \mathcal{A}~|~ {\rm Ext}^i_{\mathcal{A}}(\mathcal{M},Y)=0 ~~~ {\rm for ~all}~ 0 \lt i \lt n \rbrace. \end{align*}

Note that $\mathcal{A}$ itself is the unique 1-cluster tilting subcategory of $\mathcal{A}$.

Let $\mathcal{M}$ be an n-cluster tilting subcategory of Λ-mod. Assume that $\lbrace U_{\alpha}~|~ \alpha \in J \rbrace$ is a complete set of non-isomorphic indecomposable modules in $\mathcal{M}$. Set $U=\bigoplus_{\alpha \in J} U_{\alpha}$ and for each $\alpha \in J$ let $e_{\alpha}=\pi_{\alpha} \varepsilon_{\alpha}$, where $\pi_{\alpha}:U \rightarrow U_{\alpha}$ is the canonical projection and $\varepsilon_{\alpha}:U_{\alpha} \rightarrow U$ is the canonical injection. For each object X of Λ-Mod, we define as in [Reference Menini32, p. 242],

\begin{equation*} \widehat{\mathrm{Hom}}_{\Lambda}(U,X)=\lbrace f \in {\rm Hom}_{\Lambda}(U,X)~|~ e_{\alpha}f =0~ {\rm for ~almost~ all}~ \alpha \in J \rbrace. \end{equation*}

For X = U, we write $T:= \widehat{{\rm Hom}}_{\Lambda}(U, U)=\widehat{{\rm End}}_{\Lambda}(U)$. Following [Reference Fuller22, p. 40] (see also [Reference Yamagata43, p. 370]), $T=\widehat{{\rm End}}_{\Lambda}(U)$ is called functor ring (or Gabriel ring) of $\mathcal{M}$.

A ring S (not necessary with unit) has enough idempotents if there exists a family $\lbrace q_{\alpha}~|~\alpha \in I \rbrace$ of pairwise orthogonal idempotents of S such that $S=\bigoplus_{\alpha \in I}Sq_{\alpha}=\bigoplus_{\alpha \in I}q_{\alpha}S$ (see [Reference Fuller22, p. 39]). From [Reference Menini32, proposition 2.2(6)], we can see that $T=\bigoplus_{\alpha \in J}Te_{\alpha}=\bigoplus_{\alpha \in J}e_{\alpha}T$ is a ring with enough idempotents. Moreover, the assignment $X \mapsto \widehat{\mathrm{Hom}}_{\Lambda}(U,X)$ defines a covariant left exact functor $\widehat{\mathrm{Hom}}_{\Lambda}(U,-): \Lambda$-Mod $ \rightarrow T$Mod (see [Reference Menini32, proposition 2.2(5)]). The functor $\widehat{\mathrm{Hom}}_{\Lambda}(U,-): \Lambda$-Mod $\rightarrow T$Mod preserves coproducts and induces an additive equivalence between the full subcategory ${\rm Add}(\mathcal{M})$ of Λ-Mod and the full subcategory ${\rm Proj}(T)$ of TMod with the inverse equivalence $U \otimes_T -$ (see also [Reference Fuller22, pp. 40–41]). Hence, we can see that $\widehat{\mathrm{Hom}}_{\Lambda}(U,-): \mathcal{M} \rightarrow {\rm proj}(T)$ is an additive equivalence with the inverse equivalence $U \otimes_{T}-$. By [Reference Fazelpour and Nasr-Isfahani20, theorem 3.6], the functor ring of $\mathcal{M}$ uniquely determines up to Morita equivalence (see also the proof of [Reference Garcia and Simson25, theorem 3.3]).

It is shown in [Reference Gabriel24, proposition 2, p. 347] (see also [Reference Yamagata43, p. 370]) that there exists an additive equivalence ${\rm Mod}(\mathcal{M}) \rightarrow T{\rm Mod}$ which preserves and reflects finitely generated projective objects and exact sequences. Hence proposition 2.2 is an immediate consequence of [Reference Beligiannis11, theorem 8.23]. However, we provide an alternative proof for proposition 2.2 using the functor ring technique.

