2 Torsion pairs and $\tau $ -tilting theory

In this section, we recall necessary background on torsion pairs and $\tau $ -tilting theory. Throughout this article, $\Lambda $ shall always denote a finite-dimensional basic algebra over a field K, and $\mathsf {mod} \Lambda $ denotes the category of finitely generated left $\Lambda $ -modules. Furthermore, the Auslander–Reiten (AR) translate in $\mathsf {mod} \Lambda $ is denoted by $\tau $ .

The study of $\tau $ -tilting theory has become instrumental in the study of finitely generated $\Lambda $ -modules since its inception in [Reference Adachi, Iyama and Reiten1]. We follow the notation of [Reference Buan and Marsh9], and denote $\mathcal C(\mathsf {mod}\Lambda ):= \mathsf {mod}\Lambda \sqcup \mathsf {mod}\Lambda [1] \subseteq \mathcal {D}^b(\mathsf {mod}\Lambda )$ , where $\mathcal {D}^b(\mathsf {mod}\Lambda )$ denotes the bounded derived category of $\Lambda $ . An (usually assumed basic) object $U = M\sqcup P[1]\in \mathcal C(\mathsf {mod}\Lambda )$ is called a support $\tau $ -rigid pair if:

1. 1. $M \in \mathsf {mod}\Lambda $ satisfies $\operatorname {{\mathrm {Hom}}}(M,\tau M) = 0$ .

2. 2. $P \in \mathsf {mod}\Lambda $ is projective and satisfies $\operatorname {{\mathrm {Hom}}}(P,M) = 0$ .

If U is basic, we denote by $\mathsf {rk}(U)$ the number of indecomposable direct summands of U (up to isomorphism). If $\mathsf {rk}(U) = \mathsf {rk}(\Lambda )$ , then U is called support $\tau $ -tilting. When $P = 0$ , the module $U = M$ can be referred to as a $\tau $ -rigid (or $\tau $ -tilting if it is sincere and $\mathsf {rk}(U) = \mathsf {rk}(\Lambda )$ ) module.

By a subcategory of $\mathsf {mod}\Lambda $ , we shall always mean a full subcategory which is closed under isomorphisms. Given such a subcategory $\mathcal {A}\subseteq \mathsf {mod}\Lambda $ , we denote by $\mathcal P(\mathcal {A})$ the category of modules which are ext-projective in $\mathcal {A}$ . That is, $Q \in \mathcal P(\mathcal {A})$ if and only if $\operatorname {{\mathrm {Ext}}}^1(Q,X) = 0$ for all $X \in \mathcal {A}$ .

Moreover, given a subcategory $\mathcal {A}\subseteq \mathsf {mod}\Lambda $ , we denote by $\mathsf {Gen}(\mathcal {A})$ (resp. $\mathsf {Cogen}(\mathcal {A})$ ) the subcategory of $\mathsf {mod}\Lambda $ consisting of objects which are factors (resp. subobjects) of direct sums of objects in $\mathcal {A}$ . We likewise denote by $\mathsf {Filt}(\mathcal {A})$ the subcategory of modules which admit finite filtrations whose subsequent subfactors all lie in $\mathcal {A}$ . Given a module $X \in \mathsf {mod}\Lambda $ , we define $\mathsf {Gen} X:= \mathsf {Gen}(\mathsf {add} X)$ , and so forth, where $\mathsf {add} X$ is the subcategory of direct summands of finite direct sums of X.

For any subcategory $\mathcal {A}\subseteq \mathsf {mod}\Lambda $ , we associate two additional subcategories:

Given a module $X \in \mathsf {mod}\Lambda $ , we likewise have $X^{\perp } = (\mathsf {add} X)^{\perp }$ and

.

Finally, we recall that a subcategory $\mathcal {A} \subseteq \mathsf {mod}\Lambda $ is called functorially finite if for all $X \in \mathsf {mod}\Lambda $ :

1. 1. There exists $A_X \in \mathcal {A}$ and $a_X:A_X\rightarrow X$ such that every morphism with source in $\mathcal {A}$ and target X factors through $a_X$ . The morphism $a_X$ is called a right $\mathcal {A}$ -approximation.

2. 2. There exists $A^X \in \mathcal {A}$ and $a^X:X\rightarrow A^X$ such that every morphism with source X and target in $\mathcal {A}$ and target X factors through $a^X$ . The morphism $a^X$ is called a left $\mathcal {A}$ -approximation.

We are now ready to discuss torsion pairs. A torsion pair is a pair $({\mathcal T},{\mathcal F})$ of subcategories of $\mathsf {mod}\Lambda $ such that ${\mathcal T}^{\perp } = {\mathcal F}$ and . In this case, we call ${\mathcal T}$ a torsion class and ${\mathcal F}$ a torsion-free class. It is well known that a pair $({\mathcal T},{\mathcal T}^{\perp })$ (resp. ) is a torsion pair if and only if ${\mathcal T}$ is closed under extensions and quotients (resp. ${\mathcal F}$ is closed under extensions and subobjects). Moreover, given a torsion pair $({\mathcal T},{\mathcal F})$ , we have that ${\mathcal T}$ is functorially finite if and only if ${\mathcal F}$ is functorially finite (see [Reference Smalø25]).

