Triangular matrix categories over quasi-hereditary categories

Abstract

In this paper, we prove that the lower triangular matrix category $\Lambda =\left [ \begin{smallmatrix} \mathcal{T}&0\\ M&\mathcal{U} \end{smallmatrix} \right ]$, where $\mathcal{T}$ and $\mathcal{U}$ are $\textrm{Hom}$-finite, Krull–Schmidt $K$-quasi-hereditary categories and $M$ is an $\mathcal{U}\otimes _K \mathcal{T}^{op}$-module that satisfies suitable conditions, is quasi-hereditary. This result generalizes the work of B. Zhu in his study on triangular matrix algebras over quasi-hereditary algebras. Moreover, we obtain a characterization of the category of the $_\Lambda \Delta$-filtered $\Lambda$-modules.

1. Introduction

The notion of quasi-hereditary algebra and highest weight category were introduced and studied by E. Cline, B. Parshall, and L. Scott [8, 24, 27]. Highest weight categories arise in the representation theory of Lie algebras and algebraic groups. For the setting of finite dimensional algebras, quasi-hereditary algebras were amply studied by V. Dlab and C. M. Ringel in [9, 11, 12, 25]. In addition, they introduced the set of standard modules ${}_\Lambda \Delta$ associated with an algebra $\Lambda .$ In [25], C. M. Ringel studied the homological properties of the category $\mathcal{F}({}_\Lambda \Delta )$ of ${}_\Lambda \Delta$ -filtered $\Lambda$ -modules and constructed the characteristic tilting module ${}_\Lambda T$ associated with $\mathcal{F}(_\Lambda \Delta ).$ In [29], B. Zhu studied the triangular matrix algebra $\Lambda =\left [ \begin{smallmatrix} T &0\\ M& U\end{smallmatrix} \right ]$ where $T$ and $U$ are quasi-hereditary algebras, and he proved that under suitable conditions on $M$ , $\Lambda$ is a quasi-hereditary algebra.

On the other hand, B. Mitchell developed the idea that additive categories can be thought as rings with several objects and he showed that a substantial amount of non-commutative ring theory is still true in this generality [22]. Recently, R. Martínez-Villa and M. Ortíz studied tilting theory in arbitrary functor categories, in [19, 20]. They proved that most of the properties that are satisfied by a tilting module over an artin algebra also hold for functor categories. Following the line of the above-mentioned works, M. Ortíz introduced in [23] the concept of quasi-hereditary category to study the Auslander–Reiten components of a finite dimensional algebra $\Lambda .$ Similarly, as the standard modules appear in the theory of quasi-hereditary algebras, the concept of standard functors appears in this context. We note that the notion of standard functor is a generalization of the notion of standard module. As a consequence, a connection is obtained between highest weight categories and quasi-hereditary categories as stated by H. Krause in [14].

Finally, the concept of the triangular matrix category is introduced in [15, 16], as the analog of the triangular matrix algebra to the context of rings with several objects, and they obtain some applications to path categories given by infinite quivers, the construction of recollements, and the study of functorially finite subcategories in functor categories.

The aim of this paper is to show that the triangular matrix category $\Lambda =\left [ \begin{smallmatrix} \mathcal{T}&0\\ M&\mathcal{U} \end{smallmatrix} \right ]$ , where $\mathcal{T}$ and $\mathcal{U}$ are $\textrm{Hom}$ -finite, Krull–Schmidt $K$ -quasi-hereditary categories and $M$ is a $\mathcal{U}\otimes _K \mathcal{T}^{op}$ -module that satisfies suitable conditions, is quasi-hereditary in the sense of [15] and [23], generalizing some of the results obtained by B. Zhu in [29].

It is worth mentioning that recently in [18], similar results have been obtained in the context of standardly stratified lower triangular $\mathbb K$ -algebras with enough idempotents.

We outline the content of the paper section-by-section as follows.

In Section 1, we recall basic results about path categories, functor categories, quasi-hereditary categories, and triangular matrix categories.

In Section 2, we give the main result of this article (see Theorem 3.7), which is a generalization of a result given in [29, Theorem 3.1] as stated above (see Remark 3.8). Moreover, we obtain a characterization of the category of the $_\Lambda \Delta$ -filtered $\Lambda$ -modules, and we give an example a triangular matrix category $\Lambda$ , which is quasi-hereditary with respect to certain filtration $\{\Lambda _j\}_{j\ge 0 }$ .

2. Preliminaries

2.1. $K$ -categories, path categories, representations, and functor categories

In this part, we recall some basic definitions to approach this work. The reader can consult [1] and [6] for more details.

K-Categories. Let $K$ be a field. A category $\mathcal{C}$ is a $K$ -category if for each pair of objects $X$ and $Y$ in $\mathcal{C}$ , the set of morphisms $\mathcal{C}(X,Y)$ is equipped with a $K$ -vector space structure such that the composition $\circ$ of morphisms in $\mathcal{C}$ is a $K$ -bilinear map. A $K$ -category $\mathcal{C}$ is called Hom-finite if $\textrm{dim}_K\mathcal{C}(X,Y)\lt \infty$ .

A Krull–Schmidt category is an additive category such that each object decomposes into a finite direct sum of indecomposable objects with local endomorphism rings.

Ideals . Let $\mathcal{C}$ be an additive $K$ -category. A class $I$ of morphisms of $\mathcal{C}$ is a two-sided ideal in $\mathcal{C}$ if:

1. (a) the zero morphism $0_X\in \mathcal{C}(X,X)$ belongs to $I$ ;

2. (b) if $f,g\,:\,X\rightarrow Y$ are morphisms in $I$ and $\lambda$ , $\mu \in K$ , then $\lambda f+ \mu g\in I$ ;

3. (c) if $f \in I$ and $g$ is a morphism in $\mathcal{C}$ that is left-composable with $f$ , then $g\circ f\in I$ and

4. (d) if $f \in I$ and $h$ is a morphism in $\mathcal{C}$ that is right-composable with $f$ , then $ f\circ h\in I$ .

Equivalently, a two-sided ideal $I$ of $\mathcal{C}$ can be considered as a subfunctor $I (-,-) \subseteq \mathcal{C} (-,- ) \,:\, \mathcal{C}^{op} \times \mathcal{C}\rightarrow \textrm{Mod}\ K$ , defined by assigning to each pair $(X,Y)$ of objects $X$ , $Y$ of $\mathcal{C}$ a $K$ -subspace $I(X,Y )$ of $\mathcal{C}(X,Y )$ such that if $f \in I(X,Y )$ , $g\in \mathcal{C}(Y,Z)$ and $h\in \mathcal{C}(U,X)$ then $gfh \in I(U,Z)$ .

Given a two-sided ideal $I$ in an additive $K$ -category $\mathcal{C}$ , the quotient category $\mathcal{C}/I$ is the category whose objects are the same as the objects of $\mathcal{C}$ and the space of morphisms from $X$ to $Y$ in $\mathcal{C}/I$ is the quotient space $(\mathcal{C}/I)(X,Y) = \mathcal{C}(X,Y)/I(X,Y)$ of $\mathcal{C}(X,Y)$ . It is easy to see that the quotient category $ \mathcal{C}/I$ is an additive $K$ -category, and the projection functor $ \pi \,:\, \mathcal{C}\rightarrow \mathcal{C}/I$ assigning to each $f \,:\, X \rightarrow Y$ in $\mathcal{C}$ the coset $f + I \in (\mathcal{C}/I) (X, Y )$ is a $K$ -linear functor. Moreover, $\pi$ is full and dense, and $\textrm{Ker} (\pi ) =I$ .

The (Jacobson) radical of an additive $K$ -category $\mathcal{C}$ is the two-sided ideal $\textrm{rad}_{\mathcal{C}}(-,-)$ in $\mathcal{C}$ defined by the formula

\begin{equation*}\textrm {rad}_{\mathcal {C}}(X,Y)=\{h\in \mathcal {C}(X,Y)\,:\,1_X-gh \text { is invertible for any } g\in \mathcal {C}(Y,X)\},\end{equation*}

for all objects $X$ and $Y$ of $\mathcal{C}$ .

Tensor Product of K-Categories . Let $\mathcal{C}$ and $\mathcal{C}^{\prime}$ be $K$ -categories. The tensor product $\mathcal{C}\otimes _K \mathcal{C}^{\prime}$ is the category whose class of objects is $\textrm{Obj } \ \mathcal{C}\times \textrm{Obj }\ \mathcal{C}^{\prime}$ , where the set of morphisms from $(p_1,q_1)$ to $(p_2,q_2)$ is the ordinary tensor product $\mathcal{C}(p_1,p_2)\otimes \mathcal{C}^{\prime}(q_1,q_2)$ . The composition

\begin{equation*}\mathcal {C}(p_1,p_2)\otimes \mathcal {C}^{\prime}(q_1,q_2)\times \mathcal {C}(p_2,p_3)\otimes \mathcal {C}^{\prime}(q_2,q_3)\rightarrow \mathcal {C}(p_1,p_3)\otimes \mathcal {C}^{\prime}(q_1,q_3)\end{equation*}

is given by the rule $((f_1\otimes g_1),(f_2\otimes g_2))\mapsto (f_2f_1\otimes g_2 g_1)$ . This composition is bilinear; see [22].

Quivers, Path Algebras and Path Categories . A quiver is an oriented graph, formally denoted by a quadruple $Q=(Q_0,Q_1,s,t)$ , with a set of vertices $Q_0$ and a set of arrows $Q_1$ , and two maps $s,t\,:\, Q_1\rightarrow Q_0$ , called source and target, defined by $s(a\rightarrow b)=a$ and $t(a\rightarrow b)=b, b)=b$ , respectively, if $\alpha \,:\,a\rightarrow b$ is an arrow in $Q_1$ .

A path of length $l\ge 1$ from $a$ to $b$ in a quiver $Q$ is of the form $(a|\alpha _1,\ldots,\alpha _l|b)$ with arrows $\alpha _i$ satisfying $t(\alpha _i)=s(\alpha _{i+1})$ for all $1\le i\le l$ and $a=s(\alpha _1)$ as well as $b=t(\alpha _l)$ . In addition, for any vertex $a$ in $Q_0$ , a path of length 0 from $a$ to itself is denoted by $\epsilon _a$ .

Given a quiver $Q$ , its path category $K\mathcal{Q}$ is an additive category, with objects being direct sums of indecomposable objects. The indecomposable objects in the path category are given by the set $Q_0$ , and given $a,b\in Q_0$ , the set of maps from $a$ to $b$ is given by the $K$ -vector space with basis the set of all paths from $a$ to $b$ . The composition of maps is induced from the usual composition of paths:

(2.1) \begin{equation} (a|\alpha _1,\ldots,\alpha _l|b)(b|\beta _1,\ldots,\beta _s|c)=(a|\alpha _1,\ldots,\alpha _l\beta _1,\ldots,\beta _s|c), \end{equation}

where $(a|\alpha _1,\ldots,\alpha _l|b)$ is a path from $a$ to $b$ and $(b|\beta _1,\ldots,\beta _s|c)$ is a path from $b$ to $c$ .

Similarly, the path algebra of $Q$ denoted by $KQ$ , is the $K$ -vector space with basis the set of all paths in $Q$ , and the product of two paths is defined by (2.1) if they are composable, and it is zero if they are non-composable. In $KQ$ , any ideal $I$ is generated by a set of paths $\{\rho _i| i\}$ , that is $I=\langle \rho _i| i\ \rangle$ . Let $I$ be an ideal in $KQ$ , then given a pair of finite sets of vertices $\{X_i\}_{i=1}^n$ , $\{Y_j\}_{j=1}^m$ we set $\mathcal{I}(\oplus _{i=1}^n{X_i}, \oplus _{j=1}^m{Y_j})=\{(f_{ij})\in K\mathcal{Q}(\oplus _{i=1}^n{X_i}, \oplus _{j=1}^m{Y_j})|f_{ij}\in I\}$ . This allows us to define an ideal $\mathcal{I}$ in $K\mathcal{Q}$ , and we refer to it as the ideal generated by $I$ . If $I\subset KQ$ is generated by the set of paths $\{\rho _i| i\}$ , we say that $\mathcal{I}$ is generated by the set $\{\rho _i| i\}$ .

The ideal generated by all arrows is denoted by $KQ^{+}$ . Note that $(KQ^{+})^n$ is the ideal generated by all paths of length $\ge n$ . Given vertices $a,b\in Q_0$ , a finite linear combination $\sum _{w}c_w w$ with $c_w\in K$ where $w$ are paths of lengths $\ge 2$ from $a$ to $b$ is called a relation on $Q$ . Any ideal $I\subseteq (K Q^{+})^2$ can be generated, as an ideal, by relations. An ideal $I\subset KQ$ is called admisible if it is generated by a set of relations. We then say that an ideal $\mathcal{I}$ in $K\mathcal{Q}$ is admissible if it is generated by an admissible ideal in $KQ$ .

Representations of Quivers . A representation of a quiver $Q$ is a pair $V=\left ( (V_i)_{i\in Q_0}, (V_\alpha )_{\alpha \in Q_1}\right )$ , where each element of the family $\{V_i\}_{i\in Q_0}$ is a vector space and $V_\alpha \,:\, V_{s(\alpha )}\rightarrow V_{t(\alpha )}$ is a $K$ -linear map. Let $V$ and $ W$ be two representations of $ Q$ . A morphism from $V$ to $W$ is a family of linear maps $f=(f_i\,:\,V_i\rightarrow W_i)_{i\in Q_0}$ such that for each arrow $\alpha \,:\, i\rightarrow j$ we have $f_jV_\alpha =W_\alpha f_i$ . We denote by $\textrm{rep}\ Q$ the abelian category that has as objects the representations of $Q$ and as morphisms just the morphisms of representations. Let $\rho =(a|\alpha _1,\ldots,\alpha _l|b)$ a path in $Q$ , we set $V_\rho =V_{\alpha _l}\circ \cdots \circ V_{\alpha _1}$ . Let $I\subset KQ$ be an ideal, we then say the representation $V$ is bounded by $I$ if $V_\rho =0$ for all $\rho \in I$ . The full subcategory of $\textrm{rep}\ Q$ consisting of representations bounded by $I$ is denoted by $\textrm{rep} \ (Q,I)$ .

Strongly Locally Finite Quivers . Let $Q$ be a quiver. For $x\in Q_0$ , we denote by $x^+$ and $x^{-}$ the set of arrows starting in $x$ and the set of arrows ending in $x$ , respectively. Recall that $x$ is a sink vertex or a source vertex if $x^{+}=\emptyset$ or $x^-=\emptyset$ . One says that $Q$ is locally finite if $x^+$ and $x^{-}$ are finite sets and interval finite if the set of paths from $x$ to $y$ is finite for any $x, y \in Q_0$ . For short, we say that Q is strongly locally finite if it is locally finite and interval finite. In particular, $Q$ contains no oriented cycle in case it is interval finite. Note that under these conditions, if $Q$ is a strongly locally finite quiver, the path category $K\mathcal{Q}$ is a $\textrm{Hom}$ -finite Krull–Schmidt $K$ -category; see [7].