Proof. Let $\lbrace U_{\alpha}~|~ \alpha \in J \rbrace$ be a complete set of non-isomorphic indecomposable modules in $\mathcal{M}$ and $T=\widehat{{\rm End}}_{\Lambda}(U)$, where $U=\bigoplus_{\alpha \in J} U_{\alpha}$. Let $h: Q' \rightarrow Q$ be a morphism in ${\rm proj}(T)$. We show that ${\rm pd}({\rm Ker}(h)) \leq n-1$. Since the functor $\widehat{\mathrm{Hom}}_{\Lambda}(U,-): \Lambda$-Mod $ \rightarrow T$Mod is left exact and induces an additive equivalence between the full subcategory $\mathcal{M}$ of Λ-Mod and the full subcategory ${\rm proj}(T)$ of TMod, we have the following commutative diagram

where $f: M' \rightarrow M$ is a morphism in $\mathcal{M}$ and $\overline{f}=\widehat{\mathrm{Hom}}_{\Lambda}(U, f)$. Hence ${\rm Ker}(h) \cong \widehat{\mathrm{Hom}}_{\Lambda}(U, {\rm Ker}(f))$ as T-modules. Therefore, it is enough to show that ${\rm pd}(\widehat{\mathrm{Hom}}_{\Lambda}(U, {\rm Ker}(f))) \leq n-1$. If n = 1, then $\mathcal{M}=\Lambda$-mod. Since ${\rm Ker}(f)$ is a finitely generated left Λ-module, $\widehat{\mathrm{Hom}}_{\Lambda}(U, {\rm Ker}(f)) \in {\rm proj}(T)$ and so the result follows. Now let $n \geq 2$. Since ${\rm Ker}(f)$ is finitely generated, by [Reference Iyama29, proposition 2.3], we have an exact sequence

\begin{equation*}0 \rightarrow M_{n-1} \overset{\beta_{n-1}}{\rightarrow} \cdots \rightarrow M_0 \overset{\beta_{0}}{\rightarrow} {\rm Ker}(f) \rightarrow 0\end{equation*}

with terms in $\mathcal{M}$ such that the sequence

\begin{equation*} 0 \!\rightarrow {\rm Hom}_{\mathcal{M}}(-,M_{n-1})\! \overset{(-,\beta_{n-1})}{\rightarrow}\! \cdots \rightarrow\! {\rm Hom}_{\mathcal{M}}(-,M_0)\! \overset{(-,\beta_{0})}{\rightarrow}\! {\rm Hom}_{\mathcal{M}}(-,{\rm Ker}(f))\! \rightarrow\! 0 (\star) \end{equation*}

is exact. Now we show that the following sequence is exact

\begin{equation*}0 \rightarrow \widehat{\mathrm{Hom}}_{\Lambda}(U,M_{n-1}) \overset{\overline{\beta_{n-1}}}{\rightarrow} \cdots \rightarrow \widehat{\mathrm{Hom}}_{\Lambda}(U,M_0) \overset{\overline{\beta_{0}}}{\rightarrow} \widehat{\mathrm{Hom}}_{\Lambda}(U,{\rm Ker}(f)) \rightarrow 0, (\star \star)\end{equation*}