If $({\mathcal T},{\mathcal F})$ is a torsion pair, then every $M \in \mathsf {mod}\Lambda $ admits a unique exact sequence of the form

(1) $$ \begin{align} 0\rightarrow t_{\mathcal T}(M)\xrightarrow{\iota} M\xrightarrow{q} f_{\mathcal F}(M)\rightarrow 0 \end{align} $$

with $t_{\mathcal T}(M) \in {\mathcal T}$ and $f_{\mathcal F}(M) \in {\mathcal F}$ . In particular, the map $\iota $ is a minimal right ${\mathcal T}$ -approximation and the map q is a minimal left ${\mathcal F}$ -approximation. We note that the operations $t_{\mathcal T}(-)$ and $f_{{\mathcal F}}(-)$ are both functorial.

We will need the following observation for our discussion of Example 4.9.

We now turn our attention to the well-established relationship between torsion pairs and support $\tau $ -rigid objects.

Torsion pairs are closely related to support $\tau $ -rigid objects, as the following shows.

Before continuing, we recall the following characterization of Auslander and Smalø, which will be useful in several of our proofs.

It is implicit in Theorem 2.2 that any basic support $\tau $ -rigid object is the direct summand of at least one support $\tau $ -tilting object. In particular, we have the following.

The module $B_U$ in Theorem 2.4 is called the Bongartz complement of U. Following, for example, [Reference Buan and Marsh10] and [Reference Demonet, Iyama, Reading, Reiten and Thomas11], we refer to $C_U$ in Theorem 2.4 as the co-Bongartz complement of U.

We next recall two results which give us a “canonical decomposition” of a support $\tau $ -tilting pair. The first can be seen as a combination of [Reference Ingalls and Thomas20, Lem. 2.8] and [Reference Marks and Šťovíček23, Lem. 3.7]. We recall that a module X in a subcategory $\mathcal {A}\subseteq \mathsf {mod}\Lambda $ is called split projective (in $\mathcal {A}$ ) if every epimorphism in $\mathcal {A}$ with target X is split.

The second result relates the direct summands $M_s$ and $M_{ns}$ to Bongartz and co-Bongartz complements.

Proposition 2.7. Consider the setup in Lemma 2.6. Then $M_s$ is the Bongartz complement of $M_{ns}\sqcup P[1]$ and $M_{ns}\sqcup P[1]$ is the co-Bongartz complement of $M_s$ . In particular, we have

We conclude this section with a brief description of the dual theory of $\tau ^{-1}$ -tilting. In order to state this in our context, given an indecomposable stalk complex $M[m] \in \mathcal {D}^b(\mathsf {mod}\Lambda )$ with $M \in \mathsf {mod}\Lambda $ , we denote

$$ \begin{align*}\overline{\tau}(M[m]) := \begin{cases} \tau M[m], & M \notin \mathcal P(\mathsf{mod}\Lambda),\\ \nu M[m-1], & M \in \mathcal P(\mathsf{mod}\Lambda),\end{cases}\end{align*} $$

where $\nu $ denotes the Nakayama functor. We then say a (usually assumed basic) object $U = I[-1]\sqcup M \in \mathcal C(\mathsf {mod}\Lambda )[-1]$ is support $\tau ^{-1}$ -rigid if:

1. 1. $M \in \mathsf {mod}\Lambda $ and $\operatorname {{\mathrm {Hom}}}(\tau ^{-1} M, M) = 0$ .

2. 2. $I \in \mathsf {mod}\Lambda $ is injective and $\operatorname {{\mathrm {Hom}}}(M,I) = 0$ .

We likewise say U is $\tau ^{-1}$ -tilting if U is basic and $\mathsf {rk}(U) = \mathsf {rk}(\Lambda )$ . It is shown in [Reference Adachi, Iyama and Reiten1, Sec. 2.2] that $U \in \mathcal C(\mathsf {mod}\Lambda )[-1]$ is support $\tau ^{-1}$ -rigid (resp. support $\tau ^{-1}$ -tilting) if and only if there exists some support $\tau $ -rigid (resp. support $\tau $ -tilting) $V \in \mathcal C(\mathsf {mod}\Lambda )$ such that $U = \overline {\tau }V$ .

3 Wide subcategories

In this section, we recall the definition and basic properties of wide subcategories and discuss important classes of examples of functorially finite wide subcategories: $\tau $ -perpendicular subcategories, left-/right-finite wide subcategories, and Serre subcategories.

Recall that wide subcategories are exactly embedded abelian subcategories, and that a subcategory ${\mathcal W}\subseteq \mathsf {mod}\Lambda $ is wide if and only if it is closed under extensions, kernels, and cokernels. It is well known that a wide subcategory ${\mathcal W}\subseteq \mathsf {mod}\Lambda $ is functorially finite if and only if it is equivalent to $\mathsf {mod}\Lambda _{\mathcal W}$ for some basic finite-dimensional algebra $\Lambda _{\mathcal W}. $ (This is made explicit in [Reference Enomoto12, Prop. 4.12].) Given such a wide subcategory, we denote by $\mathsf {rk}({\mathcal W})$ the number of simple objects in ${\mathcal W}$ (or equivalently simple modules in $\mathsf {mod}\Lambda _{\mathcal W}$ ) up to isomorphism. We note that if $P \in {\mathcal W}$ is basic and $\mathcal P({\mathcal W}) = \mathsf {add}(P)$ , then $\mathsf {rk}({\mathcal W}) = \mathsf {rk}(P)$ . In particular, $\mathsf {rk}(\Lambda ) = \mathsf {rk}(\mathsf {mod}\Lambda )$ .