Functor Categories . Recall that a category $\mathcal{C}$ is said to be skeletally small if it has a small dense subcategory $\mathcal{C}^{\prime}$ , see [2]. Let $\mathcal{C}$ be a $\textrm{Hom}$ -finite Krull–Schmidt and skeletally small $K-$ category. The abelian category $(\mathcal{C}, \textbf{Ab})$ is the category of all additive covariant functors from $\mathcal{C}$ to the category of abelian groups, which we will call $\mathcal{C}$ -modules. Given two $\mathcal{C}$ -modules $F$ and $G$ , the set of morphisms $\textrm{Hom}_{(\mathcal{C}, \textbf{Ab})}(F,G)$ is denoted simply by $\textrm{Hom}_{\mathcal{C}}(F,G)$ . Following [2, 3], $(\mathcal{C}, \textbf{Ab})$ is denoted by $\textrm{Mod}(\mathcal{C})$ . A $\mathcal{C}$ -module $M$ is finitely presented if an exact sequence $P_1\rightarrow P_0\rightarrow M\rightarrow 0$ of $\mathcal{C}$ -modules exist where $P_0$ and $P_1$ are finitely generated projective $\mathcal{C}$ -modules. We recall that a $\mathcal{C}$ -module $P$ is finitely generated projective if $P$ is a direct summand of a finite coproduct of representable functors. We denote by $\textrm{mod}(\mathcal{C})$ the full subcategory of $\textrm{Mod}(\mathcal{C})$ consisting of finitely presented $\mathcal{C}$ -modules. Let $M$ be a $\mathcal{C}$ -module, so each $C$ in $\mathcal{C}$ the abelian group $M(C)$ has a structure as a $\textrm{End}_{\mathcal{C}}(C)$ -module and hence as a $K$ -module since $\textrm{End}_{\mathcal{C}}(C)$ is a $K$ -algebra. We denote by $(\mathcal{C}, \textrm{mod} \ K)$ the full subcategory of $\textrm{Mod}(\mathcal{C})$ of all $\mathcal{C}$ -modules such that $M(C)$ is a finitely generated $K$ -module. The category $(\mathcal{C}, \textrm{mod} \ K)$ is an abelian category with the property that the inclusion $(\mathcal{C}, \textrm{mod} \ K)\rightarrow \textrm{Mod}(\mathcal{C})$ is exact and contains $\textrm{mod}(\mathcal{C})$ as a full subcategory. Let $Q$ be a quiver and $I$ be an ideal $I\subset KQ$ . Set $\mathcal{C}=K\mathcal{Q}/\mathcal{I}$ . Then each representation $V=\left ( (V_i)_{i\in Q_0}, (V_\alpha )_{\alpha \in Q_1}\right )$ in $\textrm{rep} \ (Q,I)$ defines a $\mathcal{C}$ -module $\tilde V$ in $(\mathcal{C}, \textrm{mod} \ K)$ by setting $\tilde V(i)=V_i$ and $\tilde V(\alpha )=V_\alpha$ .

In general, the functor $D\,:\, (\mathcal{C}, \textrm{mod} \ K)\rightarrow (\mathcal{C}^{op}, \textrm{mod} \ K)$ given by

\begin{equation*}D(M)(X)=\textrm {Hom}_K (M(X),K)\end{equation*}

for all $X$ in $\mathcal{C}$ defines a duality between $(\mathcal{C}, \textrm{mod} \ K)$ and $(\mathcal{C}^{op}, \textrm{mod} \ K)$ , and we refer to it as the standard duality.

2.2. Quasi-hereditary categories and triangular matrix categories

Assume $\mathcal{C}$ is a $\textrm{Hom}$ -finite Krull–Schmidt $K$ -category. In order to generalize the notion of quasi-hereditary algebra to $K$ -categories, the notion of heredity ideal and heredity chain is introduced in [23].

Remark 2.1. We note that definition of quasi-hereditary category in [23, Definition 3.4] is given for contravariant functors; however, by considering the opposite category $\mathcal{C}^{op}$ we have that contravariant functors over $\mathcal{C}^{op}$ coincide with covariant functors over $\mathcal{C}$ . So, we can translate all the results in [23] to the setting of covariant functors.

H Eredity Ideals. A two-sided ideal $I$ in $\mathcal{C}$ is called (left) heredity if the following conditions hold:

1. (i) $I^2=I$ , i.e, $I$ is an idempotent ideal;

2. (ii) $I\textrm{rad}\ \mathcal{C}(-,?)I=0$ , and

3. (iii) $I(X,-)$ is a projective finitely generated $\mathcal{C}-$ module for all $X\in \mathcal{C}$ .

A chain of two-sided ideals

\begin{equation*}0=I_0\subset I_1\subset \cdots \subset \mathcal {C}(-,?),\end{equation*}

is exhaustive if $\cup _{j\in J}I_j=\mathcal{C}(-,?)$ . The category $\mathcal{C}$ is called quasi-hereditary if there exists an exhaustive chain $\{I_j\}_{j\in J}$ , where $J$ is at most countable, of two-sided ideals

\begin{equation*}0=I_0\subset I_1\subset \cdots \subset \mathcal {C}(-,?),\end{equation*}

such that $I_{j}/I_{j-1}$ is heredity in the quotient category $\mathcal{C}/I_{j-1}$ . Such a chain is called a heredity chain.

Remark 2.2.

1. (a) We note that we consider exhaustive chain of ideals because as in the classical case, we need to reach $\mathcal{C}(-,?)$ in some way and if the set $J$ is infinite we can do that by requiring the equality $\cup _{j\in J}I_j=\mathcal{C}(-,?)$ .

2. (b) In the spirit of the Remark 2.1, we should have called left heredity ideal and left quasi-hereditary category to the notions given above for covariant functors and the notions given for contravarian functors in [23] should have been called right heredity ideal and right quasi-hereditary category. However, in order to avoid overloading the notation, we will not write the left adjective in all those notions.

3. (c) By Remark 2.1 we have that $\mathcal{C}$ is quasi-hereditary in the sense given above if and only if $\mathcal{C}^{op}$ is quasi-hereditary in the sense of [23].

Let $\mathcal{B}$ be a full additive subcategory of $\mathcal{C}$ . Given $C,C^{\prime}\in \mathcal{C}$ , we denote by $I_{\mathcal{B}}(C,C^{\prime})$ the subset of $\mathcal{C}(C,C^{\prime})$ consisting of morphisms which factor through some object in $\mathcal{B}$ . This allows us to define the two-sided ideal $I_{\mathcal{B}}(-,?)$ which is an idempotent ideal in $\mathcal{C}$ .

Moreover, we denote by $\textrm{Tr}_{\{\mathcal{C}(E,-)\}_{E\in \mathcal{B}}} \mathcal{C}(X,-)$ with $X\in \mathcal{C}$ , the trace of $\{\mathcal{C}(E,-)\}_{E\in \mathcal{B}}$ in $\mathcal{C}(X,-)$ , that is,

\begin{equation*}\displaystyle \textrm {Tr}_{\{\mathcal {C}(E,-)\}_{E\in \mathcal {B}}} \mathcal {C}(X,-)=\sum \left\{\textrm {Im}(\psi )\,:\,\psi \in \textrm {Hom}_{\textrm {Mod}(\mathcal {C})}\left(\mathcal {C}(E,-),\mathcal {C}(X,-)\right),\,E\in \mathcal {B}\right\}.\end{equation*}

By the covariant version of [23, Lemma 3.1], we have that:

Lemma 2.3. Let $\mathcal{B}$ be a full additive subcategory of $\mathcal{C}$ . Then for $X\in \mathcal{C}$ we have that:

\begin{equation*}\textrm {Tr}_{\{\mathcal {C}(E,-)\}_{E\in \mathcal {B}}} \mathcal {C}(X,-)=I_{\mathcal {B}}(X,-).\end{equation*}

Quasi-Hereditary Categories . In the previous section, we gave a definition of quasi-hereditary category respect to a chain $\{I_j\}_{j\in J}$ of two-sided ideals. Now, in order to produce a chain of two sided ideals of $\mathcal{C}$ , we will need a filtration of the category $\mathcal{C}$ . So, assume we have an exhaustive filtration

\begin{equation*}\{0\}=\mathcal {B}_{0}\subseteq \mathcal {B}_{1}\subseteq \mathcal {B}_{2}\subseteq \cdots \subseteq \mathcal {C}\end{equation*}

of $\mathcal{C}$ into additive full subcategories (that is, $\cup _{j\ge 0} \mathcal{B}_j=\mathcal{C}$ ). We then have an exhaustive chain of two-sided idempotent ideals:

\begin{equation*} \{0\}=I_{\mathcal {B}_0} \subset I_{\mathcal {B}_1} \subset \cdots I_{\mathcal {B}_{j-1}} \subset I_{\mathcal {B}_j} \subset \cdots \subset \mathcal {C}(-,?). \end{equation*}

Note that $\frac{I_{\mathcal{B}_j}}{I_{\mathcal{B}_{j-1}}}$ is an idempotent ideal in the quotient category $\frac{\mathcal{C}}{I_{\mathcal{B}_{j-1}}}$ since $I_{\mathcal{B}_{j}}$ and $I_{\mathcal{B}_{j-1}}$ are idempotents in $\mathcal{C}$ and

\begin{equation*} \left (\frac {I_{\mathcal {B}_j}}{I_{\mathcal {B}_{j-1}}}\right )^2= \frac {I_{\mathcal {B}_j}}{I_{\mathcal {B}_{j-1}}}\frac {I_{\mathcal {B}_j}}{I_{\mathcal {B}_{j-1}}}=\frac {I_{\mathcal {B}_j}^2+I_{\mathcal {B}_{j-1}}}{I_{\mathcal {B}_{j-1}}}=\frac {I_{\mathcal {B}_j}}{I_{\mathcal {B}_{j-1}}}. \end{equation*}

The above motivates us to introduce the definition of quasi-hereditary category respect to an exhaustive $\{\mathcal{B}_j\}_{j\geq 0}$ filtration of $\mathcal{C}$ .

Definition 2.4. [23, Definition 3.4 (b)] Let $\mathcal{C}$ be a $\textrm{Hom}$ -finite Krull–Schmidt $K$ -category. Assume that $\{\mathcal{B}_j\}_{j\geq 0}$ is an exhaustive filtration of $\mathcal{C}$ into full additive subcategories. We say that $\mathcal{C}$ is quasi-hereditary with respect to $\{\mathcal{B}_j\}_{j\geq 0}$ if

\begin{equation*}\{0\}=I_{\mathcal {B}_0} \subset I_{\mathcal {B}_1} \subset \cdots I_{\mathcal {B}_{j-1}} \subset I_{\mathcal {B}_j} \subset \cdots \subset \mathcal {C}(-,?) \end{equation*}

is a heredity chain.

Remark 2.5.

1. (a) We note that the advantage of using the Definition 2.4 instead of the more general definition of quasi-hereditary category given previously, is that exhaustive filtrations are easier to compute than idempotent ideals in general, and exhaustive filtrations induce an exhaustive chain of idempotent ideals.

2. (b) Now, by [23, Lemma 3.9] we can see that if $\textrm{rad}^{\infty }(\mathcal{C})=0$ then all idempotent ideals of $\mathcal{C}$ are of the form $I_{\mathcal{B}}$ for some additive subcategory of $\mathcal{C}$ ; and hence all the exhaustive chains $\{I_{i}\}_{i\geq 0}$ of idempotent ideals are constructed in this way. Unfortunately, there are categories where all the idempotent ideals are not of the form $\mathcal{I}_{\mathcal{B}}$ for some $\mathcal{B}$ . For example, if $A$ is a wild algebra over an algebraically closed field, then the transfinite radical $\textrm{rad}^{\ast }$ is not zero (see [28, Proposition 4]) and $\textrm{rad}^{\ast }$ is an idempotent ideal of $\textrm{mod}(A)$ which does not contain any identity morphisms (see [28, Lemma 1]); and hence it is not of the form $\mathcal{I}_{B}$ .

Let $\mathcal{B}$ be an additive subcategory of $\mathcal{C}$ , we denote by $\textrm{add}(\mathcal{B})$ the full subcategory of $\mathcal{C}$ whose objects are the direct summands of finite coproducts of objects in $\mathcal{B}$ . A subcategory $\mathcal{B}$ of $\mathcal{C}$ is closed under direct summands if $\textrm{add}(\mathcal{B})=\mathcal{B}$ . Let $\mathcal{B}$ be an additive subcategory of $\mathcal{C}$ , we denote by ${\textbf{Ind}}(\mathcal{B})$ the class of all the indecomposable objects belonging to $\mathcal{B}$ . The following result given in [23] will be useful in the remainder of this work.

Theorem 2.6. [23, Theorem 3.6] Let $\mathcal{C}$ be a $\textrm{Hom}$ -finite Krull–Schmidt $K$ -category and let $\{\mathcal{B}_j\}_{j\geq 0}$ be a family of closed under direct summands additive subcategories of $\mathcal{C}$ . Suppose that $\{\mathcal{B}_j\}_{j\geq 0}$ is an exhaustive filtration of $\mathcal{C}$ . Then $\mathcal{C}$ is quasi-hereditary with respect to $\{\mathcal{B}_j\}_{j\geq 0}$ if and only if the following conditions hold:

1. (i) $\textrm{rad}_{\mathcal{C}}(E,E^{\prime})=I_{\mathcal{B}_{j-1}}(E,E^{\prime})$ , for all $E,E^{\prime}\in \textrm{ Ind} \ \mathcal{B}_{j}-\textrm{ Ind} \ \mathcal{B}_{j-1}$ ;

2. (ii) and for all $X \in \mathcal{C}$ and $j\geq 1$ , there exists an exact sequence

   

   with $E_{j} \in \mathcal{B}_{j}$ and $E_{j-1} \in \mathcal{B}_{j-1}$ .

Remark 2.7. (a). Let us see why the Definition 2.4 is a generalization of the classical notion of quasi-hereditary algebra. Let $A$ be a finite dimensional $K$ -algebra. In this case, we consider the $K$ -category $\mathcal{C}\,:\!=\,\textrm{proj}(A)$ , that is, the full subcategory of $\textrm{mod}(A)$ whose objects are the finitely generated projective $A$ -modules. Since $A$ is semiperfect we have that $\mathcal{C}=\textrm{proj}(A)$ is a Krull–Schmidt category (see [13, Proposition 4.1]).

Suppose that $\textrm{proj}(A)$ is a quasi-hereditary category in the sense of Definition 2.4, with respect to the exhaustive filtration

\begin{equation*}\{0\}=\mathcal {B}_{0}\subseteq \mathcal {B}_{1}\subseteq \cdots \subseteq \mathcal {B}_{m-1}\subseteq \mathcal {B}_{m}=\mathcal {C}=\textrm {proj}(A)\end{equation*}

of $\textrm{proj}(A)$ into additive full subcategories and closed under direct summands. Then we have an exhaustive chain of two-sided idempotent ideals of $\mathcal{C}(-,?)$ :

\begin{equation*} \{0\}=I_{\mathcal {B}_0} \subset I_{\mathcal {B}_1} \subset \cdots I_{\mathcal {B}_{m-1}} \subset I_{\mathcal {B}_m} =\mathcal {C}(-,?). \end{equation*}

By [14, Lemma 4.5], we have the corresponding chain of idempotent ideals of $A^{op}$

\begin{equation*}0=J_{0}\subseteq J_{1}\subseteq \cdots \subseteq J_{m-1}\subseteq J_{m}\simeq A^{op}\end{equation*}

where $J_{i}\,:\!=\,I_{\mathcal{B}_{i}}(A,A)$ for all $i$ . In this case, $J_{i}=Ae_{i}A$ for some idempotent $e_{i}\in A$ and by [14, Lemma 4.5], we have that $\mathcal{B}_{i}=\textrm{add}(Ae_{i})$ for all $i\geq 0$ .