where $\overline{\beta_{i}}=\widehat{\mathrm{Hom}}_{\Lambda}(U,\beta_{i})$ for each $i \in \lbrace 0, \ldots, n-1 \rbrace$. Since the functor $\widehat{\mathrm{Hom}}_{\Lambda}(U,-): \Lambda$-Mod $ \rightarrow T$Mod is left exact, the sequence $(\star \star)$ is exact at $\widehat{\mathrm{Hom}}_{\Lambda}(U,M_{n-1})$ and $\widehat{\mathrm{Hom}}_{\Lambda}(U,M_{n-2})$. Let $l \in \lbrace -1, 0, \ldots, n-3\rbrace$. Set $M_{-1}= {\rm Ker}(f)$ and $\beta_{-1}=0$. It is sufficient to show that it is exact at $\widehat{\mathrm{Hom}}_{\Lambda}(U,M_l)$. Let $g \in \widehat{\mathrm{Hom}}_{\Lambda}(U,M_l)$ such that $g \beta_{l} = 0$. For each $\alpha \in J$ let $e_{\alpha}=\pi_{\alpha} \varepsilon_{\alpha}$, where $\pi_{\alpha}:U \rightarrow U_{\alpha}$ is the canonical projection and $\varepsilon_{\alpha}:U_{\alpha} \rightarrow U$ is the canonical injection. Since $g \in \widehat{\mathrm{Hom}}_{\Lambda}(U,M_l)$, there exists a finite subset A of J such that $e_{\alpha}g = 0$ for all $\alpha \in J \setminus A$. Consider the following diagram

where $M_{-2}=0$ and $\varepsilon: \bigoplus_{\alpha \in A}U_{\alpha} \rightarrow U$ and $\pi: U \rightarrow \bigoplus_{\alpha \in A}U_{\alpha}$ are the canonical injection and projection, respectively. Since $\bigoplus_{\alpha \in A}U_{\alpha} \in \mathcal{M}$, $\varepsilon g \beta_{l} = 0$, and the sequence $(\star)$ is exact, there exists a morphism $\gamma: \bigoplus_{\alpha \in A}U_{\alpha} \rightarrow M_{l+1}$ such that $\gamma \beta_{l+1} = \varepsilon g$ and so $\pi \gamma \beta_{l+1} = \pi \varepsilon g$. It is not difficult to see that $\pi \varepsilon g =g$ and $\pi \gamma \in \widehat{\mathrm{Hom}}_{\Lambda}(U,M_{l+1})$. This implies that the sequence $(\star \star)$ is exact and so the result follows.

Let R be a ring with enough idempotents. A unitary left R-module N is called flat if the functor $- \otimes_{R}N: {\rm Mod}R \rightarrow \mathfrak{Ab}$ is an exact functor, where $\mathfrak{Ab}$ is the category of abelian groups (see [Reference Garcia and Simson25, p. 115]). Note that a unitary left R-module N is flat if and only if N is a direct limit of finitely generated projective unitary left R-modules (see [Reference Wisbauer42, proposition 49.5]). Moreover, we denote by ${\rm Flat}(T)$ the full subcategory of $T{\rm Mod}$ consisting of flat unitary left T-modules.

Given a unitary left R-module M, the flat dimension of M which is denoted by ${\rm fd}(M)$ is less than or equal to m if there exists a finite flat resolution

\begin{equation*}0 \rightarrow F_m \rightarrow \cdots \rightarrow F_1 \rightarrow F_0 \rightarrow M \rightarrow 0.\end{equation*}

If no such finite resolution exists, then ${\rm fd}(M)=\infty$. The weak global dimension of R which is denoted by ${\rm w. gl. dim}(R)$ is defined as a supremum of the flat dimension of all unitary left R-modules (see [Reference Del Rio13, p. 210]).

Let $\mathcal{M}$ is an n-cluster tilting subcategory of Λ-mod. Beligiannis in [Reference Beligiannis11, theorem 8.23] proved that the projective dimension of any object in ${\rm mod}(\mathcal{M})$ is less or equal to n + 1. Using the functor ring technique, in theorem 2.3, we show that flat dimension of any object in ${\rm Mod}(\mathcal{M})$ is less or equal to n + 1.