As wide subcategories are abelian categories in their own right, they have their own wide subcategories, torsion classes, and torsion-free classes. We will be concerned with such subcategories in the sequel, and so the following well-known and easily proved fact is useful.

We are now ready to define our main categories of interest.

Definition 3.2. A full subcategory ${\mathcal W} \subseteq \mathsf {mod}\Lambda $ is called a $\tau $ -perpendicular subcategory if there exists a support $\tau $ -rigid object $U = M\sqcup P[1] \in \mathcal C(\mathsf {mod}\Lambda )$ such that ${\mathcal W} = \mathcal J(U)$ , where

(2)

Such categories were first considered by Jasso [Reference Jasso22], who proved that they are equivalent to module categories, and hence, they are functorially finite. Actually, Jasso explicitly dealt with the case $P=0$ , but his proofs and statements can be easily modified. This is mentioned explicitly in [Reference Demonet, Iyama, Reading, Reiten and Thomas11], where it is also shown that such categories are in fact wide. Summarizing, we have the following.

By identifying a $\tau $ -perpendicular subcategory ${\mathcal W} = \mathcal J(U)$ with a module category, one can consider the $\tau $ -tilting theory of ${\mathcal W}$ . That is, we consider the category $\mathcal C({\mathcal W}) := {\mathcal W}\sqcup {\mathcal W}[1] \subseteq \mathcal C(\mathsf {mod}\Lambda )$ . We then say an object $N\sqcup Q[1] \in \mathcal C({\mathcal W})$ is support $\tau $ -rigid in ${\mathcal W}$ if:

1. 1. $N \in {\mathcal W}$ satisfies $\operatorname {{\mathrm {Hom}}}(N,\tau _{\mathcal W} N) = 0$ , where $\tau _W$ denotes the AR translate in ${\mathcal W}$ .

2. 2. $Q \in {\mathcal W}$ is projective in ${\mathcal W}$ and satisfies $\operatorname {{\mathrm {Hom}}}(Q,N) = 0$ .

We emphasize that, in general, objects which are projective in ${\mathcal W}$ may not be projective in $\mathsf {mod}\Lambda $ . Likewise, we may have that $\tau _{\mathcal W} N \not \cong \tau N$ , so in general, we can have modules which are not $\tau $ -rigid in $\mathsf {mod} \Lambda $ , but still are $\tau $ -rigid in ${\mathcal W}$ . However, it is a direct consequence of Proposition 2.3 and the fact that ${\mathcal W}$ is an exactly embedded subcategory that $\tau $ -rigid (or projective) objects in $\mathsf {mod} \Lambda $ remain $\tau $ -rigid (or projective) in ${\mathcal W}$ .

Note also that since ${\mathcal W}$ is an exact subcategory of $\mathsf {mod} \Lambda $ , the category $\mathcal C({\mathcal W})$ can be considered as sitting either inside $\mathcal C(\mathsf {mod}\Lambda ) \subseteq \mathcal {D}^b(\mathsf {mod}\Lambda )$ (as we have defined it) or inside of $\mathcal {D}^b({\mathcal W})$ . Indeed, for $X, Y \in {\mathcal W}$ , we have a canonical isomorphism

$$ \begin{align*}\operatorname{{\mathrm{Hom}}}_{\mathcal{D}^b(\mathsf{mod}\Lambda)}(X,Y[1]) = \operatorname{{\mathrm{Hom}}}_{\mathcal{D}^b({\mathcal W})}(X,Y[1]).\end{align*} $$

Continuing in this way, if $V = N\sqcup Q[1]$ is $\tau $ -rigid in ${\mathcal W}$ , we denote

(3)

We also have the following dual concept.

Definition 3.4. A subcategory ${\mathcal W} \subseteq \mathsf {mod}\Lambda $ is called a $\tau ^{-1}$ -perpendicular subcategory if there exists a support $\tau ^{-1}$ -rigid object $U = I[-1]\sqcup M \in \mathcal C(\mathsf {mod}\Lambda )[-1]$ such that

We show as part of Theorem 1.1 that $\tau ^{-1}$ -perpendicular subcategories and $\tau $ -perpendic- ular subcategories coincide. Moreover, it will be a consequence of Theorem 1.1 that not all functorially finite wide subcategories are $\tau $ -perpendicular subcategories (see Example 4.9).

We proceed to discuss another central class of functorially finite wide subcategories. These arise from applying the so-called Ingalls–Thomas bijections [Reference Ingalls and Thomas20, Reference Marks and Šťovíček23] to functorially finite torsion classes and torsion-free classes. We recall these constructions now.