Now, let us see that $J_{1}$ is a heredity ideal of $A^{op}$ .

$(i)$ . By Theorem 2.6(i), we have that $\textrm{rad}_{\mathcal{C}}(E,E^{\prime})=I_{\mathcal{B}_{0}}(E,E^{\prime})=0,$ for all pair of objects $E,E^{\prime}\in \textrm{ Ind} \ \mathcal{B}_{1}$ , since $\mathcal{B}_{0}=\{0\}$ . Hence, $\textrm{rad}(\mathcal{B}_{1})=0$ and thus $e_{1}\textrm{rad}(A)e_{1}=0$ . We conclude that $J_{1}A^{op}J_{1}=0$ .

$(ii)$ By Theorem 2.6(ii) we have an exact sequence

with $Y\in \mathcal{B}_{1}=\textrm{add}(Ae_{1})$ and $Z\in \mathcal{B}_{0}=\{0\}$ .

Hence, $J_{1}=I_{\mathcal{B}_{1}}(A,A)\simeq \mathcal{C}(Y,A)=\textrm{proj}(Y,A)=\textrm{Hom}_{A}(Y,A)$ which is projective over $A^{op}\simeq \textrm{proj}(A,A)$ .

Hence, $J_{1}$ is a heredity ideal of $A^{op}$ . We can proceed inductively and conclude that $J_{i}/J_{i-1}$ is a heredity ideal of $A^{op}/J_{i-1}$ for all $i\geq 1$ . Therefore, $A^{op}$ is quasi-hereditary and hence by [11, Statement 9] we conclude that $A$ is quasi-hereditary. Similarly, by using [14, Lemma 4.5], it can be proved that if $A$ is a quasi-hereditary algebra then $\textrm{proj}(A)$ is a quasi-hereditary category.

(b). Suppose that $A$ is quasi-hereditary, then there exists a chain of idempotent ideals

\begin{equation*}0=J_{0}\subseteq J_{1}\subseteq \cdots \subseteq J_{m-1}\subseteq J_{m}=A\end{equation*}

such that $J_{t}/J_{t-1}$ is a heredity ideal of $A/J_{t-1}$ for all $t$ .

By [4, Proposition 6.1], there exists projective $A$ -modules $P_{1},\cdots, P_{m}$ such that $J_{i}=\textrm{Tr}_{P_{1}\oplus \cdots \oplus P_{i}}(A)$ for $i=1,\cdots, m$ . Moreover, if $\mathcal{B}_{i}=\textrm{add}(P_{1}\oplus \cdots \oplus P_{i})$ , then for each $i=1,\cdots, m$ there exists an exact sequence in $\textrm{mod}(A)$

such that $P_{i,0}\in \mathcal{B}_{i}$ and $P_{i,1}\in \mathcal{B}_{i-1}$ .

We note that these exact sequences are the analogous of the exact sequences given in the Theorem 2.6(ii):

with $P_{i,0} \in \mathcal{B}_{i}$ , $P_{i,1} \in \mathcal{B}_{i-1}$ , and $X\in \mathcal{C}$ .

It is well known that a semiprimary ring $A$ is quasi-hereditary if and only if $A^{op}$ is quasi-hereditary (see [11, Statement 9] in p. 288). We have somehow a similar result for the context of quasi-hereditary categories.

We recall the following notions. Let $\mathcal{A}$ be an arbitrary category and $\mathcal{B}$ a full subcategory of $\mathcal{A}$ . The full subcategory $\mathcal{B}$ is contravariantly finite if for every $A\in \mathcal{A}$ there exists a morphism $f_{A}\,:\,B\longrightarrow A$ with $B\in \mathcal{B}$ such that if $f^{\prime}\,:\,B^{\prime}\longrightarrow A$ is other morphism with $B^{\prime}\in \mathcal{B}$ , then there exist a morphism $g\,:\,B^{\prime}\longrightarrow B$ such that $f^{\prime}=f_{A} g$ . The morphism $f_{A}$ is called a $\textbf{right}$ $\boldsymbol{\mathcal{B}}$ $\textbf{-approximation}$ of $A$ . A right $\mathcal{B}$ -approximation $f_{A}\,:\,B\longrightarrow A$ of $A$ is $\text{minimal}$ if whenever $g\,:\,B\longrightarrow B$ is a morphism such that $g f_{A}=f_{A}$ then $g$ is an isomorphism. Dually, is defined the notion of $\textbf{covariantly finite}$ and $\textbf{left minimal}$ $\boldsymbol{\mathcal{B}}$ $\textbf{-approximation}$ . We say that $\mathcal{B}$ is $\textbf{functorially finite}$ if $\mathcal{B}$ is contravariantly finite and covariantly finite.

Proposition 2.8. Let $\mathcal{C}$ be a Hom-finite and Krull–Schmidt category and let $\{\mathcal{B}_j\}_{j\geq 0}$ be a family of closed under direct summands additive subcategories of $\mathcal{C}$ ; and suppose that each $\mathcal{B}_{j}$ is covariantly finite. If $\mathcal{C}$ is quasi-hereditary in the sense of [23, Definition 3.4], then $\mathcal{C}^{op}$ is quasi-hereditary in the sense of [23, Definition 3.4].

Proof. Suppose that $\mathcal{C}$ is quasi-hereditary with respect to $\{\mathcal{B}_j\}_{j\geq 0}$ . By [23, Theorem 3.6], there exist exact sequences for all $X \in \mathcal{C}$ and $j\geq 1$ :

with $E_{j} \in \mathcal{B}_{j}$ and $E_{j-1}\in \mathcal{B}_{j-1}$ . Now, since $B_{j}$ is covariantly finite, there exists an epimorphism $\mathcal{C}(X,-)\longrightarrow I_{\mathcal{B}_{j}}(C,-)$ for every $C\in \mathcal{C}$ by using the proof of [26, Proposition 4.12]. Then, this implies that $B_{j}$ is functorially finite by [26, Proposition 4.12].

We assert that for all $X\in \mathcal{C}$ there exists a monic right $\mathcal{B}_{1}$ -approximation of $X$ . Indeed, by taking, $j=1$ , from the above exact sequence we get that $\mathcal{C}(-,E_{1})\simeq I_{\mathcal{B}_{1}}(-,X)$ . By Yoneda’s Lemma we get a morphism $\gamma \,:\,E_{1} \longrightarrow X$ and this morphism is a right $\mathcal{B}_{1}$ -approximation of $X$ . Now, let $\alpha \,:\,Y \longrightarrow E_{1}$ such that $\gamma \alpha =0$ . Since $\mathcal{C}(-,\gamma )\,:\,\mathcal{C}(-,E_{1})\longrightarrow I_{\mathcal{B}_{1}}(-,X)$ is an isomorphism, we conclude that $\alpha =0$ , this implies that $\gamma$ is a monomorphism.

Now, since $\mathcal{B}_{1}$ is covariantly finite, $\textrm{add}(\mathcal{B}_{1})=\mathcal{B}_{1}$ and $\mathcal{C}$ is a Krull–Schmidt category, every object of $\mathcal{C}$ has a left minimal $\mathcal{B}_{1}$ -approximation. So, let $\theta \,:\,X\longrightarrow E^{\prime}_{1}$ be a left minimal $\mathcal{B}_{1}$ -approximation of $X$ .

We assert that $\theta$ is an epimorphism. Indeed, consider $\beta \,:\,E^{\prime}_{1}\longrightarrow Y$ a morphism such that $\beta \theta =0$ . By the first assertion above, there exists $\lambda \,:\,E\longrightarrow Y$ a monic right $\mathcal{B}_{1}$ -approximation of $Y$ ; and hence $\beta =\lambda \delta$ for some $\delta \,:\,E^{\prime}_{1}\longrightarrow E$ . Since, $0=\beta \theta =\lambda \delta \theta$ and $\lambda$ is a monomorphism we get that $\delta \theta =0$ .

Now, for all $g\,:\,E \longrightarrow E^{\prime}_{1}$ we have that $(1_{E^{\prime}_{1}}-g \delta )\theta =\theta$ . Since $\theta$ is minimal we get that $(1_{E^{\prime}_{1}}-g\delta )$ is an isomorphism; and thus we conclude that $\delta \in \textrm{rad}_{\mathcal{C}}(E^{\prime}_{1},E)=\textrm{rad}_{\mathcal{B}_{1}}(E^{\prime}_{1},E)=0$ (see [23, Theorem 3.6(i)]). Hence, $\delta =0$ and we obtain that $\beta =\lambda \delta =0$ . This proves that $\theta$ is an epimorphism. Now, $\mathcal{C}(\theta,-)$ is an epimorphism as established in the proof of [26, Proposition 4.12]. Therefore, we get an isomorphism

\begin{equation*} \mathcal {C}(\theta,-)\,:\,\mathcal {C}(E^{\prime}_{1},-)\longrightarrow I_{\mathcal {B}_{1}}(X,-).\end{equation*}

We note that $I_{\mathcal{B}_{j}}^{op}=I_{\mathcal{B}_{j}^{op}}$ is a bilateral ideal of $\mathcal{C}^{op}$ for all $j$ . Thus, we get that $I_{\mathcal{B}_{1}^{op}}(-,X)=I_{\mathcal{B}_{1}}(X,-)\simeq \mathcal{C}(E^{\prime}_{1},-)\simeq \mathcal{C}^{op}(-,E^{\prime}_{1})$ is a projective $\mathcal{C}^{op}$ -module. Since $\textrm{rad}\mathcal{C}(-,-)$ is a bilateral ideal we have that $I_{\mathcal{B}_{1}}\textrm{rad}\ \mathcal{C}(-,?)I_{\mathcal{B}_{1}}=0$ implies that $I_{\mathcal{B}_{1}^{op}}\textrm{rad}\mathcal{C}^{op}(-,?)I_{\mathcal{B}_{1}^{op}}=0.$ Hence, we obtain that $I_{\mathcal{B}_{1}^{op}}(-,X)$ is a heredity ideal of $\mathcal{C}^{op}$ . Proceeding inductively, we conclude that

\begin{equation*}\{0\}=I_{\mathcal {B}_0^{op}}\subset I_{\mathcal {B}_1^{op}} \subset \cdots \subset I_{\mathcal {B}_{j-1}^{op}} \subset I_{\mathcal {B}_j^{op}} \subset \cdots \subset \mathcal {C}^{op}(-,?) \end{equation*}

is a heredity chain. Hence, $\mathcal{C}^{op}$ is quasi-hereditary with respect to $\{\mathcal{B}_j^{op}\}_{j\geq 0}$ in the sense of [23, Definition 3.4].

Corollary 2.9. Let $\mathcal{C}$ be a Hom-finite and Krull–Schmidt category and let $\{\mathcal{B}_j\}_{j\geq 0}$ be a family of closed under direct summands additive subcategories of $\mathcal{C}$ and suppose that each $\mathcal{B}_{j}$ is functorially finite. Then $\mathcal{C}$ is quasi-hereditary in the sense of [23 , Definition 3.4] if and only if $\mathcal{C}^{op}$ is quasi-hereditary in the sense of [23 , Definition 3.4].

Proof. It follows from Proposition 2.8 and its dual.

By the Corollary above and Remark 2.2(c), we conclude the following.

Corollary 2.10. Let $\mathcal{C}$ be a Hom-finite and Krull–Schmidt category and let $\{\mathcal{B}_j\}_{j\geq 0}$ be a family of closed under direct summands additive subcategories of $\mathcal{C}$ and suppose that each $\mathcal{B}_{j}$ is functorially finite. Then $\mathcal{C}$ is quasi-hereditary in the sense of [23 , Definition 3.4] if and only if $\mathcal{C}$ is quasi-hereditary in the sense of Definition 2.4 .

Standard $\mathcal{C}$ -Modules . Let $\mathcal{C}$ be a quasi-hereditary category with respect to a family of additively closed subcategories $\{\mathcal{B}_j\}$ . Each module

\begin{equation*}_{\mathcal {C}}\Delta _E(j)\,:\!=\, \mathcal {C}(E,-)/I_{\mathcal {B}_{j-1}}(E, -)\end{equation*}

with $E\in \textrm{Ind} \mathcal{B}_j-\textrm{Ind} \mathcal{B}_{j-1}$ is called standard, and $_{\mathcal{C}}\Delta (j)$ denotes the category consisting of the standard $\mathcal{C}$ -modules $_{\mathcal{C}}\Delta _E(j)$ . In addition, $_{\mathcal{C}}\Delta$ denotes the full subcategory consisting of the standard $\mathcal{C}$ -modules.

Filtered $\mathcal{C}$ -Modules. Let $\mathcal{A}$ be an abelian category, and $\mathcal{X}\subseteq \mathcal{A}.$ We denote by $\mathcal{X}^{\amalg }$ the class of objects of $\mathcal{A}$ , which are a finite direct sum of objects in $\mathcal{X}.$ We say that $M\in \mathcal{A}$ is $\mathcal{X}$ - $\textbf{filtered}$ if there exists a chain $\{M_{j}\}_{j\ge 0}$ of subobjects of $M$ such that $M_{j+1}/M_{j}\in \mathcal{X}^{\amalg }$ for $j\ge 0.$ In case $M=M_n$ for some $n\in \mathbb N$ , we say that $M$ has a finite $\mathcal{X}$ -filtration of length $n$ . We denote by $\mathcal{F}(\mathcal{X})$ the class of objects that are $\mathcal{X}$ -filtered and by $\mathcal{F}_f(\mathcal{X})$ the class of objects that have a finite filtration. For $M \in \mathcal{F}_f(\mathcal{X})$ , the $\mathcal{X}$ -length of $M$ can be defined as follows $l_{\mathcal{X}}(M)\,:\!=\,\textrm{min}\{ n\in \mathbb{N} \,:\, M \text{ has an $\mathcal{X}$-filtration of length $n$}\}$ .

By using the notion of $\mathcal{X}$ -length and induction, the following useful remark can be proven.

Remark 2.11. Let $\mathcal{X}$ be a class of objects in an abelian category $A$ . Then, the class $\mathcal{F}_f (\mathcal{X})$ is closed under extensions.

Given $F$ a $\mathcal{C}-$ module, its trace filtration with respect to $\{\mathcal{B}_j\}_{j\geq 0}$ is given by

\begin{equation*} \{0\}=F^{[0]} \subset F^{[1]} \subset F^{[2]} \subset \cdots \subset F^{[j-1]} \subset F^{[j]} \subset \cdots, \end{equation*}

where $F^{[j]}\,:\!=\,\textrm{Tr}_{\{\mathcal{C}(E,-)\}_{E\in \mathcal{B}_j}}F$ and $F= \bigcup \limits _{j\geq 0}F^{[j]}$ .

It is of interest to study the $ \mathcal{C}$ -modules $F$ that possesses a trace filtration that satisfies $ \frac{F ^{[j]}}{F ^{[j-1]}} \in{}_{\mathcal{C}}\Delta (j) ^{\amalg }$ for all $ j \geq 1$ . It then follows that these $ \mathcal{C}$ -modules are $ \Delta$ -filtered. We denote the full subcategory of the $ \Delta$ -filtered modules by $ \mathcal{F} (_{\mathcal{C}}\Delta )$ .