Proof. For any $N \in T$Mod, we take an exact sequence

\begin{equation*}0 \rightarrow K \rightarrow Q_n \rightarrow \cdots \rightarrow Q_1 \overset{g}{\rightarrow} Q_0 \rightarrow N \rightarrow 0\end{equation*}

in TMod, where each Q_i is projective in TMod. By applying the technique used in the proof of [Reference Fazelpour and Nasr-Isfahani21, theorem 3.1(1)], we can see that ${\rm Ker}(g)\cong {\underrightarrow{\lim}} {\rm Ker}(g_i)$ as T-modules, where each g_i is a morphism in ${\rm proj}(T)$. By proposition 2.2, ${\rm pd}({\rm Ker}(g_i)) \leq n-1$. Let X be a unitary right T-module. Then ${\rm Tor}_n^{T}(X, {\rm Ker}(g_i))=0$. Since ${\rm Tor}_n^{T}(X, {\rm Ker}(g)) \cong {\underrightarrow{\lim}} {\rm Tor}_n^{T}(X, {\rm Ker}(g_i))$, we have ${\rm Tor}_n^{T}(X, {\rm Ker}(g)) = 0$. This yields K is a flat unitary left T-module. Therefore, ${\rm fd}(N) \leq n+1$ and so ${\rm w.gl.dim}(T) \leq n+1$.

As a consequence, we have the following results.

Λ is called of finite type if Λ-mod has an additive generator.

3. Main results

In this section, we prove the main results of the article. A full subcategory $\mathcal{M}$ of an abelian category $\mathcal{A}$ is called n-rigid, if ${\rm Ext}_{\mathcal{A}}^{k}(\mathcal{M}, \mathcal{M})=0$ for every $k \in \lbrace 1, \ldots, n-1\rbrace$ (see [Reference Beligiannis11, p. 443]). Clearly, any n-cluster tilting subcategory $\mathcal{M}$ of $\mathcal{A}$ is n-rigid.

In the following proposition, we list some basic properties of the subcategory ${\underrightarrow{\lim}}\mathcal{M}$ of Λ-Mod when $\mathcal{M}$ is an n-cluster tilting subcategory of Λ-mod. Note that proposition 3.1(a) and (b) are consequences of [Reference Ebrahimi and Nasr-Isfahani17, theorem 3.4(i)].

Proof. (a) We already know that every module is a factor module of a free module and every module is an essential submodule of an injective module. From $\Lambda , D(\Lambda) \in \mathcal{M}$ and the fact that injective modules are precisely direct summands of direct sums of copies of the $D(\Lambda)$, for each left Λ-module M, we obtain homomorphisms $P \rightarrow M$ and $M \rightarrow E$ which are epimorphism and monomorphism, respectively, where $P, E \in {\rm Add}(\mathcal{M})$. Hence ${\underrightarrow{\lim}}\mathcal{M}$ is a generating-cogenerating subcategory of Λ- ${\rm Mod}$.

(b) Since $\mathcal{M}$ is a covariantly finite subcategory of Λ-mod, by [Reference Krause31, proposition 3.11], ${\underrightarrow{\lim}}\mathcal{M}$ is a covariantly finite subcategory of Λ- ${\rm Mod}$. Also by [Reference El Bashir18, theorem 3.2], ${\underrightarrow{\lim}}\mathcal{M}$ is a contravariantly finite subcategory of Λ- ${\rm Mod}$ (see also [Reference Ebrahimi and Nasr-Isfahani17, theorem 3.4(i)]).

(c) By using [Reference Rotman37, proposition 7.21] and [Reference Enochs and Jenda19, proposition 3.1.16], for any $N \in {\rm Add}(\mathcal{M})$ and $M \in {\underrightarrow{\lim}}\mathcal{M}$, we have ${\rm Ext}_{\Lambda}^k(N,M) \cong \prod_{i \in A}{\underrightarrow{\lim}}_{j \in J} {\rm Ext}_{\Lambda}^k(N_i,M_j)$, where each $N_i, M_j \in \mathcal{M}$. Since $\mathcal{M}$ is n-rigid, ${\rm Ext}_{\Lambda}^{k}(N, M)=0$ for every $k \in \lbrace 1,\ldots,n-1 \rbrace$ and the result follows.