One of the key results of [Reference Ingalls and Thomas20] (hereditary case) and [Reference Marks and Šťovíček23] (general case) is that for any wide subcategory ${\mathcal W}$ , one has

$$ \begin{align*}{\mathcal W}_L(\mathsf{Filt}\mathsf{Gen}({\mathcal W})) = {\mathcal W} = {\mathcal W}_R(\mathsf{Filt}\mathsf{Cogen}({\mathcal W})).\end{align*} $$

They also show that $\mathsf {Filt}\mathsf {Gen}(W_L({\mathcal T})) = {\mathcal T}$ (resp. $\mathsf {Filt}\mathsf {Cogen}{\mathcal W}_R({\mathcal F}) ={\mathcal F}$ ), when ${\mathcal T}$ (resp. ${\mathcal F}$ ) is functorially finite. Following Asai [Reference Asai2], a wide subcategory ${\mathcal W} \subseteq \mathsf {mod}\Lambda $ is called left finite (resp. right finite) if it is of the form ${\mathcal W}_L({\mathcal T})$ (resp. ${\mathcal W}_R({\mathcal F})$ ) for some functorially finite torsion class ${\mathcal T}$ (resp. torsion-free class ${\mathcal F}$ ). It is straightforward that if ${\mathcal W}$ is either left finite or right finite, then it is functorially finite. The converse, however, does not hold in general (see [Reference Asai2, Exam. 3.13] or Example 4.9).

We conclude this section by discussing a well understood class of functorially finite wide subcategories, namely, the Serre subcategories. A subcategory $\mathcal {S}$ is Serre if for any short exact sequence

$$ \begin{align*}0\rightarrow X\rightarrow Y \rightarrow Z\rightarrow 0\end{align*} $$

in $\mathsf {mod}\Lambda $ , we have $Y \in \mathcal {S}$ if and only if $X,Z \in \mathcal {S}$ . That is, $\mathcal {S}$ is closed under extensions, quotients, and subobjects.

Serre subcategories are indeed examples of wide subcategories. In fact, they are also both torsion classes and torsion-free classes, as the following shows.

As useful consequences, we obtain the following corollaries.

4 Characterizing $\tau $ -perpendicular subcategories

In this section, we give the proof of Theorem 1.1, restated as Theorem 4.5. This characterizes $\tau $ -perpendicular subcategories as precisely the Serre subcategories of left-finite and right-finite wide subcategories of $\mathsf {mod}\Lambda $ .

The following technical result will be useful for the proof.

We now start building toward our proof of Theorem 1.1 with the following lemmas.

We note that (1) also appears as [Reference Yurikusa26, Eq. 1.2].

Proof. (1) It is shown in [Reference Marks and Šťovíček23, Lem. 3.8] that . Moreover, by Proposition 2.7, we have that . This proves the result.

(2) First, let X be a simple object of ${\mathcal W}_R({\mathcal F})$ . (Note that X is not necessarily simple in $\mathsf {mod}\Lambda $ .) We will show that $X \in \mathcal J(M_s)$ . Since $\mathcal J(M_s)$ is closed under extensions, this will imply that ${\mathcal W}_R({\mathcal F}) \subseteq \mathcal J(M_s)$ .

We first note that ${\mathcal W}_R({\mathcal F}) \subseteq {\mathcal F} = M_s^{\perp }$ , so we need only show that $\operatorname {{\mathrm {Hom}}}(X,\tau M_s) = 0$ . Suppose to the contrary that $\operatorname {{\mathrm {Hom}}}(X,\tau M_s) \neq 0$ . By Proposition 2.3, this means there exists $X' \in \mathsf {Gen} X$ and a nonsplit exact sequence of the form

$$ \begin{align*}0 \rightarrow X' \rightarrow E\rightarrow M_s\rightarrow 0.\end{align*} $$

By Proposition 4.1, we note that $X'$ cannot be a proper quotient of X. Indeed, if this were the case, we would have , a contradiction. Therefore, we can assume that $X' = X$ . Applying Proposition 4.1 again, this implies that $E \in {\mathcal T}$ . Since $M_s$ is split projective in ${\mathcal T}$ , this is a contradiction.

Now, let $Y \in \mathcal J(M_s)$ . It is clear that $Y \in {\mathcal F} = M_s^{\perp }$ . Thus, let $Z \in {\mathcal F} = M_s^{\perp }$ and $g:Y\rightarrow Z$ . We then have an exact sequence

$$ \begin{align*}0 = \operatorname{{\mathrm{Hom}}}(M_s,Z)\rightarrow \operatorname{{\mathrm{Hom}}}(M_s,\operatorname{{\mathrm{coker}}} g)\rightarrow \operatorname{{\mathrm{Ext}}}^1(M_s,\operatorname{{\mathrm{image}}} g) = 0,\end{align*} $$

where the last term is zero by Proposition 2.3 and the fact that $\operatorname {{\mathrm {image}}} g$ is a quotient of Y. We conclude that $\operatorname {{\mathrm {coker}}} g \in {\mathcal F}$ , and therefore, $Y \in {\mathcal W}_R({\mathcal F})$ . This completes the proof.

Proof. Let $U = M\sqcup P[1] \in \mathcal C(\mathsf {mod}\Lambda )$ be support $\tau $ -rigid, and let $\mathcal {S}$ be a Serre subcategory of $\mathcal J(U)$ . By Proposition 3.6, there exists $Q \in \mathcal P(\mathcal J(U))$ so that $\mathcal {S} = Q^{\perp } \cap \mathcal J(U)$ . It follows from [Reference Jasso22, Prop. 3.14] and Theorem 2.2 that $Q = f_{M^{\perp }}(B)$ for some direct summand B of the Bongartz complement $B_U$ of U. We then have an exact sequence

$$ \begin{align*}0 \to t_{\mathsf{Gen} M}(B) \to B \to Q \to 0,\end{align*} $$

and since $t_{\mathsf {Gen} M}(B)$ is in $\mathsf {Gen} M$ , we have $\operatorname {{\mathrm {Hom}}}(t_{\mathsf {Gen} M}(B), \mathcal J(U)) = 0$ . Hence, we have $\mathcal {S} = Q^{\perp } \cap \mathcal J(U) = B^{\perp } \cap \mathcal J(U) = \mathcal J(B\sqcup U)$ , and so $\mathcal {S}$ is a $\tau $ -perpendicular subcategory of $\mathsf {mod}\Lambda $ .