The following result will be very useful, the item $(b)$ is a generalization of a result of Dlab (see [9, Proposition A.3.2]).

Lemma 2.12. [23, Lemmas 3.17, 3.18] Let $F \in \mathcal{F}(\Delta )$ .

1. (a) For all $j\geq 0$ , $F^{[j]}$ has a presentation

   (2.2) \begin{equation} \mathcal{C}(E_{j-1},-) \rightarrow \mathcal{C}(E_{j},-) \rightarrow F^{[j]} \rightarrow 0,\ E_{j-1} \in \mathcal{B}_{j-1}, E_{j}\in \mathcal{B}_{j}. \end{equation}

2. (b) $F^{[j]}\cong \mathcal{C} \otimes _{\mathcal{B}_j}(F|_{\mathcal{B}_{j}})$ .

3. (c) If the filtration $0=\mathcal{B}_{0}\subseteq \mathcal{B}_{1}\subseteq \cdots \subseteq \mathcal{B}_{n}=\mathcal{C}$ is finite, then $\mathcal{F}(\Delta )$ consists of finitely presented functors.

Triangular Matrix Categories. In this section, we recall some results from the article [15]. The notion of triangular matrix categories is introduced in [15, 16] in order to define the analogous of the triangular matrix algebras to the context of rings with several objects.

Definition 2.13. [15, Definition 3.5] Given $\mathcal{T}$ and $\mathcal{U}$ additive $K$ -categories and a functor $M\,:\,\mathcal{U}\otimes _K \mathcal{T}^{op}\rightarrow \textrm{Mod}(K)$ , the triangular matrix category $\Lambda = \left (\begin{smallmatrix} \mathcal{T}& \quad\!\! 0 \\[2pt] M & \quad\!\! \mathcal{U} \end{smallmatrix}\right )$ is the additive $K$ -category whose collection of objects are the matrices $\left (\begin{smallmatrix} T& \quad\!\! 0 \\[2pt] M & \quad\!\! U \end{smallmatrix}\right )$ , where $U$ and $T$ are objects in $\mathcal{U}$ and $\mathcal{T}$ , respectively.

Let $X=\left (\begin{smallmatrix} T& \quad\!\! 0 \\[2pt] M & \quad\!\! U \end{smallmatrix}\right )$ , $X^{\prime}=\left (\begin{smallmatrix} T^{\prime}& \quad\!\! 0 \\[2pt] M & \quad\!\! U^{\prime} \end{smallmatrix}\right )$ , and $X^{\prime\prime}=\left (\begin{smallmatrix} T^{\prime\prime}& \quad\!\! 0 \\[2pt] M & \quad\!\! U^{\prime\prime} \end{smallmatrix}\right )$ be objects in $\Lambda$ . The set of morphisms from $X$ to $X^{\prime}$ is $\Lambda (X,X^{\prime})=\left ( \begin{smallmatrix} \mathcal{T}(T,T^{\prime})& \quad\!\! 0 \\[2pt] M (U^{\prime},T)& \quad\!\! \mathcal{U}(U,U^{\prime}) \end{smallmatrix}\right )$ , where $\left ( \begin{smallmatrix} \mathcal{T}(T,T^{\prime})& \quad\!\! 0 \\[2pt] M (U^{\prime},T)& \quad\!\! \mathcal{U}(U,U^{\prime}) \end{smallmatrix}\right )\,:\!=\,\left \{\left ( \begin{smallmatrix} f& \quad\!\! 0 \\[2pt] m & \quad\!\! g\end{smallmatrix}\right )\,:\, f\in \mathcal{T}(T,T^{\prime}),\right.$ $\left. g\in \mathcal{U}(U,U^{\prime}), m\in M(U^{\prime},T)\right \}$ , and the composition $\Lambda (X^{\prime},X^{\prime\prime})\times \Lambda (X,X^{\prime})\rightarrow \Lambda (X,X^{\prime\prime})$ is defined by

\begin{equation*} \hspace {1.2cm}\left (\left (\begin {matrix} f_2 & \quad 0 \\[3pt] m_2 & \quad g_2 \end {matrix}\right ), \left (\begin {matrix} f_1 & \quad 0 \\[3pt] m_1 & \quad g_1 \end {matrix}\right )\right ) \mapsto \left (\begin {matrix} f_2 \circ f_1 & \quad 0 \\[3pt] m_2 \bullet f_1+g_2 \bullet m_1 & \quad g_2 \circ g_1 \end {matrix}\right ) \end{equation*}

with $m_2 \bullet f_1\,:\!=\,M(1_{U^{\prime\prime}}\otimes f_1^{op})(m_2)$ and $g_2 \bullet m_1\,:\!=\,M(g_2\otimes 1_{T})(m_1)$ .

We note that

\begin{equation*}M\big(1_{U^{\prime\prime}}\otimes f_1^{op}\big)\,:\,M(U^{\prime\prime},T^{\prime})\rightarrow M(U^{\prime\prime},T),\end{equation*}

\begin{equation*}M\big(g_2\otimes 1_T\big)\,:\,M(U^{\prime}, T)\rightarrow M(U^{\prime\prime},T)\end{equation*}

are morphisms in $\textrm{Mod}(K)$ .

Let $ M\in \textrm{Mod}(\mathcal{U}\otimes _{K} \mathcal{T}^{op})$ . Then, there exists a covariant functor

\begin{equation*}E\,:\,\mathcal {T}\longrightarrow \textrm {Mod}(\mathcal {U})^{op},\end{equation*}

defined as follows:

$(a)$ . For $T\in \mathcal{T}$ , we have a covariant functor $E(T)\,:\!=\,M_{T}\,:\,\mathcal{U} \longrightarrow \textbf{Ab}$ given as:

1. (i) $ M_{T}(U)\,:\!=\,M(U,T)$ , for all $ U\in \mathcal{U}$ .

2. (ii) $ M_{T}(u)\,:\!=\,M(u\otimes 1_{T})$ , for all $ u\in \textrm{Hom}_{\mathcal{U}}(U,U^{\prime}).$

$(b)$ . Now, given a morphism $t\,:\,T\longrightarrow T^{\prime}$ in $\mathcal{T}$ we define the natural transformation $E(t)\,:\!=\,\bar{t}\,:\,M_{T^{\prime}}\longrightarrow M_{T}$ where $ \bar{t}=\lbrace [\bar{t}]_{U}\,:\,M_{T^{\prime}}(U)\longrightarrow M_{T}(U)\rbrace _{ U\in \mathcal{U}}$ with $ [\bar{t}]_{U}=M(1_{U}\otimes t^{op})\,:\,M(U,T^{\prime})\longrightarrow M(U,T)$ .

Now, let $Y\,:\,\textrm{Mod}(\mathcal{U})\longrightarrow \textrm{Mod}\Big (\textrm{Mod}(\mathcal{U})^{op}\Big )$ be the Yoneda functor given as follows $Y(B)\,:\!=\,\textrm{Hom}_{\textrm{Mod}(\mathcal{U})}(-,B)$ ; and consider the following functor $I\,:\,\textrm{Mod}\Big (\textrm{Mod}(\mathcal{U})^{op}\Big )\longrightarrow \textrm{Mod}(\mathcal{T})$ , which is induced by composing with $E\,:\,\mathcal{T}\longrightarrow \textrm{Mod}(\mathcal{U})^{op}$ . We set

\begin{equation*}\mathbb {G}\,:\!=\,I\circ Y\,:\, \textrm {Mod}(\mathcal {U})\longrightarrow \textrm {Mod}(\mathcal {T}).\end{equation*}

Hence, we have the comma category $ \Big ( \textrm{Mod}(\mathcal{T}),\mathbb{G}(\textrm{Mod}(\mathcal{U}))\Big )$ whose objects are the triples $ (A,f,B)$ with $A\in \textrm{Mod}(\mathcal{T}), B\in \textrm{Mod}(\mathcal{U}),$ and $ f\,:\,A\longrightarrow \mathbb{G}(B)$ is a morphism of $ \mathcal{T}$ -modules. A morphism between two objects $ (A,f,B)$ and $ (A^{\prime},f^{\prime},B^{\prime})$ is a pair of morphism $(\alpha,\beta )$ where $\alpha \,:\,A\longrightarrow A^{\prime}$ is a morphism of $\mathcal{T}$ -modules and $\beta \,:\,B\longrightarrow B^{\prime}$ is a morphism of $\mathcal{U}$ -modules such that the diagram

commutes. Now, given $(A,f,B) \in \Big (\textrm{Mod}(\mathcal{T}),\mathbb{G}\textrm{Mod}(\mathcal{U})\Big )$ , we can construct a functor

\begin{equation*}A\amalg _f B\,:\, \boldsymbol{\Lambda } \rightarrow \textbf {Ab}\end{equation*}

defined as follows:

1. (a) For $\left [\begin{smallmatrix} T& \quad\!\! 0 \\[2pt] M & \quad\!\! U \\[2pt] \end{smallmatrix} \right ]\in \boldsymbol{\Lambda }$ we set $\Big (A\amalg _f B\Big )\left(\left (\begin{smallmatrix} T& \quad\!\! 0 \\[2pt] M & \quad\!\! U \\[2pt] \end{smallmatrix} \right )\right) \,:\!=\,A (T)\amalg B(U)\in \textbf{Ab}.$

2. (b) If $\left (\begin{smallmatrix} t& \quad\!\! 0 \\[2pt] m& \quad\!\! u \\[2pt] \end{smallmatrix} \right )\in \textrm{Hom}_{\boldsymbol{\Lambda }}\left(\left (\begin{smallmatrix} T& \quad\!\! 0 \\[2pt] M & \quad\!\! U \\[2pt] \end{smallmatrix} \right ), \left (\begin{smallmatrix} T^{\prime}& \quad\!\! 0 \\[2pt] M & \quad\!\! U^{\prime} \\[2pt] \end{smallmatrix} \right )\right)$ we define the map

   \begin{equation*}\Big (A\amalg _f B\Big )\left(\left (\begin {matrix} t& \quad 0 \\[2pt] m& \quad u \\[2pt] \end {matrix} \right )\right)\,:\!=\,\left (\begin {matrix} A(t)& \quad 0 \\[2pt] m & \quad B(u) \\[2pt] \end {matrix} \right )\,:\, A(T)\amalg B(U)\rightarrow A(T^{\prime})\amalg B(U^{\prime})\end{equation*}
   given by $\left (\begin{smallmatrix} A(t)& 0 \\[2pt] m& B(u) \\[2pt] \end{smallmatrix} \right )\left (\begin{smallmatrix} x \\[2pt] y \\[2pt] \end{smallmatrix} \right )=\left (\begin{smallmatrix} A(t)(x) \\[2pt] m\cdot x+ B(u)(y) \\[2pt] \end{smallmatrix} \right )$ for $(x,y)\in A(T)\amalg B(U)$ , where $m\cdot x\,:\!=\,[f_{T}(x)]_{U^{\prime}}(m)\in B(U^{\prime})$ .

We define $\mathfrak{f}\,:\, \left (\textrm{Mod}(\mathcal{T}),\mathbb{G}(\textrm{Mod}(\mathcal{U}))\right ) \rightarrow \textrm{Mod}(\Lambda )$ as $\mathfrak{f}((A,f,B))=A\amalg _f B$ and it is a covariant functor (see [15, Proposition 3.12]). We have the following Theorem.

Theorem 2.14. [15, Theorem 3.17] There exists an equivalence of categories

\begin{equation*}\mathfrak {f}\,:\, \left (\textrm {Mod}(\mathcal {T}),\mathbb {G}(\textrm {Mod}(\mathcal {U}))\right ) \rightarrow \textrm {Mod}(\Lambda ).\end{equation*}

Hence, given a $\Lambda$ -module $C$ , there exists a pair of functors $C^{(1)}\,:\, \mathcal{T} \rightarrow \textbf{Ab}$ , $C^{(2)}\,:\, \mathcal{U} \rightarrow \textbf{Ab}$ and $C^{(1)} \overset{f}{\longrightarrow } \mathbb{G}(C^{(2)})$ such that $\mathfrak{f}((C^{(1)},f,C^{(2)}))\cong C$ , where we have that $C^{(1)}(T^{\prime})\,:\!=\,C\left (\left( \begin{smallmatrix} T^{\prime} & \quad\!\! 0 \\[2pt] M & \quad\!\! 0 \end{smallmatrix} \right)\right )$ and $C^{(2)}(U^{\prime})\,:\!=\,C\left ( \left(\begin{smallmatrix} 0 & \quad\!\! 0 \\[2pt] M & \quad\!\! U^{\prime} \end{smallmatrix}\right) \right )$ .

Now, we have the following Proposition.

Proposition 2.15. Let $C\in \textrm{Mod}(\Lambda )$ and consider $C^{(1)}\,:\, \mathcal{T} \rightarrow \textbf{Ab}$ and $C^{(2)}\,:\, \mathcal{U} \rightarrow \textbf{Ab}$ and $C^{(1)} \overset{f}{\longrightarrow } \mathbb{G}(C^{(2)})$ such that $\mathfrak{f}((C^{(1)},f,C^{(2)}))\cong C$ . Suppose that there exists exact sequences

Then there exists an exact sequence in $\textrm{Mod}(\Lambda )$

\begin{equation*} {\Lambda \left ( \left (\begin {matrix} T_1 & \quad 0 \\ M & \quad U_1 \end {matrix}\right ), - \right )} \longrightarrow {\Lambda \left ( \left (\begin {matrix} T_0 & \quad 0 \\ M & \quad U_0 \end {matrix}\right ), - \right )} \longrightarrow \mathfrak {f}\left (\left (C^{(1)},f,C^{(2)}\right )\right ) \longrightarrow 0, \end{equation*}

where $P_j\,:\!=\,\Lambda \left (\left ( \begin{smallmatrix} T_j & 0 \\ M & U_j \end{smallmatrix}\right ), - \right )$ is a projective $\Lambda -$ module for $j=0,1$ .

Proof. It is similar to the proof of [15, Proposition 6.3]).

Proposition 2.16 (15, Proposition 5.5). The subcategory $\textrm{proj}(\textrm{Mod}(\Lambda ))$ of $\textrm{Mod}(\Lambda )$ consisting of finitely generated projective $\Lambda$ -modules is equivalent to the subcategory of $( \textrm{Mod}(\mathcal{T}), \mathbb{G}\textrm{Mod}(\mathcal{U}))$ consisting of morphism of $\mathcal{T}$ -modules

\begin{equation*}g\,:\,\mathcal {T}(T,-)\longrightarrow \mathbb {G}( M_T\amalg \mathcal {U}(U,-))\end{equation*}

given by $g\,:\!=\,\big \{[g]_{T^{\prime}}\,:\, \mathcal{T}(T,T^{\prime})\longrightarrow \textrm{Hom}_{\textrm{Mod}(\mathcal{U})}\big (M_{T^{\prime}},M_{T}\amalg \mathcal{U}(U,-)\big )\big \}_{T^{\prime}\in \mathcal{T}}$ , with $[g]_{T^{\prime}}(t)\,:\!=\,\left [ \begin{smallmatrix} \overline{t} \\[2pt] 0 \end{smallmatrix} \right ]\,:\,M_{T^{\prime}}\rightarrow M_{T}\amalg \mathcal{U}(U,-)$ for all $t\in \mathcal{T}(T,T^{\prime})$ , where for $U^{\prime}\in \mathcal{U}$ , the map $\left[\left [ \begin{smallmatrix} \overline{t} \\[2pt] 0 \end{smallmatrix} \right ]\right]_{U^{\prime}}\,:\,M(U^{\prime},T^{\prime})\rightarrow M(U^{\prime},T)\amalg \mathcal{U}(U,U^{\prime})$ is defined by $\left[\left [ \begin{smallmatrix} \overline{t} \\[2pt] 0 \end{smallmatrix} \right ]\right]_{U^{\prime}}(m)\,:\!=\,(M(1_{U}\otimes t^{op})(m),0)=(m\bullet t,0), \forall m\in M(U^{\prime},T).$

Proposition 2.17 (15, Proposition 5.6). A sequence of maps

is exact in $\big ( \textrm{Mod}(\mathcal{T}), \mathbb{G}(\textrm{Mod}(\mathcal{U}))\big )$ if and only if the following are exact sequences

in $\textrm{Mod}(\mathcal{T})$ and $\textrm{Mod}(\mathcal{U})$ , respectively.