(d) Let $\lbrace U_{\alpha}~|~ \alpha \in J \rbrace$ be a complete set of representative of the isomorphic classes of indecomposable modules in $\mathcal{M}$ and $T=\widehat{{\rm End}}_{\Lambda}(U)$, where $U=\bigoplus _{\alpha \in J} U_{\alpha}$. Assume that X is a left Λ-module such that ${\rm Ext}_{\Lambda}^{k}(\mathcal{M}, X)=0$ for all $k \in \lbrace 1, \ldots,n-1 \rbrace$. We show that $X \in {\underrightarrow{\lim}}\mathcal{M}$. Consider the minimal injective resolution of X, namely

\begin{equation*}0 \rightarrow X \rightarrow E_0 \rightarrow E_1 \rightarrow \cdots \rightarrow E_n.\end{equation*}

Applying the functor $\widehat{\mathrm{Hom}}_{\Lambda}(U,-)$ and using the fact that ${\rm Ext}_{\Lambda}^{k}(\mathcal{M}, X)=0$ for all ${k \in \lbrace 1, \ldots, n-1 \rbrace}$, we obtain an exact sequence

\begin{equation*} 0 \rightarrow \widehat{\mathrm{Hom}}_{\Lambda}(U,X) \rightarrow \widehat{\mathrm{Hom}}_{\Lambda}(U,E_0) \rightarrow \widehat{\mathrm{Hom}}_{\Lambda}(U,E_1) \rightarrow \cdots \rightarrow \widehat{\mathrm{Hom}}_{\Lambda}(U,E_n). \end{equation*}

Since each $E_l \in {\rm Add}(\mathcal{M})$, each $\widehat{\mathrm{Hom}}_{\Lambda}(U,E_l) \in {\rm Proj}(T)$ and so by theorem 2.3, we can see that $\widehat{\mathrm{Hom}}_{\Lambda}(U,X) \in {\rm Flat}(T)$. Hence $U \otimes_T \widehat{\mathrm{Hom}}_{\Lambda}(U,X) \in {\underrightarrow{\lim}}\mathcal{M}$. On the other hand, by using the fact that $\Lambda \in \mathcal{M}$, it is not difficult to see that $U \otimes_T \widehat{\mathrm{Hom}}_{\Lambda}(U,X) \cong X$. Therefore, $X \in {\underrightarrow{\lim}}\mathcal{M}$ and the result follows.

As a consequence, we have the following result.

Let $\mathcal{M}$ be an n-cluster tilting subcategory of Λ-mod with $n \geq 2$. Then by the proof of proposition 3.1(a), ${\rm Add}(\mathcal{M})$ is a generating-cogenerating subcategory of Λ-Mod. By [Reference Angeleri-Hugel1, p. 651], ${\rm Add}(\mathcal{M})$ is a contravariantly finite subcategory of Λ-Mod. Also, by proposition 3.1(c), ${\rm Add}(\mathcal{M})$ is an n-rigid subcategory of Λ-Mod. Therefore, it is natural to ask when ${\rm Add}(\mathcal{M})$ is a maximal n-rigid subcategory of Λ-Mod. We present the following proposition based on [Reference Ebrahimi and Nasr-Isfahani16, theorem 4.12] and [Reference Ebrahimi and Nasr-Isfahani17, proposition 4.4].

Let $\mathcal{M}$ be an additive full subcategory of Λ-mod which is closed under direct summands. A sequence $0 \rightarrow A \overset{f}{\rightarrow} B \overset{g}{\rightarrow} C \rightarrow 0$ in ${\underrightarrow{\lim}}\mathcal{M}$ (i.e., a pair of maps f and g with fg = 0) is said to be pure-exact if the sequence

\begin{equation*}0 \rightarrow {\rm Hom}_{\Lambda}(M, A) \overset{{\rm Hom}_{\Lambda}(M, f)}{\rightarrow} {\rm Hom}_{\Lambda}(M, B) \overset{{\rm Hom}_{\Lambda}(M, g)}{\rightarrow} {\rm Hom}_{\Lambda}(M, C) \rightarrow 0\end{equation*}

is exact for each $M \in \mathcal{M}$ (see [Reference Crawley-Boevey12, p. 1653]).