We are now ready to prove our first main result.

Proof. $(1\iff 2):$ Recall that if $U \in \mathcal C(\mathsf {mod}\Lambda )$ is support $\tau $ -rigid, then $\overline {\tau }U \in \mathcal C(\mathsf {mod}\Lambda )[-1]$ is support $\tau ^{-1}$ -rigid, and moreover, that every support $\tau ^{-1}$ -rigid object in $\mathcal C(\mathsf {mod}\Lambda )[-1]$ occurs in this way. Thus, suppose $U = M\sqcup P[1]$ is support $\tau $ -rigid and write $M \cong M_p\sqcup M_{np}$ , where $M_p \in \mathcal P(\mathsf {mod}\Lambda )$ and $M_{np}$ has no projective direct summand. We note that

and

. Moreover, we have $\overline {\tau }(U) = (\nu M_p)[-1]\sqcup \tau M_{np} \sqcup \nu P$ , where $\tau M_{np}$ has no injective direct summand. This means

This proves the result.

$(1\implies 5)$ : This follows from Theorem 2.2 and the definition of $\mathcal J(U)$ .

$(5\implies 3)$ : Write ${\mathcal W} = {\mathcal T}\cap {\mathcal F}$ with ${\mathcal T}$ a functorially finite torsion class and ${\mathcal F}$ a functorially finite torsion-free class. We will first show that ${\mathcal W}\subseteq {\mathcal W}_L({\mathcal T})$ using an argument similar to [Reference Marks and Šťovíček23, Lem. 3.8]. Let $X \in {\mathcal W},$ and let $g:Y\rightarrow X$ be a morphism in ${\mathcal T}$ . Note that $\operatorname {{\mathrm {image}}} g \in {\mathcal W}$ since it is a subobject of X and a quotient of Y. Now, consider the canonical exact sequence with respect to the torsion pair

:

By assumption, we have $f_{\mathcal F}(Y) \in {\mathcal T}\cap {\mathcal F} = {\mathcal W}$ and

. In particular, we have that

, and so $\operatorname {{\mathrm {image}}} g$ is a quotient of $f_{\mathcal F}(Y)$ . That is, we obtain the following diagram with rows and columns exact.

Now,

since ${\mathcal W}$ is closed under kernels. Therefore, $\ker g \in {\mathcal T}$ since ${\mathcal T}$ is closed under extensions. We conclude that $X \in {\mathcal W}_L({\mathcal T})$ .

We will now show that ${\mathcal W}$ is a Serre subcategory of ${\mathcal W}_L({\mathcal T})$ . Let

$$ \begin{align*}0\rightarrow X\rightarrow Y\rightarrow Z\rightarrow 0\end{align*} $$

be a short exact sequence in ${\mathcal W}_L({\mathcal T})$ . It is clear that if $X,Z \in {\mathcal W},$ then $Y \in {\mathcal W}$ . Thus, suppose $Y \in {\mathcal W} = {\mathcal T}\cap {\mathcal F}$ . It follows that $X \in {\mathcal W}$ since it is in ${\mathcal W}_L({\mathcal T}) \subseteq {\mathcal T}$ and ${\mathcal F}$ is closed under subobjects. Since ${\mathcal W}$ is closed under cokernels, it follows that $Z \in {\mathcal W}$ as well.

$(5\implies 4)$ : The proof is dual to that of $(5\implies 3)$ , but we include the details here for convenience. We will first show that ${\mathcal W} \subseteq {\mathcal W}_R({\mathcal F})$ . Let $X \in {\mathcal W}$ , and let $g:X\rightarrow Y$ be a morphism in ${\mathcal F}$ . We note that $\operatorname {{\mathrm {image}}} g \in {\mathcal W}$ since it is a quotient of X and a subobject of Y. Now consider the canonical exact sequence with respect to the torsion pair $({\mathcal T},{\mathcal T}^{\perp })$ :

$$ \begin{align*}0 \rightarrow t_{\mathcal T}(Y) \rightarrow Y \rightarrow f_{({\mathcal T}^{\perp})}(Y)\rightarrow 0.\end{align*} $$

By assumption, we have $t_{\mathcal T}(Y) \in {\mathcal T}\cap {\mathcal F} = {\mathcal W}$ and $f_{({\mathcal T}^{\perp })}(Y) \in {\mathcal T}^{\perp } \subseteq {\mathcal F}$ . In particular, we have $\operatorname {{\mathrm {image}}} g \subseteq t_{\mathcal T}(T)$ . Therefore, we have an exact sequence

$$ \begin{align*}0\rightarrow t_{\mathcal T}(Y)/\operatorname{{\mathrm{image}}} g \rightarrow \operatorname{{\mathrm{coker}}} f \rightarrow f_{({\mathcal T}^{\perp})}(Y)\rightarrow 0.\end{align*} $$

Since ${\mathcal W}$ is wide, we have $t_{\mathcal T}(Y)/\operatorname {{\mathrm {image}}} g \in {\mathcal W}\subseteq {\mathcal F}$ . Since ${\mathcal F}$ is closed under extensions, this implies that $\operatorname {{\mathrm {coker}}} g \in {\mathcal F}$ . We conclude that $X \in {\mathcal W}_R({\mathcal F})$ .