3. Triangular matrix categories over quasi-hereditary categories

In this section, we will prove the main result of this paper. First, we recall by [15, Proposition 3.7], that if $\mathcal{U}$ and $\mathcal{T}$ have finite coproducts, then $\Lambda = \left (\begin{smallmatrix} \mathcal{T} & \quad\!\! 0 \\[2pt] M & \quad\!\! \mathcal{U} \end{smallmatrix}\right )$ has finite coproducts. For the convenience of the reader, we will give an idea of how construct finite coproducts in $\Lambda$ . Let $\left (\begin{smallmatrix}T_{1} & \quad\!\! 0 \\[2pt] M & \quad\!\! U_{1} \end{smallmatrix}\right )$ , $\left (\begin{smallmatrix}T_{2} & \quad\!\! 0 \\[2pt] M & \quad\!\! U_{2} \end{smallmatrix}\right )$ two objects in $\Lambda$ . Consider the coproduct $\{u_{i}\,:\,U_{i}\longrightarrow U_{1}\oplus U_{2}\}_{i=1}^{2}$ of the family $\{U_{i}\}_{i=1}^{2}$ in $\mathcal{U}$ and the coproduct $\{v_{i}\,:\,T_{i}\longrightarrow T_{1}\oplus T_{2}\}_{i=1}^{2}$ of the family $\{T_{i}\}_{i=1}^{2}$ in $\mathcal{T}$ . Then we can construct the following morphisms in $\Lambda$

\begin{equation*}\left\{\left (\begin {matrix} u_{i} & \quad 0 \\[2pt] 0 & \quad v_{i} \end {matrix}\right )\,:\, \left (\begin {matrix} U_{i} & \quad 0 \\[2pt] 0 & \quad T_{i} \end {matrix}\right )\longrightarrow \left (\begin {matrix} U_{1}\oplus U_{2} & \quad 0 \\[2pt] 0 & \quad T_{1}\oplus T_{2} \end {matrix}\right )\right\}_{i=1}^{2},\end{equation*}

and it is straightforward to see that this family is a coproduct for the family of objects $\left\{\left (\begin{smallmatrix}T_{i} & \quad\!\! 0 \\[2pt] M & \quad\!\! U_{i} \end{smallmatrix}\right )\right\}_{i=1}^{2}$ . Now, let $\mathcal{U}$ and $\mathcal{T}$ be $\textrm{Hom}$ -finite Krull–Schmidt quasi-hereditary $K$ -categories with respect to filtrations $\{\mathcal{U}_j\}_{0\le j\le n}$ and $\{\mathcal{T}_j\}_{j\ge 0}$ , respectively, consisting of full additive subcategories which are closed under direct summands. Assume that the $\mathcal{U}$ -module $M_T=M(-,T)$ lies in $\mathcal{F}(_{\mathcal{U}}\Delta )$ for all $T\in \mathcal{T}$ ; therefore, $M_T$ is finitely presented since the filtration of $\mathcal{U}$ is finite (see Lemma 2.12 (c)). Thus, $\Lambda = \Big(\begin{smallmatrix} \mathcal{T} & \quad\!\! 0 \\[2pt] M & \quad\!\! \mathcal{U} \end{smallmatrix}\Big)$ is a $\textrm{Hom}$ -finite Krull–Schmidt $K$ -category (see [15, Proposition 6.9]).

Consider the filtration of $\Lambda$ into subcategories $\{\Lambda _j\}_{j\ge 0}$ given by

(3.1) \begin{eqnarray} \Lambda _0&=&\left (\begin{matrix} 0 & \quad 0 \\[2pt] M & \quad 0 \end{matrix}\right );\notag \\ \Lambda _j&=&\left (\begin{matrix} 0 & \quad 0 \\ M & \quad \mathcal{U}_j \end{matrix}\right )\,:\!=\,\left\{\left ( \begin{matrix} 0 & \quad 0 \\ M & \quad U \end{matrix} \right )\,:\, U \in \mathcal{U}_j \right\}, \text{ if $1\le j\le n$};\\[2pt] \Lambda _{n+j}&=&\left (\begin{matrix} \mathcal{T}_{j} & \quad 0 \\ M & \quad \mathcal{U} \end{matrix}\right )=\left\{ \left (\begin{matrix} T & \quad 0 \\ M & \quad U \end{matrix}\right ) \,:\,T\in \mathcal{T}_j, U \in \mathcal{U} \right\}, \text{ if $j\ge 1$}\notag . \end{eqnarray}

It is clear that $\Lambda _j\subseteq \Lambda$ is an additive full subcategory for all $j\ge 0$ . For and object $\left ( \begin{smallmatrix} 0 & \quad\!\! 0 \\[2pt] M & \quad\!\! U \end{smallmatrix} \right )\in \Lambda$ with $U\in \textrm{Ind}\,\,\mathcal{U}_{j}$ we have that

\begin{equation*}\textrm {Hom}_{\Lambda }\left(\left ( \begin {matrix} 0 & \quad 0 \\[2pt] M & \quad U \end {matrix} \right ),\left ( \begin {matrix} 0 & \quad 0 \\[2pt] M & \quad U \end {matrix} \right )\right)=\left ( \begin {matrix} 0 & \quad 0 \\[2pt] 0 & \quad \mathcal {U}(U,U) \end {matrix} \right )\simeq \mathcal {U}(U,U)\end{equation*}

is local. Hence, we get that $\left ( \begin{smallmatrix} 0 & \quad\!\! 0 \\[2pt] M & \quad\!\! U \end{smallmatrix} \right )$ is an indecomposable object in $\Lambda _{j}$ . Similarly, for an object $\left ( \begin{smallmatrix} T & \quad\!\! 0 \\[2pt] M & \quad\!\! 0 \end{smallmatrix} \right )\in \Lambda$ with $T \in \textrm{Ind}\,\,\mathcal{T}_{j}$ , we have that $\left ( \begin{smallmatrix} T & \quad\!\! 0 \\[2pt] M & \quad\!\! 0\end{smallmatrix} \right )$ is an indecomposable object in $\Lambda _{n+j}$ . It follows that

\begin{eqnarray*} \textrm{Ind}\ \Lambda _j& = & \left\{ \left (\begin{matrix} 0 & \quad 0 \\[2pt] M & \quad U \end{matrix}\right ) \,:\, U \in \textrm{Ind} \ \mathcal{U}_j \right\}, \text{ if $1\le j\le n$, and};\\[2pt] \textrm{Ind} \ \Lambda _{n+j}& =& \left\{ \left (\begin{matrix} 0 & \quad 0 \\[2pt] M & \quad U \end{matrix}\right ) \,:\, U \in \textrm{Ind} \ \mathcal{U} \right\}\cup \left\{ \left (\begin{matrix} T & \quad 0 \\[2pt] M & \quad 0 \end{matrix}\right ) \,:\,T\in \textrm{Ind} \ \mathcal{T}_j \right\}, \text{ if $j\ge 1$}. \end{eqnarray*}

In this way, we have that $\Lambda _j$ , for $j\ge 0$ , is closed under direct summands. Moreover,

\begin{eqnarray*} \textrm{Ind}\ \Lambda _{j } - \textrm{Ind} \ \Lambda _{j-1}= \begin{cases} \left \{ \left (\begin{matrix} 0 & \quad 0 \\[2pt] M & \quad U\end{matrix}\right ) \,:\, U\in \textrm{Ind} \ \mathcal{U}_j- \textrm{Ind}\ \mathcal{U}_{j-1}\right\}, \text{ if $1\le j\le n$}, \\[10pt] \left \{ \left (\begin{matrix} T & \quad 0 \\[2pt] M & \quad 0\end{matrix}\right ) \,:\, T\in \textrm{Ind} \ \mathcal{T}_{j-n}- \textrm{Ind}\ \mathcal{T}_{j-n-1}\right \}, \text{ if $j\gt n$}. \end{cases} \end{eqnarray*}

One of the main results of this section is a generalization of a Theorem due to B. Zhu (see [29, Theorem 3.1]). The proof of such Theorem 3.7 will be a consequence of a series of results that are presented below.

Lemma 3.1. Let $\mathcal{U}$ be a quasi-hereditary category with respect to a filtration $\{\mathcal{U}_j\}_{j\ge 0}$ . Let $M$ be a $\mathcal{U}$ -module, and set

\begin{equation*}M^{[j]}\,:\!=\, \textrm {Tr}_{\{\mathcal {U}(U,-)\}_{U\in \mathcal {U}_j } }(M).\end{equation*}

Assume that $M\in \mathcal{F}(_{\mathcal{U}}\Delta )$ . Then for all $U^{\prime}\in \mathcal{U}$ we have that

\begin{equation*}M^{[j]}(U^{\prime})=\left\{ m\,:\, m= M(s)(m^{\prime})\,\, \text {for}\,\, s\in \mathcal {U} (U^{\prime\prime}, U^{\prime}), U^{\prime\prime} \in \mathcal {U}_j\,\,\text {and}\,\, m^{\prime}\in M(U^{\prime\prime}) \right\}.\end{equation*}

Proof. ( $\subseteq$ ). By Yoneda’s isomorphism $Y^{U^{\prime}}\,:\,\textrm{Nat}\left ((U^{\prime},-), M^{[j]}\right )\cong M^{[j]}(U^{\prime})$ , $\eta \mapsto \eta _{U^{\prime}}(1_{U^{\prime}})$ . Let $m\in M^{[j]}(U^{\prime})$ , then there exists $\eta ^m\,:\,\mathcal{U}(U^{\prime},-)\rightarrow M^{[j]}$ such that $\eta ^m_{U^{\prime}}(1_{U^{\prime}})=m$ . On the other hand, by Lemma 2.12, there exists $p\,:\, \mathcal{U}(U^{\prime\prime},-) \rightarrow M^{[j]} \rightarrow 0$ , with $U^{\prime\prime}\in \mathcal{U}_j$ and since $\mathcal{U}(U^{\prime},-)$ is a projective $\mathcal{U}$ -module, there exists a morphism $s\,:\,U^{\prime\prime}\rightarrow U^{\prime}$ for which the following diagram is commutative:

Again by Yoneda’s lemma we have the following commutative diagram:

(3.2)

Let $m^{\prime}\,:\!=\,Y^{U^{\prime\prime}}(p)$ . Since $M^{[j]}$ is a subfunctor of $M$ , we obtain the equality $M^{[j]}(s)(m^{\prime})=M(s)(m^{\prime})$ and hence by the above commutative diagram we have that

\begin{align*} M(s)(m^{\prime})=M^{[j]}(s)(m^{\prime})=M^{[j]}\left(Y^{U^{\prime\prime}}(p)\right) & =Y^{U^{\prime}}\left(\textrm{Nat}(\mathcal{U}(s,-),M^{[j]})(p)\right)\\ & =Y^{U^{\prime}}(p\circ \mathcal{U}(s,-))\\ & =Y^{U^{\prime}}(\eta ^{m})=m. \end{align*}

( $\supseteq$ ). Let $m\in M(U^{\prime})$ and assume there exists $s\,:\,U^{\prime\prime}\rightarrow U^{\prime}$ , with $U^{\prime\prime}\in \mathcal{U}_j$ , such that $ m=M(s)(m^{\prime})$ for some $ m^{\prime}\in M(U^{\prime\prime})$ . Then there exist two morphisms $\eta ^m\,:\,\mathcal{U}(U^{\prime},-)\rightarrow M$ and $p^{m^{\prime}}\,:\,\mathcal{U}(U^{\prime\prime},-) \rightarrow M$ satisfying the following: $Y^{U^{\prime}}(\eta ^{m})=\eta ^m_{U^{\prime}}(1_{U^{\prime}})=m$ and $Y^{U^{\prime\prime}}\left(p^{m^{\prime}}\right)=p^{m^{\prime}}_{U^{\prime\prime}}(1_{U^{\prime\prime}})=m^{\prime}$ . Thus, by using the diagram (3.2) (with $M$ instead of $M^{[j]}$ ), we have that

\begin{align*} Y^{U^{\prime}}\left(p^{m^{\prime}}\circ \mathcal{U}(s,-)\right) =Y^{U^{\prime}}\left(\textrm{Nat}(\mathcal{U}(s,-),M)\left(p^{m^{\prime}}\right)\right) & = M(s)\left(Y^{U^{\prime\prime}}\left(p^{m^{\prime}}\right)\right)\\ & = M(s)(m^{\prime})=m. \end{align*}

Since $Y^{U^{\prime}}$ is an isomorphism, we conclude that $p^{m^{\prime}}\circ \mathcal{U}(s,-)=\eta ^m$ . Note that $\textrm{Im}\left(p^{m^{\prime}}\right)$ is a subfunctor of $M$ and is generated by $\mathcal{U}(U^{\prime\prime},-)$ . Since $U^{\prime\prime}\in \mathcal{U}_j$ , $\textrm{Im}\left(p^{m^{\prime}}\right)$ is contained in the largest $\mathcal{U}$ -submodule of $M$ generated by $\{\mathcal{U}(U,-)\,:\,U\in \mathcal{U}_j\}$ , namely $M^{[j]}$ , we obtain that $\textrm{Im}\left(p^{m^{\prime}}\right)\subset M^{[j]}$ . It follows that

\begin{equation*} m=\eta ^m_{U^{\prime}}(1_{U^{\prime}})=p^{m^{\prime}}_{U^{\prime}}\circ \mathcal {U}(s,-)_{U^{\prime}}(1_{U^{\prime}})=p^{m^{\prime}}_{U^{\prime}}(s) \in \textrm {Im}\left(p^{m^{\prime}}\right)(U^{\prime}) \subset M^{[j]}(U^{\prime}), \end{equation*}

that is, $m \in M^{[j]}(U^{\prime})$ .

Let $\mathcal{U}$ and $\mathcal{T}$ be $\textrm{Hom}$ -finite Krull–Schmidt categories and consider exhaustive filtrations $\{\mathcal{U}_j\}_{0\le j\le n}$ and $\{\mathcal{T}_j\}_{j\ge 0}$ of $\mathcal{U}$ and $\mathcal{T}$ , consisting of full additive subcategories which are closed under direct summands. In the remainder of this section, we will assume that the categories $\mathcal{U}$ and $\mathcal{T}$ are quasi-hereditary categories with respect to the filtrations $\{\mathcal{U}_j\}_{0\le j\le n}$ and $\{\mathcal{T}_j\}_{j\ge 0}$ , respectively, and $M_T\in \mathcal{F}(_{\mathcal{U}}\Delta )$ for all $T\in \mathcal{T}$ .