A homomorphism $g: M \rightarrow X$ is called right minimal if any endomorphism $h: M \rightarrow M$ such that hg = g is an isomorphism. Let $\mathcal{X}$ be a full subcategory of Λ-Mod. A left Λ-module N has an $\mathcal{X}$-cover if there is a right minimal right $\mathcal{X}$-approximation $X \rightarrow N$ (see [Reference Bazzoni and Positselski9, Section 4]).

Remark 3.5. Let $\mathcal{M}$ be an n-cluster tilting subcategory of Λ-mod with $n \geq 2$. Motivated by proposition 3.3 (see also [Reference Ebrahimi and Nasr-Isfahani17, proposition 4.4]), it is natural to ask whether ${\rm Add}(\mathcal{M})$ is closed under direct limits. This question is equivalent to Iyama’s question. Let $\lbrace U_{\alpha}~|~ \alpha \in J \rbrace$ be a complete set of non-isomorphic indecomposable modules in $\mathcal{M}$ and set $U=\bigoplus_{\alpha \in J} U_{\alpha}$. Let $0 \rightarrow N' \rightarrow N \rightarrow N'' \rightarrow 0$ be a pure exact sequence, where $N \in {\rm Add}(U)$. By [Reference Krause31, lemma 3.10], $N' \in {\underrightarrow{\lim}}\mathcal{M}$. Hence by proposition 3.1, ${\rm Ext}_{\Lambda}^{1}(U, N')=0$. This implies that the sequence of abelian groups

\begin{equation*}0 \rightarrow {\rm Hom}_{\Lambda}(U, N') \rightarrow {\rm Hom}_{\Lambda}(U, N) \rightarrow {\rm Hom}_{\Lambda}(U, N'') \rightarrow 0\end{equation*}

is exact. Therefore by using [Reference Bazzoni and Positselski9, corollary 9.6], Iyama’s question is equivalent to the following question.

• • Let $\mathcal{M}$ be an n-cluster tilting subcategory of Λ-mod with $n \geq 2$. Does any object of ${\underrightarrow{\lim}}\mathcal{M}$ have an ${\rm Add}(\mathcal{M})$-cover?

Let $\mathcal{M}$ be an additive full subcategory of Λ-mod which is closed under direct summands. Following [Reference Simson40, Reference Simson41], the category ${\underrightarrow{\lim}}\mathcal{M}$ is defined to be pure semisimple if any pure-exact sequence in ${\underrightarrow{\lim}}\mathcal{M}$ splits (see also [Reference Crawley-Boevey12, Section 3]).

In the following theorem, we give equivalence conditions for n-rigidity of filtered colimits of n-cluster tilting subcategories of module categories. Note that the equivalence between (b) and (c) and implication (a) implies (b) is already known by [Reference Ebrahimi and Nasr-Isfahani16, Reference Ebrahimi and Nasr-Isfahani17].

Proof. $(a) \Rightarrow (b)$ follows from proposition 3.3 and [Reference Ebrahimi and Nasr-Isfahani17, proposition 4.4].

$(b) \Leftrightarrow (c)$ follows from [Reference Ebrahimi and Nasr-Isfahani17, theorem 3.6].

$(c) \Rightarrow (d)$ is clear.