We will now show that ${\mathcal W}$ is a Serre subcategory of ${\mathcal W}_R({\mathcal F})$ . Let

$$ \begin{align*}0\rightarrow X\rightarrow Y\rightarrow Z\rightarrow 0\end{align*} $$

be a short exact sequence in ${\mathcal W}_R({\mathcal F})$ . It is clear that if $X,Z \in {\mathcal W}$ then $Y \in {\mathcal W}$ . Thus suppose $Y \in {\mathcal W} = {\mathcal T}\cap {\mathcal F}$ . It follows that $Z \in {\mathcal W}$ since it is in ${\mathcal W}_R({\mathcal F}) \subseteq {\mathcal F}$ and ${\mathcal T}$ is closed under subobjects. Since ${\mathcal W}$ is closed under kernels, it follows that $X \in {\mathcal W}$ as well.

$(3\implies 1)$ : Let ${\mathcal W}\subseteq {\mathcal W}_L({\mathcal T})$ be a Serre subcategory of a left-finite wide subcategory of $\mathsf {mod}\Lambda $ . By Lemma 4.3, we have that ${\mathcal W}_L({\mathcal T})$ is $\tau $ -perpendicular in $\mathsf {mod}\Lambda $ . It then follows from Lemma 4.4 that ${\mathcal W}$ is $\tau $ -perpendicular in $\mathsf {mod}\Lambda $ as well.

$(4\implies 1)$ : The proof is analogous to that of $(3\implies 1)$ .

We conclude this section by tabulating several consequences of Theorem 4.5 and the preceding lemmas.

Proof. The implications $(1\implies 3)$ and $(2\implies 3)$ are contained in Lemma 4.3. Thus, assume (3). By Theorem 4.5, there exists a left-finite wide subcategory $\mathcal V \subseteq \mathsf {mod}\Lambda $ such that ${\mathcal W}$ is a Serre subcategory of $\mathcal V$ . Lemma 4.3 and Theorem 3.3 then imply that

$$ \begin{align*}\mathsf{rk}(\Lambda) - 1 = \mathsf{rk}({\mathcal W}) \leq \mathsf{rk}(\mathcal V) \leq \mathsf{rk}(\Lambda).\end{align*} $$

Now, if $\mathsf {rk}({\mathcal W}) = \mathsf {rk}(\mathcal V)$ , then ${\mathcal W} = \mathcal V$ as a consequence of the same lemma and theorem. In particular, ${\mathcal W}$ is a left-finite wide subcategory of $\mathsf {mod}\Lambda $ in this case. Otherwise, $\mathcal V = \mathsf {mod}\Lambda $ by the same argument, and so ${\mathcal W}$ is a left-finite wide subcategory of $\mathsf {mod}\Lambda $ by Corollary 3.8. We conclude that (3) implies (1). The proof that (3) implies (2) is identical.

In [Reference Asai2, Exam. 3.13], Asai gives an example of a functorially finite wide subcategory which is right finite but not left finite. (It is, however, a Serre subcategory of a left-finite wide subcategory, consistent with Theorem 4.5.) By modifying Asai’s example, we obtain an example of a functorially finite wide subcategory which is not a $\tau $ -perpendicular subcategory.

Example 4.9. Consider the quiver

and let $\Lambda = KQ/(\beta _2\alpha _1,\beta _1\alpha _2)$ . Consider the $\Lambda $ -modules

We will demonstrate that ${\mathcal W}:= \mathsf {Filt}(X_1,X_2)$ is a functorially finite wide subcategory of $\mathsf {mod}\Lambda $ which is not a $\tau $ -perpendicular subcategory.

We first note that $\operatorname {{\mathrm {Hom}}}(X_1,X_2) = 0 = \operatorname {{\mathrm {Hom}}}(X_2,X_1)$ and that $\operatorname {{\mathrm {End}}}(X_1) \cong K \cong \operatorname {{\mathrm {End}}}(X_2)$ . Moreover, it is straightforward to show that

In particular, $\operatorname {{\mathrm {Hom}}}(X_1,\tau X_2) = 0 = \operatorname {{\mathrm {Hom}}}(X_2,\tau X_1)$ and every morphism $X_1\rightarrow \tau X_1$ (or $X_2\rightarrow \tau X_2$ ) factors through the injective $I(2)$ . By the AR formulas, we conclude that $\operatorname {{\mathrm {Ext}}}^1(X_i,X_j)$ for any $i,j \in \{1,2\}$ . This means ${\mathcal W}$ is a wide subcategory equivalent to the module category of a semisimple algebra, so in particular, ${\mathcal W}$ is functorially finite.