Proposition 3.2. Let $E=\left (\begin{smallmatrix} T & \quad\!\! 0 \\[2pt] M & \quad\!\! U \end{smallmatrix}\right )$ and $E^{\prime}=\left (\begin{smallmatrix} T^{\prime} & \quad\!\! 0 \\[2pt] M & \quad\!\! U^{\prime} \end{smallmatrix}\right )$ in $\Lambda$ . Then,

\begin{eqnarray*} I_{\Lambda _j}(E,E^{\prime})&=&\begin{pmatrix} 0 & \quad 0 \\[4pt] M^{[j]}_T(U^{\prime}) & \quad I_{\mathcal{U}_j}(U,U^{\prime}) \end{pmatrix}, \text{if } 0\leq j\leq n, \text{and }\\ I_{\Lambda _j}(E,E^{\prime})&=& \begin{pmatrix} I_{\mathcal{T}_{j-n}}(T,T^{\prime}) & \quad 0 \\[4pt] M_T(U^{\prime}) & \quad \mathcal{U}(U,U^{\prime}) \end{pmatrix}, \text{ if } j\gt n. \end{eqnarray*}

Proof. Let $\left (\begin{smallmatrix}f & 0 \\ m & h \end{smallmatrix}\right )\in \textrm{Hom}_{\Lambda }(E,E^{\prime})$ . Therefore, $f\in \textrm{ Hom}_{\mathcal{T}}(T,T^{\prime})$ , $m\in M(U^{\prime},T)$ , and $h\in \textrm{Hom}_{\mathcal{U}}(U,U^{\prime})$ .

Consider the case $0\leq j\leq n$ . The morphism $\left (\begin{smallmatrix}f & 0 \\ m & h \end{smallmatrix}\right )$ lies in $I_{\Lambda _j}(E,E^{\prime})$ if and only if there is a commutative diagram

with $\left (\begin{smallmatrix} 0 & \quad\!\! 0 \\[2pt] M & \quad\!\! U^{\prime\prime} \end{smallmatrix}\right ) \in \Lambda _j$ , $1\leq j \leq n$ . Thus, $U^{\prime\prime} \in \mathcal{U}_j$ and $\left (\begin{smallmatrix}0 & \quad\!\! 0 \\[2pt] 0 & \quad\!\! s \end{smallmatrix}\right )\left (\begin{smallmatrix}0 & \quad\!\! 0 \\[2pt] m^{\prime} & \quad\!\! r \end{smallmatrix}\right )$ = $\left (\begin{smallmatrix}f & \quad\!\! 0 \\[2pt] m & \quad\!\! h \end{smallmatrix}\right )$ ; therefore, $f=0$ , $m=s\bullet m^{\prime}$ and $h=s\circ r$ . It is clear that $h \in I_{\mathcal{U}_j}(U,U^{\prime})$ because $U^{\prime\prime} \in \mathcal{U}_j$ . In this way, we conclude that $\left (\begin{smallmatrix}f & \quad\!\! 0 \\[2pt] m & \quad\!\! h \end{smallmatrix}\right ) \in I_{\Lambda _j}(E,E^{\prime})$ if and only if $h\in I_{\mathcal{U}_j}(U,U^{\prime})$ and $m=s\bullet m^{\prime}= M(s\otimes 1_{T})(m^{\prime})=M_T(s)(m^{\prime})$ (see Definition 2.13) and $f=0$ where $h\,:\, U\xrightarrow{r}U^{\prime\prime}\xrightarrow{s} U^{\prime}$ and $U^{\prime\prime} \in \mathcal{U}_j$ . In other words, $h\in I_{\mathcal{U}_j}(U,U^{\prime})$ and $m\in M_T^{[j]}(U^{\prime})$ , by Lemma 3.1. Thus, $I_{\Lambda _j}(E,E^{\prime})=\left (\begin{smallmatrix} 0 & \quad\!\! 0 \\[2pt] M^{[j]}_T(U^{\prime}) & \quad\!\! I_{\mathcal{U}_j}(U,U^{\prime}) \end{smallmatrix}\right )$ .

Now, consider the case $j\gt n$ . We have that $\left (\begin{smallmatrix}f & \quad\!\! 0 \\[2pt] m & \quad\!\! h \end{smallmatrix}\right )\in I_{\Lambda _j}(E,E^{\prime})$ if and only if there is a commutative diagram

with $\left (\begin{smallmatrix} T^{\prime\prime} & \quad\!\! 0 \\[2pt] M & \quad\!\! U^{\prime\prime} \end{smallmatrix}\right ) \in \Lambda _j$ , for $j=n+(j-n)\gt n$ , that is, $T^{\prime\prime}\in \mathcal{T}_{j-n}$ and $U^{\prime\prime} \in \mathcal{U}$ .

Since $\left (\begin{smallmatrix}s & \quad\!\! 0 \\[2pt] m_2 & \quad\!\! h_2 \end{smallmatrix}\right )\left (\begin{smallmatrix}r & \quad\!\! 0 \\[2pt] m_1 & \quad\!\! h_1 \end{smallmatrix}\right )=\left (\begin{smallmatrix}f & \quad\!\! 0 \\[2pt] m & \quad\!\! h \end{smallmatrix}\right )$ , we get that $f=s\circ r$ , $m=m_2\bullet r+h_2\bullet m_1$ and $h=h_2\circ h_1$ ; moreover, $m\in M(U^{\prime},T)$ , $h\in \mathcal{U}(U,U^{\prime})$ , and $f\in I_{\mathcal{T}_{j-n}}(T,T^{\prime})$ since $T^{\prime\prime}\in \mathcal{T}_{j-n}$ . Therefore, $\left (\begin{smallmatrix}f & \quad\!\! 0 \\[2pt] m & \quad\!\! h \end{smallmatrix}\right )\in I_{\Lambda _j}(E,E^{\prime})$ if and only if $m\in M(U^{\prime},T)$ , $h\in \mathcal{U}(U,U^{\prime})$ , and $f\in I_{\mathcal{T}_{j-n}}(T,T^{\prime})$ .

Proposition 3.3. For each pair $E,E^{\prime}\in \textrm{Ind} \hspace{.1cm} \Lambda _j- \textrm{Ind} \hspace{.1cm} \Lambda _{j-1}$ , we have

\begin{equation*} \textrm {rad}_{\Lambda }(E,E^{\prime})= I_{ \Lambda _{j-1}}(E,E^{\prime}),\quad \forall j\geq 0. \end{equation*}

Proof. The proof is divided in two cases.

Case $1\leq j \leq n$ . Let $E=\left (\begin{smallmatrix} 0 & \quad\!\! 0 \\[2pt] M & \quad\!\! U \end{smallmatrix}\right )$ and $E^{\prime}=\left (\begin{smallmatrix} 0 & \quad\!\! 0 \\[2pt] M & \quad\!\! U^{\prime} \end{smallmatrix}\right )$ for which $U,U^{\prime}\in \textrm{Ind} \hspace{.1cm} \mathcal{U}_j- \textrm{Ind} \hspace{.1cm} \mathcal{U}_{j-1}$ . Therefore, by [15, Proposition 3.8]

\begin{equation*} \textrm {rad}_{\Lambda }\left (\begin {pmatrix} 0 & \quad 0 \\[2pt] M & \quad U \end {pmatrix},\begin {pmatrix} 0 & \quad 0 \\[2pt] M & \quad U^{\prime} \end {pmatrix}\right )=\begin {pmatrix} \textrm {rad}_{\mathcal {T}}(0,0) & \quad 0 \\[2pt] M(U^{\prime},0) & \quad \textrm {rad}_{\mathcal {U}}(U,U^{\prime}) \end {pmatrix}=\begin {pmatrix} 0 & \quad 0 \\[2pt] 0 & \quad \textrm {rad}_{\mathcal {U}}(U,U^{\prime}) \end {pmatrix}. \end{equation*}

On the other hand, by Theorem 2.6, we have $\textrm{rad}_{\mathcal{U}}(U,U^{\prime})=I_{\mathcal{U}_{j-1}}(U,U^{\prime})$ . Therefore, by Proposition 3.2, we conclude that

\begin{equation*} \textrm {rad}_{\Lambda }\left (\begin {pmatrix} 0 & \quad 0 \\[2pt] M & \quad U \end {pmatrix},\begin {pmatrix} 0 & \quad 0 \\[2pt] M & \quad U^{\prime} \end {pmatrix}\right )=I_{\Lambda _{j-1}}\left (\begin {pmatrix} 0 & \quad 0 \\[2pt] M & \quad U \end {pmatrix},\begin {pmatrix} 0 & \quad 0 \\[2pt] M & \quad U^{\prime} \end {pmatrix}\right ). \end{equation*}

Case $j\gt n$ . Let $E=\left (\begin{smallmatrix} T & 0 \\[2pt] M & 0 \end{smallmatrix}\right )$ and $E^{\prime}=\left (\begin{smallmatrix} T^{\prime} & 0 \\[2pt] M & 0 \end{smallmatrix}\right )$ such that the objects $T,T^{\prime}\in \textrm{Ind}\hspace{.1cm} \mathcal{T}_{j-n}-\textrm{Ind}\hspace{.1cm} \mathcal{T}_{j-n-1}$ . By [15, Proposition 3.8] we have

\begin{equation*} \textrm {rad}_{\Lambda }\left (\begin {pmatrix} T & \quad 0 \\[2pt] M & \quad 0 \end {pmatrix},\begin {pmatrix} T^{\prime} & \quad 0 \\[2pt] M & \quad 0 \end {pmatrix}\right )=\begin {pmatrix} \textrm {rad}_{\mathcal {T}}(T,T^{\prime}) & \quad 0 \\[2pt] M(0,T) & \quad \textrm {rad}_{\mathcal {U}}(0,0) \end {pmatrix}=\begin {pmatrix} \textrm {rad}_{\mathcal {T}}(T,T^{\prime}) & \quad 0 \\[2pt] 0 & \quad 0 \end {pmatrix}. \end{equation*}

Again by Theorem 2.6 we know that $\textrm{rad}_{\mathcal{T}}(T,T^{\prime})=I_{\mathcal{T}_{j-n-1}}(T,T^{\prime})$ . It follows from Proposition 3.2 that

\begin{equation*} \textrm {rad}_{\Lambda }\left (\begin {pmatrix} T & \quad 0 \\[2pt] M & \quad 0 \end {pmatrix},\begin {pmatrix} T^{\prime} & \quad 0 \\[2pt] M & \quad 0 \end {pmatrix}\right )=I_{\Lambda _{j-1}}\left (\begin {pmatrix} T & \quad 0 \\[2pt] M & \quad 0 \end {pmatrix},\begin {pmatrix} T^{\prime} & \quad 0 \\[2pt] M & \quad 0 \end {pmatrix}\right ). \end{equation*}

Remark 3.4. For $X= \left (\begin{smallmatrix} T & \quad\!\! 0 \\[2pt] M & \quad\!\! U \end{smallmatrix}\right )\in \Lambda$ , we consider $I_{\Lambda _{j}}(X,-)\in \textrm{Mod}(\Lambda )$ . By Theorem 2.14, there exists $I_{\Lambda _{j}}^{(1)}(X,-)\in \textrm{Mod}(\mathcal{T})$ and $I_{\Lambda _{j}}^{(2)}(X,-)\in \textrm{Mod}(\mathcal{U})$ and a morphism of $\mathcal{T}$ -modules $f\,:\,I_{\Lambda _{j}}^{(1)}(X,-)\longrightarrow \mathbb{G}\left(I_{\Lambda _{j}}^{(2)}(X,-)\right)$ such that

\begin{equation*}I_{\Lambda _{j}}(X,-)\simeq \mathfrak {f}\Big (I_{\Lambda _{j}}^{(1)}(X,-),f,I_{\Lambda _{j}}^{(2)}(X,-)\Big ).\end{equation*}

As a direct result of the above proposition, we have:

Lemma 3.5. Let $X= \left (\begin{smallmatrix} T & \quad\!\! 0 \\[2pt] M & \quad\!\! U \end{smallmatrix}\right )\in \Lambda$ . Let us identify $I_{\Lambda _j}(X,-)$ with its corresponding object $\left (I_{\Lambda _j}^{(1)}(X,-), f, I_{\Lambda _j}^{(2)}(X,-)\right )$ in $(\textrm{Mod}(\mathcal{T}), \mathbb{G}(\textrm{Mod}(\mathcal{U})))$ . Then the following holds:

1. (i) $ I_{\Lambda _j}^{(1)}(X,-)=0$ and $I_{\Lambda _j}^{(2)}(X,-)\cong M_T^{[j]} \coprod I_{\mathcal{U}_j}(U,-)$ , if $0\le j\le n$ ;

2. (ii) $ I_{\Lambda _j}^{(1)}(X,-)\cong I_{\mathcal{T}_{j-n}}(T,-)$ and $I_{\Lambda _j}^{(2)}(X,-)\cong M_T \coprod \mathcal{U}(U,-)$ , if $j\gt n$ .

Proof. Let $T^{\prime}\in \mathcal{T}$ and $U^{\prime}\in \mathcal{U}$ . The proof is divided into two cases.

Case $1\leq j \leq n$ . By Proposition 3.2, we have that

\begin{align*} (1)\quad I_{\Lambda _j}^{(1)}(X,T^{\prime})=I_{\Lambda _j}\left ( X, \left ( \begin{matrix} T^{\prime} & \quad 0 \\[2pt] M & \quad 0 \end{matrix}\right ) \right )=\left (\begin{matrix} 0 & \quad 0 \\[2pt] 0 & \quad 0 \end{matrix}\right ),\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \end{align*}

where the first equality is by Theorem 2.14, the and similarly by Proposition 3.2 and Theorem 2.14 we have that

\begin{align*} (2)\quad I_{\Lambda _j}^{(2)}(X,U^{\prime}) = I_{\Lambda _j}\left ( X, \left (\begin{matrix} 0 & \quad 0 \\[2pt] M & \quad U^{\prime} \end{matrix}\right ) \right ) & =\left (\begin{matrix} 0 & \quad 0 \\[2pt] M^{[j]}(U^{\prime}) & \quad I_{\mathcal{U}_j}(U,U^{\prime}) \end{matrix}\right ) \quad \quad \quad \quad \quad \\[2pt] & \cong M^{[j]}(U^{\prime}) \coprod I_{\mathcal{U}_j}(U,U^{\prime}). \end{align*}

Case $j\gt n$ . By Theorem 2.14, and Proposition 3.2, we have that

\begin{align*} (1)\quad I_{\Lambda _j}^{(1)}(X,T^{\prime})=I_{\Lambda _j}\left ( X, \left (\begin{matrix} T^{\prime} & \quad 0 \\[2pt] M & \quad 0 \end{matrix}\right ) \right ) = \left (\begin{matrix} I_{\mathcal{T}_{j-n}}(T,T^{\prime}) & \quad 0 \\[2pt] M^{[j]}_T(0) & \quad 0 \end{matrix}\right ) \cong I_{\mathcal{T}_{j-n}}(T,T^{\prime}), \end{align*}

\begin{align*} (2)\quad I_{\Lambda _j}^{(2)}(X,U^{\prime})=I_{\Lambda _j}\left ( X, \left (\begin{matrix} 0 & \quad 0 \\[2pt] M & \quad U^{\prime} \end{matrix}\right ) \right ) & = \left (\begin{matrix} I_{\mathcal{T}_{j-n}}(T,0) & \quad 0 \\[2pt] M_T(U^{\prime}) & \quad \mathcal{U}(U,U^{\prime}) \end{matrix}\right )\\[2pt] & \cong M_T(U^{\prime}) \coprod \mathcal{U}(U,U^{\prime}).\quad \quad \quad \quad \end{align*}

Proposition 3.6. Let $X= \left (\begin{smallmatrix} T & 0 \\ M & U \end{smallmatrix}\right )\in \Lambda$ , and assume $M_T\in \mathcal{F}({}_{\mathcal{U}}\Delta )$ for all $T\in \mathcal{T}$ . Then, for all $j\geq 1$ the following exact sequence exists:

with $E_j\in \Lambda _j$ and $E_{j-1}\in \Lambda _{j-1}$ .