$(d) \Rightarrow (e)$. Let X be a module in ${\rm Add}(\mathcal{M})$ and N be a pure submodule of X. We show that N is a direct summand of X. Consider the pure exact sequence

\begin{equation*}0 \rightarrow N \rightarrow X \rightarrow X/N \rightarrow 0.\end{equation*}

$X \in {\underrightarrow{\lim}}\mathcal{M}$ and by [Reference Krause31, lemmas 3.9 and 3.10], N and $X/N$ are modules in ${\underrightarrow{\lim}}\mathcal{M}$. Since ${\rm Ext}_{\Lambda}^1({\underrightarrow{\lim}}\mathcal{M}, {\underrightarrow{\lim}}\mathcal{M})=0$, the exact sequence $0 \rightarrow N \rightarrow X \rightarrow X/N \rightarrow 0$ is split and so the result follows.

$(e) \Rightarrow (f)$. By remark 3.6, it is enough to show that every module in ${\underrightarrow{\lim}}\mathcal{M}$ is a direct sum of finitely generated modules. Let X be a module in ${\underrightarrow{\lim}}\mathcal{M}$. Then by [Reference Crawley-Boevey12, lemma 4.1], there exists a pure epimorphism $\varphi: \bigoplus_{\alpha \in A}M_{\alpha} {\rightarrow} X$ with each $M_{\alpha} \in \mathcal{M}$. This implies that ${\rm Ker}(\varphi)$ is a pure submodule of $\bigoplus_{\alpha \in A}M_{\alpha}$. Hence by assumption, X is a direct sum of finitely generated modules and so the result follows.

$(f) \Rightarrow (a)$ follows from [Reference Fazelpour and Nasr-Isfahani21, corollary 4.6].

A left Λ-module M is called Mittag-Leffler if the canonical map

\begin{equation*} (\prod_{i \in I} Q_i) \otimes_{\Lambda} M \rightarrow \prod_{i \in I} (Q_i \otimes_{\Lambda}M)\end{equation*}

is injective, where $\lbrace Q_i~ |~ i \in I\rbrace$ are arbitrary right Λ-modules (see [Reference Azumaya7, p. 19]). A direct system $(M_i, \varphi_{ij})_{i, j \in I}$ of left Λ-module is called factorizable if for every $i \in I$ there exists $j \geq i$ such that φ_ij factors through each φ_ik with $k \geq i$. Let $(M_i, \varphi_{ij})_{i, j \in I}$ be a direct system of finitely generated left Λ-modules and $M={\underrightarrow{\lim}} M_i$. Then by [Reference Raynaud and Gruson35, pp. 69–70, lemma 2.1.2 and propositions 2.1.4 and 2.1.5], M is Mittag-Leffler if and only if $(M_i, \varphi_{ij})_{i, j \in I}$ is factorizable. For artin algebras, it is known that Mittag-Leffler modules coincide with locally pure projective modules (see [Reference Angeleri-Hugel, Herbera and Trlifaj2, lemma 20]). Recall that a left Λ-module N is called locally pure projective if any pure epimorphism $g: Y \rightarrow N$ is locally split, that is, for each $x \in N$ there is a map $\varphi=\varphi_{x}: N \rightarrow Y$ such that $x = (x)\varphi g$.

Let $\mathcal{M}$ be a skeletally small additive full subcategory of Λ-mod which is closed under direct summands. Let $\lbrace M_i~|~i \in I\rbrace$ be a complete set of isomorphism classes of objects in $\mathcal{M}$ and put $M= \bigoplus_{i \in I}M_i$. We denote by $\mathscr{G}(M)$ the category of all left Λ-modules N which admit a locally split epimorphism $g: M^{(K)} \rightarrow N$ for some set K. It is clear by definition that ${\rm Add}(\mathcal{M}) \subseteq \mathscr{G}(M)$. Moreover, $\mathscr{G}(M)$ consists of all modules in ${\underrightarrow{\lim}}\mathcal{M}$ which are locally pure projective (see [Reference Angeleri-Hugel1, proposition 2.3]).

As a consequence of theorem 3.7 and [Reference Angeleri-Hugel1, theorems 3.4, 4.4, 5.1, and 5.2], we have the following result that shows the questions in remark 3.9 are equivalent to Iyama’s question.