We will now show that ${\mathcal W}$ is not a $\tau $ -perpendicular subcategory. By Corollary 4.8, the fact that $\mathsf {rk}({\mathcal W}) = \mathsf {rk}(\Lambda )-1$ means we need only show that ${\mathcal W}$ is not left finite. To see this, we note that the Serre subcategory $P(3)^{\perp }$ is equivalent to the module category of the Kronecker path algebra. Thus, we can consider the Ingalls–Thomas bijection (Definition 3.5) ${\mathcal W}_L^{P(3)^{\perp }}$ in the category $P(3)^{\perp }$ . Then the wide subcategory $\mathcal V = {\mathcal W}_L^{P(3)^{\perp }}(\mathsf {Filt}\mathsf {Gen}({\mathcal W})\cap P(3)^{\perp }) = {\mathcal W}_L^{P(3)^{\perp }}(\mathsf {Filt}\mathsf {Gen}(\tau X_1,\tau X_2))$ is the additive closure of a pair of homogeneous tubes having $\tau X_1$ and $\tau X_2$ at their mouths. It is then a well-known fact that right $\mathcal V$ -approximations will in general not exist, and hence, $\mathcal V$ is not functorially finite (in either $P(3)^{\perp }$ or in $\mathsf {mod}\Lambda $ ). In particular, this means $\mathcal V$ is not a left-finite wide subcategory of $P(3)^{\perp }$ , and so $\mathsf {Filt}\mathsf {Gen}({\mathcal W})\cap P(3)^{\perp }$ is not functorially finite. Since $P(3)^{\perp }$ is a functorially finite torsion-free class of $\mathsf {mod}\Lambda $ (see Proposition 3.6 and Corollary 3.8), Lemma 2.1 then implies that the torsion class $\mathsf {Filt}\mathsf {Gen}({\mathcal W})$ is not functorially finite; that is, that ${\mathcal W}$ is not left finite. (Note that, as a consequence of Corollary 4.8, ${\mathcal W}$ is not right finite either. This can also be seen directly using duality.)

Remark 4.10. Summarizing, we have the following inclusions of classes of subcategories of $\mathsf {mod}\Lambda $ .

Moreover, each of these inclusions can be proper. Indeed, (2) can be proper by Example 4.9 and (3) and (4) can be proper by [Reference Asai2, Exam. 3.13] and its dual. It is clear that if $\Lambda $ is not $\tau $ -tilting finite, then (1) will in general be proper. Likewise, (5) and (6) will be proper unless $\Lambda $ is semisimple or local. We note that all six of these classes coincide when $\Lambda $ is semisimple or local, and that all but the Serre subcategories coincide when $\Lambda $ is $\tau $ -tilting finite. Moreover, if $\Lambda $ is hereditary, then (2)–(4) are all equalities by [Reference Ingalls and Thomas20, Cor. 2.17].

5 Reduction of support $\tau $ -rigid objects

The proof of our second main theorem will rely on a reduction formula, which allows us to compare $\tau $ -rigid objects in $\mathcal C(\mathsf {mod}\Lambda )$ with those in $\mathcal C({\mathcal W})$ , with ${\mathcal W}\subseteq \mathsf {mod}\Lambda $ a $\tau $ -perpendicular subcategory. Theorem 5.1 extends Jasso’s reduction of $\tau $ -rigid modules (see [Reference Jasso22, Cor. 3.18]) to account for shifted projectives in the reduction. Note that there is an analogous, and related, formulation in terms of torsion classes (see [Reference Demonet, Iyama, Reading, Reiten and Thomas11, Th. 4.12] and [Reference Jasso22, Th. 3.12 and 3.13]). For the statements in this section, we recall the notation for the canonical exact sequence of a module with respect to a torsion pair from Equation (1) in Section 2.

In case ${\mathcal W} = \mathsf {mod}\Lambda $ , we will sometimes denote $\mathcal E_U:= \mathcal E_U^{\mathsf {mod}\Lambda }$ . In Theorem 6.12, we will show that these “ $\mathcal E$ -maps” satisfy an associativity property as follows: If ${\mathcal W} \subseteq \mathsf {mod}\Lambda $ is a $\tau $ -perpendicular subcategory of $\mathsf {mod}\Lambda $ and $U\sqcup V$ is basic and support $\tau $ -rigid in $\mathcal C({\mathcal W})$ , then $\mathcal E_{U\sqcup V}^{\mathcal W} = \mathcal E^{\mathcal J_{\mathcal W}(U)}_{\mathcal E^{\mathcal W}_U(V)}\circ \mathcal E^W_U$ . This result is established in [Reference Buan and Marsh9, Th. 5.9] in the case that $\Lambda $ is $\tau $ -tilting finite.

For the remainder of this section, we recall some technical results about these bijections and their relationship with Bongartz complements. In essence, these lemmas are well-known (see, e.g., [Reference Buan and Marsh10, Lem. 4.13]), but we give proofs here for completeness.

Proof. Let B be an indecomposable direct summand of $B_U$ . By Theorem 2.4, we have

Moreover, by [Reference Buan and Marsh9, Lem. 3.8], $B\sqcup M$ is $\tau $ -rigid in the Serre subcategory $P^{\perp }$ and satisfies

Since $\mathsf {Gen}(B\sqcup M)\subseteq P^{\perp }$ , it then follows by applying [Reference Adachi, Iyama and Reiten1, Prop. 2.22] in the category $P^{\perp }$ that $B \notin \mathsf {Gen} M$ .