Proof. The proof is divided into two cases.

Case $1\leq j \leq n$ . By Lemma 2.12 and Theorem 2.6, there exist exact sequences

with $U^{\prime}_{j}, U^{\prime\prime}_{j}\in \mathcal{U}_j$ , $U^{\prime}_{j-1}, U^{\prime\prime}_{j-1}\in \mathcal{U}_{j-1}$ . Since in functor categories limits and colimits are computed pointwise, we have an exact sequence

with $U_{j-1}=U^{\prime}_{j-1}\coprod U^{\prime\prime}_{j-1}\in \mathcal{U}_{j-1}$ and $U_{j}=U^{\prime}_{j}\coprod U^{\prime\prime}_{j}\in \mathcal{U}_{j}$ . By Remark 3.4, we get that $\mathfrak{f}\left ((I_{\Lambda _j}^{(1)}(X,-),f,I_{\Lambda _j}^{(2)}(X,-)\right )\simeq I_{\Lambda _j}(X,-)$ , and by Lemma 3.5 we have that $ I_{\Lambda _j}^{(1)}(X,-)=0$ and hence $\mathfrak{f}\left ((I_{\Lambda _j}^{(1)}(X,-),0,I_{\Lambda _j}^{(2)}(X,-)\right )\simeq I_{\Lambda _j}(X,-)$ . It follows by Proposition 2.15, that there exists a exact sequence of $\Lambda$ -modules

\begin{equation*}\Lambda \left ( {\left (\begin {matrix} 0 & \quad 0 \\[2pt] M & \quad U_{j-1} \end {matrix}\right )}, -\right )\rightarrow \Lambda \left ( {\left (\begin {matrix} 0 & \quad 0 \\[2pt] M & \quad U_{j} \end {matrix}\right )}, -\right )\rightarrow \mathfrak {f}\left (\left(I_{\Lambda _j}^{(1)}(X,-),0,I_{\Lambda _j}^{(2)}(X,-)\right )\right.\rightarrow 0\end{equation*}

where $\left (\begin{smallmatrix} 0 & \quad 0 \\[2pt] M & \quad U_{j-1} \end{smallmatrix}\right ) \in \Lambda _{j-1}$ and $\left (\begin{smallmatrix} 0 & 0 \\[2pt] M & \quad U_{j} \end{smallmatrix}\right ) \in \Lambda _j$ .

Case $j\gt n$ . Since $M_T\in \mathcal{F}({}_{\mathcal{U}}\Delta )$ , $M_T$ is a finitely presented $\mathcal{U}$ -module, then $M_T\coprod \mathcal{U}(U,-)\cong I_{\Lambda _j}^{(2)}(X,-)$ is finitely presented. Then, there exists an exact sequence of $\mathcal{U}$ -modules

On the other hand, by Theorem 2.6, there exists an exact sequence of $\mathcal{T}$ -modules

with $T_{j-n}\in \mathcal{T}_{j-n}$ and $T_{j-n-1}\in \mathcal{T}_{j-n-1}$ . Thus, we get an exact sequence

\begin{align*} \Lambda \left ( \left (\begin{matrix} T_{j-n-1} & \quad 0 \\[2pt] M & \quad U^{\prime\prime} \end{matrix}\right ), - \right ) \longrightarrow \Lambda \left ( \left (\begin{matrix} T_{j-n} & \quad 0 \\[2pt] M & \quad U^{\prime} \end{matrix}\right ), - \right ) \longrightarrow \mathfrak{f}\left (\left(I_{\Lambda _j}^{(1)},f,I_{\Lambda _j}^{(2)}\right)\right ) \longrightarrow 0, \end{align*}

where $\mathfrak{f}\left(\left(I_{\Lambda _j}^{(1)},f,I_{\Lambda _j}^{(2)}\right)\right)\cong I_{\Lambda _j}(X,-)$ such that $\left (\begin{smallmatrix} T_{j-n-1} & \quad\!\! 0 \\[2pt] M & \quad\!\! U^{\prime\prime} \end{smallmatrix}\right ) \in \Lambda _{j-1}$ and also $\left (\begin{smallmatrix} T_{j-n} & 0 \\[2pt] M & \quad\!\! U^{\prime} \end{smallmatrix}\right ) \in \Lambda _j$ .

Theorem 3.7. Let $\mathcal{U}$ and $\mathcal{T}$ be $\textrm{Hom}$ -finite Krull–Schmidt categories and consider exhaustive filtrations $\{\mathcal{U}_j\}_{0\le j\le n}$ and $\{\mathcal{T}_j\}_{j\ge 0}$ of $\mathcal{U}$ and $\mathcal{T}$ , consisting of full additive subcategories which are closed under direct summands. Suppose that $\mathcal{U}$ and $\mathcal{T}$ are quasi-hereditary categories with respect to the filtrations $\{\mathcal{U}_j\}_{0\le j\le n}$ and $\{\mathcal{T}_j\}_{j\ge 0}$ , respectively. Assume that $M_T\in \mathcal{F}(_{\mathcal{U}}\Delta )$ for all $T\in \mathcal{T}$ . Then $\Lambda = \left(\begin{smallmatrix} \mathcal{T} & \quad\!\! 0 \\ M & \quad\!\! \mathcal{U} \end{smallmatrix}\right)$ is quasi-hereditary with respect to the filtration $\{\Lambda _j\}_{j\ge 0 }$ given in equation (3.1).

Proof. It follows from Theorem 2.6; and Propositions 3.3 and 3.6.

Remark 3.8. Now, let us explain why Theorem 3.7 is a generalization of a Theorem given by B. Zhu in [29]. Suppose that $U$ and $T$ are quasi-hereditary algebras and that ${}_{U}M_{T}$ is a bimodule such that ${}_{U}M \in \mathcal{F}(_{U}\Delta )$ . Consider the triangular matrix algebra $A\,:\!=\,\begin{pmatrix} T & \quad\!\! 0 \\ M & \quad\!\! U \end{pmatrix}$ .

By Remark 2.7(a), we get that $\textrm{proj}(U)$ and $\textrm{proj}(T)$ are quasi-hereditary categories in the sense of the Definition 2.4. Since ${}_{U}M \in \mathcal{F}(_{U}\Delta )$ , we can see that $M$ satisfies the hypothesis in Theorem 3.7. Thus, by Theorem 3.7, we get that $\begin{pmatrix} \textrm{proj}(T) & \quad\!\! 0 \\ M & \quad\!\! \textrm{proj}(U) \end{pmatrix}$ is a quasi-hereditary category. Now, by using [5, Proposition 2.3] in p. 75, we have that $\textrm{proj}(A)= \begin{pmatrix} \textrm{proj}(T) & \quad\!\! 0 \\ M & \quad\!\! \textrm{proj}(U)\end{pmatrix}$ . Therefore, by Remark 2.7(a), we obtain that $A$ is a quasi-hereditary algebra.

3.1. The standard modules in $\textrm{Mod}(\Lambda )$ and $\mathcal{F}(_\Lambda \Delta )$

Assume that $\mathcal{U}$ and $\mathcal{T}$ are $\textrm{Hom}$ -finite Krull–Schmidt and quasi-hereditary $K-$ categories with respect to filtrations $\{\mathcal{U}_j\}_{0\le j\le n}$ and $\{\mathcal{T}_j\}_{j\ge 0}$ . If $M\,:\, \mathcal{U}\otimes _K\mathcal{T}^{op}\rightarrow \textrm{mod} (K)$ is a functor such that $M_T=M(-,T)\,:\,\textrm{Mod}(\mathcal{U})\rightarrow \textrm{mod}(K)$ is finitely presented $\mathcal{U}$ -module, then by Theorem 3.7, the triangular matrix category $\Lambda =(\begin{smallmatrix} \mathcal{T}&0\\ M&\mathcal{U}\end{smallmatrix})$ is a quasi-hereditary $K$ -category with respect to some filtration $\{\Lambda _j\}_{j\ge 0}$ . In this part, we study the relation between the full standard subcategories $_{\mathcal{U}}\Delta$ , $_{\mathcal{T}}\Delta$ , and $_\Lambda \Delta$ of $\textrm{Mod}(\mathcal{U})$ , $\textrm{Mod}(\mathcal{T})$ , and $\textrm{Mod}(\Lambda )$ , respectively. More concretely, we will show in Theorem 3.10, by using the notation of Theorem 2.14, that

\begin{equation*}\mathcal {F}_f(_{\Lambda }\Delta )=\left\{(F^{(1)}, g, F^{(2)})\,:\, F^{(1)}\in \mathcal {F}_f(_{\mathcal {T}}\Delta ) \text { and } F^{(2)}\in \mathcal {F}_f(_{\mathcal {U}}\Delta )\right\};\end{equation*}

see [29, Theorem 3.1]. Recall that given an abelian category $\mathcal{A}$ and $\mathcal{X}\subseteq \mathcal{A}.$ We denote by $\mathcal{X}^{\amalg }$ the class of objects of $\mathcal{A}$ , which are a finite direct sum of objects in $\mathcal{X}.$ We say that $M\in \mathcal{A}$ is $\mathcal{X}$ - $\textbf{filtered}$ if there exists a chain $\{M_{j}\}_{j\ge 0}$ of subobjects of $M$ such that $M_{j+1}/M_{j}\in \mathcal{X}^{\amalg }$ for $j\ge 0.$ In case $M=M_n$ for some $n\in \mathbb N$ , we say that $M$ has a finite $\mathcal{X}$ -filtration. We denote by $\mathcal{F}(\mathcal{X})$ the class of objects that are $\mathcal{X}$ -filtered and by $\mathcal{F}_f(\mathcal{X})$ the class of objects that have a finite filtration.

Regardless, we need the following result.

Proposition 3.9. The functor $\mathfrak{f}\,:\,(\textrm{Mod}(\mathcal{T}), \mathbb{G}(\textrm{Mod}(\mathcal{U}))) \rightarrow \textrm{Mod}( \Lambda )$ induce equivalences of full subcategories:

\begin{equation*} (0,0, {}_{\mathcal {U}}\Delta (j)) \longleftrightarrow \,\, {}_{\Lambda }\Delta (j), \hspace {.3cm} \text {if} \hspace {.2cm} 1\leq j \leq n,\quad and, \end{equation*}

\begin{equation*} ({}_{\mathcal {T}}\Delta (j-n),0,0) \longleftrightarrow \,\, {}_{\Lambda }\Delta (j), \hspace {.3cm} \text {if} \hspace {.2cm} j\gt n. \hspace {1.1cm} \end{equation*}

Proof. First, for $E \in \textrm{Ind} \hspace{.1cm} \Lambda _j - \textrm{Ind} \hspace{.1cm} \Lambda _{j-1}$ , consider $_{\Lambda }\Delta _E=\frac{\Lambda (E,-)}{I_{\Lambda _{j-1}}(E,-)}$ . Then there exists a triple $\left( \Delta _E^{(1)}, g, \Delta _E^{(2)}\right)$ such that $\mathfrak{f}\left(\left( \Delta _E^{(1)}, g, \Delta _E^{(2)}\right)\right)=\,_{\Lambda }\Delta _E$ where $\Delta _E^{(1)}\,:\,\mathcal{T}\rightarrow \textbf{Ab}$ and $\Delta _E^{(2)}\,:\,\mathcal{U}\rightarrow \textbf{Ab}$ . Now, we consider two cases.

Case $1\leq j \leq n$ . Let $T^{\prime}\in \mathcal{T}$ and $E= \left (\begin{smallmatrix} 0 & \quad\!\! 0 \\[2pt] M & \quad\!\! U \end{smallmatrix}\right ) \in \textrm{Ind} \hspace{.1cm} \Lambda _j - \textrm{Ind} \hspace{.1cm} \Lambda _{j-1}$ , with $U \in \textrm{Ind} \hspace{.1cm} \mathcal{U}_j - \textrm{Ind} \hspace{.1cm} \mathcal{U}_{j-1}$ . Then

\begin{equation*} _{\Lambda }\Delta _E^{(1)}(T^{\prime})=\frac {\Lambda \left ( \left (\begin {matrix} 0 & \quad 0 \\[2pt] M & \quad U \end {matrix}\right ), \left (\begin {matrix} T^{\prime} & \quad 0 \\[2pt] M & \quad 0 \end {matrix}\right ) \right )}{I_{\Lambda _{j-1}}\left ( \left (\begin {matrix} 0 & \quad 0 \\[2pt] M & \quad U \end {matrix}\right ), \left (\begin {matrix} T^{\prime} & \quad 0 \\[2pt] M & \quad 0 \end {matrix}\right ) \right )}\cong 0. \end{equation*}

On the other hand, if $U^{\prime} \in \mathcal{U}$ , we get

\begin{equation*} _{\Lambda }\Delta _E^{(2)}(U^{\prime})=\frac {\Lambda \left ( \left (\begin {matrix} 0 & \quad 0 \\[2pt] M & \quad U \end {matrix}\right ), \left (\begin {matrix} 0 & \quad 0 \\[2pt] M^{\prime} & \quad U \end {matrix}\right ) \right )}{I_{\Lambda _{j-1}}\left ( \left (\begin {matrix} 0 & \quad 0 \\[2pt] M & \quad U \end {matrix}\right ), \left (\begin {matrix} 0 & \quad 0 \\[2pt] M & \quad U^{\prime} \end {matrix}\right ) \right )}\cong \frac {\mathcal {U}(U,U^{\prime})}{I_{\mathcal {U}_{j-1}}(U,U^{\prime})}. \end{equation*}

In this way, $_{\Lambda }\Delta _E^{(1)}\cong 0$ and $_{\Lambda }\Delta _E^{(2)}\cong \frac{\mathcal{U}(U,-)}{I_{\mathcal{U}_{j-1}}(U,-)}=\,{}_{\mathcal{U}}\Delta _U$ , where the object $U$ belongs to $\textrm{Ind} \hspace{.1cm} \mathcal{U}_j - \textrm{Ind} \hspace{.1cm} \mathcal{U}_{j-1}$ .

