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 [Reference Assem, Simson and Skowrónski1] and [Reference Barot6] 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 [Reference Mitchell22].

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 [Reference Bautista, Liu and Paquette7].

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 [Reference Auslander2]. 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 [Reference Auslander2, Reference Auslander, Platzeck and Todorov3], $(\mathcal{C}, \textbf{Ab})$ 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 [Reference Ortiz23].

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.

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 [Reference Ortiz23, 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. [Reference Ortiz23, 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.

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 [Reference Ortiz23] will be useful in the remainder of this work.

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 [Reference Krause13, 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 [Reference Krause14, 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 [Reference Krause14, 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 [Reference Dlab and Ringel11, Statement 9] we conclude that $A$ is quasi-hereditary. Similarly, by using [Reference Krause14, 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 [Reference Auslander and Reiten4, 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 [Reference Dlab and Ringel11, 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.

Proof. Suppose that $\mathcal{C}$ is quasi-hereditary with respect to $\{\mathcal{B}_j\}_{j\geq 0}$ . By [Reference Ortiz23, 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 [Reference Rodríguez-Valdés, Sandoval-Miranda and Santiago-Vargas26, Proposition 4.12]. Then, this implies that $B_{j}$ is functorially finite by [Reference Rodríguez-Valdés, Sandoval-Miranda and Santiago-Vargas26, 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 [Reference Ortiz23, 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 [Reference Rodríguez-Valdés, Sandoval-Miranda and Santiago-Vargas26, 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 [Reference Ortiz23, 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.

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.

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 [Reference Dlab9, Proposition A.3.2]).

Triangular Matrix Categories. In this section, we recall some results from the article [Reference Leszczyński15]. The notion of triangular matrix categories is introduced in [Reference Leszczyński15, Reference León-Galeana, Ortíz-Morales and Santiago16] in order to define the analogous of the triangular matrix algebras to the context of rings with several objects.

Definition 2.13. [Reference Leszczyński15, 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 [Reference Leszczyński15, Proposition 3.12]). We have the following Theorem.

Theorem 2.14. [Reference Leszczyński15, 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 [Reference Leszczyński15, Proposition 6.3]).

Proposition 2.16 (Reference Leszczyński15, 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 (Reference Leszczyński15, 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.

3.1. 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 [Reference Šťovíček29, 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 $_{\Lambda }\Delta _E^{(2)}\cong 0$ .

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 [Reference León-Galeana, Ortíz-Morales and Santiago17 , 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 [Reference León-Galeana, Ortíz-Morales and Santiago17]; 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 [Reference León-Galeana, Ortíz-Morales and Santiago17], 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