Case $j\gt n$ . Let $T^{\prime} \in \mathcal{T}$ and $E= \left (\begin{smallmatrix} T & \quad\!\! 0 \\[2pt] M & \quad\!\! 0 \end{smallmatrix}\right )$ with $T \in \textrm{Ind} \hspace{.1cm} \mathcal{T}_{j-n} - \textrm{Ind} \hspace{.1cm} \mathcal{T}_{j-n-1}$ . Thus, we obtain that:

\begin{equation*} _{\Lambda }\Delta _E^{(1)}(T^{\prime})=\frac {\Lambda \left ( \left (\begin {matrix} T & \quad 0 \\[2pt] M & \quad 0 \end {matrix}\right ), \left (\begin {matrix} T^{\prime} & \quad 0 \\[2pt] M & \quad 0 \end {matrix}\right ) \right )}{I_{\Lambda _{j-1}}\left ( \left (\begin {matrix} T & \quad 0 \\[2pt] M & \quad 0 \end {matrix}\right ), \left (\begin {matrix} T^{\prime} & \quad 0 \\[2pt] M & \quad 0 \end {matrix}\right ) \right )}\cong \frac {\mathcal {T}(T,T^{\prime})}{I_{\mathcal {T}_{j-n-1}}(T,T^{\prime})}. \end{equation*}

If $U^{\prime} \in \mathcal{U}$ , we obtain

\begin{equation*} _{\Lambda }\Delta _E^{(2)}(U^{\prime})=\frac {\Lambda \left ( \left (\begin {matrix} T & \quad 0 \\[2pt] M & \quad 0 \end {matrix}\right ), \left (\begin {matrix} 0 & \quad 0 \\[2pt] M & \quad U^{\prime} \end {matrix}\right ) \right )}{I_{\Lambda _{j-1}}\left ( \left (\begin {matrix} T & \quad 0 \\[2pt] M & \quad 0 \end {matrix}\right ), \left (\begin {matrix} 0 & \quad 0 \\[2pt] M & \quad U^{\prime} \end {matrix}\right ) \right )}\cong 0. \end{equation*}

Therefore, $_{\Lambda }\Delta _E^{(1)}\cong \frac{\mathcal{T}(T,-)}{I_{\mathcal{T}_{j-n-1}}(T,-)} \cong \,{} _{\mathcal{T}}\Delta _T$ , with $T \in \textrm{Ind} \hspace{.1cm} \mathcal{T}_{n-j} - \textrm{Ind} \hspace{.1cm} \mathcal{T}_{n-j-1}$ and $_{\Lambda }\Delta _E^{(2)}\cong 0$ .

Theorem 3.10. Let $F=\mathfrak{f}\big (F^{(1)}, g, F^{(2)}\big )\in \textrm{mod}(\Lambda )$ , and consider its trace filtration $\{F^{[j]} \}_{j\ge 0}= \left\{\left(\left(F^{[j]}\right)^{(1)}, g^{[j]}, \left(F^{[j]}\right)^{(2)}\right)\right\} _{j\ge 0}$ with respect to $\{\Lambda _j\}_{j\geq 0}$ . Then the following conditions hold.

1. (i) $(F^{[j]})^{(1)}\cong 0$ if $0\le j\le n$ , and $(F^{[j]})^{(2)}\cong F^{(2)}$ if $j\gt n$ .

2. (ii) If $F=\mathfrak{f}\left(\left(F^{(1)},g,F^{(2)}\right)\right)\in \mathcal{ F}(_{\Lambda }\Delta )$ , then we have that $F^{(1)}\in \mathcal{F}(_{\mathcal{T}}\Delta )$ and $F^{(2)}\in \mathcal{F}(_{\mathcal{U}}\Delta )$ .

3. (iii) $\mathcal{ F}_f(_{\Lambda }\Delta )=\left\{\mathfrak{f}\big (F^{(1)}, g, F^{(2)}\big )\,:\, F^{(1)}\in \mathcal{F}_f(_{\mathcal{T}}\Delta ) \text{ and } F^{(2)}\in \mathcal{F}_f(_{\mathcal{U}}\Delta )\right\}$ .

Proof. (i) Since $F\in \textrm{mod}(\Lambda )$ , we have an exact sequence of $\Lambda$ -modules

(3.3) \begin{equation} (X^{\prime},-)\rightarrow (X,-) \rightarrow F\rightarrow 0 \end{equation}

with $X^{\prime}=\left (\begin{smallmatrix} T^{\prime} & \quad\!\! 0 \\[2pt] M & \quad\!\! U^{\prime} \end{smallmatrix}\right )$ and $X=\left (\begin{smallmatrix} T & \quad\!\! 0 \\[2pt] M & \quad\!\! U \end{smallmatrix}\right )$ . By applying $\textrm{Tr}_{\{\Lambda (E,-)\}_{E\in \Lambda _j}}(-)$ to the previous exact sequence and using that each $\Lambda (E,-)$ is a projective $\Lambda$ -module and by Lemma 2.3, we get an exact sequence

(3.4) \begin{equation} I_{\Lambda _j}(X^{\prime},-)\rightarrow I_{\Lambda _j}(X,-)\rightarrow F^{[j]}\rightarrow 0. \end{equation}

By Proposition 2.16 and Theorem 2.14, we identify the exact sequence in (3.3) with the following exact sequence in the comma category $\left (\textrm{Mod}(\mathcal{T}),\mathbb{G}\left(\textrm{Mod}(\mathcal{U})\right)\right )$ :

By Proposition 2.17, we have the exact sequence in $\textrm{Mod}(\mathcal{U})$ :

(3.5) \begin{equation} M_{T^{\prime}}\amalg \mathcal{U}(U^{\prime},-)\rightarrow M_{T}\amalg \mathcal{U}(U,-)\rightarrow F^{(2)}\rightarrow 0. \end{equation}

Second, by Theorem 2.14, we identify the exact sequence in (3.4) with the exact sequence in the comma category $\left (\textrm{Mod}(\mathcal{T}),\mathbb{G}(\textrm{Mod}(\mathcal{U}))\right )$ :

By Proposition 2.17, we get exact sequences

\begin{equation*} I_{{ \Lambda }_j}^{(k)}(X^{\prime},-)\rightarrow I_{{ \Lambda }_j}^{(k)}(X,-)\rightarrow (F^{[j]})^{(k)}\rightarrow 0,\,\,\text {for}\,\, k=1,2,\end{equation*}

in $\textrm{Mod}(\mathcal{T})$ and $\textrm{Mod}(\mathcal{U})$ . By Lemma 3.5(i), we have that $ I_{{ \Lambda }_j}^{(1)}(X^{\prime},-)$ $\cong 0$ and $ I_{{ \Lambda }_j}^{(1)}(X,-) \cong 0$ if $0\le j\le n$ ; therefore, $(F^{[j]})^{(1)}\cong 0$ for $0\le j\le n$ .

If $j\gt n$ , by Lemma 3.5(ii), we have $ I_{{ \Lambda }_j}^{(2)}(X^{\prime},-)\cong M_{T^{\prime}}\amalg \mathcal{U}(U^{\prime},-)$ and $ I_{{ \Lambda }_j}^{(2)}(X,-)\cong M_{T}\amalg \,\,\mathcal{U}(U,-)$ . By the exact sequence (3.5), we get $ (F^{[j]})^{(2)}\cong F^{(2)}$ if $j\gt n$ .

(ii). If $F\in \mathcal{F}(\Delta )$ , we have that $F^{[j]}/F^{[j-1]}$ is a sum of copies of elements in $_{\Lambda } \Delta (j)$ and hence by using item (i) we have that

\begin{equation*} \mathfrak {f}\left(\frac {F^{[j]}}{F^{[j-1]}}\right)\cong \begin {cases} \left (0,0, (F^{[j]})^{(2)}/(F^{[j-1]})^{(2)}\right ), \text { if $1\le j\le n$};\\[3pt] \left ((F^{[j]})^{(1)}/(F^{[j-1]})^{(1)},0, 0\right ), \text { if $j\gt n$}. \end {cases} \end{equation*}

Then (ii) follows by Proposition 3.9.

(iii) Assume that $F=\mathfrak{f}\big (F^{(1)},g,F^{(2)}\big )\in \mathcal{F}_f(_\Lambda \Delta )$ . By (ii), it only remains to prove that if $F^{(1)}\in \mathcal{F}_f(_{\mathcal{T}}\Delta )$ and $F^{(2)} \in \mathcal{F}_f(_{\mathcal{U}}\Delta )$ , then $F\in \mathcal{F}_f(_\Lambda \Delta )$ . In fact, the $\Lambda$ -modules $\mathfrak{f}\big (F^{(1)},0,0\big )$ and $\mathfrak{f}\left(0,0,F^{(2)}\right)$ are in $\mathcal{F}_f(_{\Lambda }\Delta )$ by Proposition 3.9. It follows that we have a short exact sequence

\begin{equation*} 0\rightarrow \mathfrak {f}\big (F^{(1)},0,0\big )\rightarrow \mathfrak {f}\big (F^{(1)},g,F^{(2)}\big )\rightarrow \mathfrak {f}\big ( 0,0,F^{(2)}\big )\rightarrow 0 \ . \end{equation*}

Thus, $ \mathfrak{f}\left(F^{(1)},g,F^{(2)}\right)$ is in $\mathcal{F}_f(_{\Lambda }\Delta )$ since $\mathcal{F}_f(_{\Lambda }\Delta )$ is closed under extensions by Remark 2.11.

We end this section with an example.

Example 3.11. Consider the infinite quivers

Let $\mathcal{U}=K\mathcal{Q}$ and $\mathcal{T}=K\mathcal{R}/\mathcal{J}$ be the path categories of the above quivers, where $\mathcal{J}$ is the ideal in $K\mathcal{R}$ generated by the set of relations

(3.6) \begin{equation} \{\beta _1\alpha _1 \ and\ \alpha _{t+1}\alpha _t, \beta _{t}\beta _{t+1}, \alpha _t\beta _t-\beta _{t+1}\alpha _{t+1}, t\ge 1\}, \end{equation}

First, we will see that $\mathcal{T}$ and $\mathcal{U}$ are quasi-hereditary categories.

Set $\mathcal{T}_0=\{0\}$ , and let $\mathcal{T}_j=\textrm{add} \{ t\,:\, 1\le t\le j\}$ , for $j\ge 1$ . Therefore,

\begin{equation*}\{0\}=\mathcal {T}_0\subset \mathcal {T}_1\subset T_2 \subseteq \cdots \cdots \end{equation*}

is a filtration of $\mathcal{T}$ into additively closed subcategories.

1. (i) It is clear that $\textrm{rad}_{\mathcal{T}}(1,1)=0$ because $\beta _1\alpha _1=0$ . Since we have that $\textrm{Ind} \mathcal{T}_j-\textrm{Ind} \mathcal{T}_{j-1}=\{j\}$ for all $j\ge 1$ , we conclude the following: $\textrm{rad}_{\mathcal{T}}(j,j)=(\beta _j\alpha _j)= (\alpha _{j-1}\beta _{j-1})=I_{\mathcal{T}_{j-1}}(j,j)$ .

2. (ii) $I_{\mathcal{T}_1}(1,-)\cong \mathcal{T}(1,-)$ , $I_{\mathcal{T}_1}(2,-)\cong \mathcal{T}(1,-)$ and $I_{\mathcal{T}_1}(j,-)=0$ , if $j\ge 3$ .

For $j\ge 2$ , we can readily check that there exists an exact sequence

\begin{equation*} 0\rightarrow I_{\mathcal {T}_{j-1}}(j,-)\rightarrow \mathcal {T}(j,-) \rightarrow I_{\mathcal {T}_{j}}(j+1,-)\rightarrow 0\end{equation*}

and $I_{\mathcal{T}_j}(j+t,-)=0$ if $t\ge 2$ .

By Theorem 2.6 , we have that $\mathcal{T}$ is quasi-hereditary. Now, we define the following subcategories $\mathcal{U}_0=\{0\}$ , $\mathcal{U}_1=\textrm{add}\{ j^{\prime} \in \mathbb{N}\,:\, j\,\,\text{is odd }\}$ and consider $\mathcal{U}_2=\textrm{add}\{ j^{\prime}\,:\, j \in \mathbb{N}\}=K\mathcal{Q}$ . Thus, $K\mathcal{Q}$ is quasi-hereditary with respect to the finite filtration $\{0\}=\mathcal{U}_0\subset \mathcal{U}_1\subset \mathcal{U}_2=K\mathcal{Q}$ . The condition (i) in Definition 2.6 , clearly holds since $\textrm{rad}_{\mathcal{U}}(E,E^{\prime})=I_{\mathcal{U}_{j-1}}(E,E^{\prime})=0$ for all pairs $E,E^{\prime}\in \textrm{Ind}\ \mathcal{U}_j-\textrm{Ind}\ \mathcal{U}_{j-1}$ and $j=1,2$ . On the other hand, $I_{\mathcal{U}_1}(j,-)=\mathcal{U}(j,-)$ if $j$ is odd and $I_{\mathcal{U}_1}(j,-)\cong \mathcal{U}(j-1,-)\oplus \mathcal{U}(j+1,-)=\mathcal{U}((j-1)\oplus (j+1),-)$ if $j$ is even. Then in this case we have that condition (ii) in Theorem 2.6 also holds and hence $K\mathcal{Q}$ is quasi-hereditary.

Now, by a result of Z. Leszczyński given in [17 , Lemma 1.3], we have that $(K\mathcal{Q})\otimes _{K} (K\mathcal{R}/\mathcal{J})^{op}\simeq K(\mathcal{Q}\times \mathcal{R}^{op})/0\square \mathcal{J}$ where $K(\mathcal{Q}\times \mathcal{R}^{op})$ is the product quiver given in p. 145 in [17]; and $0\square \mathcal{J}$ is generated by the sets of relations $Q_0\times \mathcal{J}$ and

\begin{equation*}\left\{(\gamma,t(\alpha ))(s(\gamma ),\alpha )-(t(\gamma ),\alpha )(\gamma,s(\alpha )), (\gamma,t(\beta ))(s(\gamma ),\beta )-(t(\gamma ),\beta )(\gamma,s(\beta ))\right\},\end{equation*}

where $\gamma \in Q_1$ and $\alpha, \beta \in R_1$ . A functor $M\,:\, K\mathcal{Q}\otimes K(\mathcal{R}/\mathcal{J})^{op}\rightarrow \textrm{mod}\ K$ can be identified with a functor $M\,:\,K(\mathcal{Q}\times \mathcal{R}^{op})/0\square \mathcal{J}\rightarrow \textrm{mod}\ K$ . In this example, we consider $M$ as the following representation on the right below:

We see that $M(1^{\prime},1)=K^2$ , $M(1^{\prime},2)=K$ . Thus, the functor $M$ satisfies that $M_T\,:\,K\mathcal{Q}\rightarrow \textrm{mod}\ K$ is projective for all $T$ since we have that $M_{1}\cong \mathcal{U}(1,-)^2$ , $M_2\cong \mathcal{U}(1,-)$ , and $M_t\cong 0$ , for all $t\gt 2$ , which are all in $\mathcal{F}(_{\mathcal{U}}\Delta )$ because $\mathcal{U}$ is quasi-hereditary.

In this way using results in [17], we can see that the matrix category $\left ( \begin{smallmatrix} \mathcal{T}&0\\ M&\mathcal{U} \end{smallmatrix} \right )$ is equivalent to the path category of the quiver $Q^{\prime\prime}$ given below, modulo the ideal generated by the set of relations (3.6) and is quasi-hereditary with respect to the filtration

\begin{equation*}\{0\}=\Lambda _0\subset \Lambda _1\subset \Lambda _2\subset \Lambda _3\cdots, \end{equation*}

where $\Lambda _1=\mathcal{U}_1$ , $\Lambda _2=\mathcal{U}_2$ and $\Lambda _{j+2}= \textrm{add}\left ( \{ j^{\prime}\,:\, j \in \mathbb{N}\}\cup \{ t\in \mathbb N\,:\, 1\le t\le j\} \right )$ and