6 $\tau $ -cluster morphism categories

In this section, we apply our previous results to extend the definition of the $\tau $ -cluster morphism category to arbitrary finite-dimensional algebras. This is a small category whose objects correspond to the $\tau $ -perpendicular subcategories of $\mathsf {mod}\Lambda $ and whose morphisms are indexed by support $\tau $ -rigid pairs in these subcategories (see Definition 6.1). This category was defined by Igusa and Todorov for hereditary algebras in [Reference Igusa and Todorov18] under the name “cluster morphism category.” A combinatorial interpretation in Dynkin type A using noncrossing partitions and binary forests was also given by Igusa in [Reference Igusa17]. The definition was extended to $\tau $ -tilting finite algebras by Marsh and the first author in [Reference Buan and Marsh9] under the name “a category of wide subcategories” and given the name “ $\tau $ -cluster morphism category” in [Reference Hanson and Igusa14]. We state our extension of this definition to arbitrary finite-dimensional algebras now.

The main goal of this section is to prove that the $\tau $ -cluster morphism category is indeed a well-defined category (Theorem 1.3, restated as Theorem 6.13). As with [Reference Buan and Marsh9, Reference Igusa17, Reference Igusa and Todorov18], the main technicality is in showing that the composition law is well-defined and associative. In the present article, this will be a consequence of the following generalization of [Reference Buan and Marsh9, Th. 4.3].

Theorem 6.4. Let ${\mathcal W}\subseteq \mathsf {mod}\Lambda $ be a $\tau $ -perpendicular subcategory of $\mathsf {mod}\Lambda $ . Let $U\sqcup V$ be basic and support $\tau $ -rigid in $\mathcal C({\mathcal W})$ . Then

$$ \begin{align*}\mathcal J_{\mathcal W}(U\sqcup V) = \mathcal J_{\mathcal J_{{\mathcal W}}(U)}(\mathcal E^{\mathcal W}_U(V)).\end{align*} $$

Our proof of Theorem 6.4 is largely contained in the two technical lemmas which follow.

Proof. The equality $(1) = (2)$ follows immediately from the fact that $\mathcal J(U) \subseteq M^{\perp }$ .

We next show that $(2) = (3)$ . Note that by definition

$$ \begin{align*}f_{(M^{\perp})}(\mathsf{Gen}(M\sqcup N)) \subseteq (\mathsf{Gen}(M\sqcup N))\cap M^{\perp}.\end{align*} $$

Moreover, we have that since $U\sqcup N$ is support $\tau $ -rigid. Now, if $X \in (\mathsf {Gen}(M\sqcup N))\cap \mathcal J(U)$ , then in particular, $X \in M^{\perp }$ and so $f_{(M^{\perp })}(X) = X$ . We conclude that $(2) = (3)$ .

We now show that $(3) = (4)$ . It is shown in [Reference Buan and Marsh9, Lem. 5.5] that

Moreover, since $N \in P^{\perp }$ and $f_{(M^{\perp })}(N) \in \mathsf {Gen} N$ , we have that $\mathsf {Gen}(f_{(M^{\perp })}(N)) \subseteq P^{\perp }$ . It follows that $(3) = (4)$ .

It remains to show that $(4) = (5)$ . This follows from the definition of $\mathcal E_U$ (see Theorem 5.1) and the fact that $f_{(M^{\perp })}(N) = f_{(M^{\perp })}(\overline {N})$ .

Proof. The equality $(1) = (2)$ follows immediately from the fact that . Likewise, the equality $(2) = (3)$ follows immediately from the definition of the Bongartz complement (Theorem 2.4).

We next show that $(3) = (4)$ . By Theorem 5.1 and Lemma 5.3, we note that $\mathcal E_U(B) \in \mathsf {mod}\Lambda $ . Moreover, Theorem 5.1 also implies (see Remark 5.2) that $\mathcal E_U(B) \sqcup L \sqcup R[1] = \mathcal E_U(B \sqcup V)$ is support $\tau $ -tilting in $\mathcal C(\mathcal J(U))$ . This means $(3) = (4)$ is a special case of equation $(2) = (5)$ in Lemma 6.5.

We proceed to show that $(4) \subseteq (5)$ . First, note that since R is projective in $\mathcal J(U)$ , we have that is closed under factors in $\mathcal J(U)$ , and hence, it suffices to show that . This follows from the fact that $\mathcal E_U(B) \sqcup L \sqcup R[1]$ is support $\tau $ -tilting in $\mathcal C(\mathcal J(U))$ .

We will conclude by showing that $(5) \subseteq (2)$ . Let $B'$ be the Bongartz complement of $\mathcal E_U(V)$ in $\mathcal J(U)$ . Since $B'$ is a module, Theorem 5.1 implies that $\overline {B}:= \mathcal E_U^{-1}(B') \in \mathsf {mod}\Lambda $ and that $\overline {B}\sqcup U \sqcup V$ is support $\tau $ -tilting. We claim that

The first equality follows from Theorem 2.4, and the second from the equality $(2)= (5)$ in Lemma 6.5. The inclusion follows from the fact that $\overline {B}\sqcup U \sqcup V$ is support $\tau $ -tilting, using that $(2)$ is closed under factors in $\mathcal J(U)$ .

We now proceed with our proof of Theorem 6.4.

Proof of Theorem 6.4

Since ${\mathcal W}$ is equivalent to the module category of a basic finite-dimensional algebra, it suffices to consider the case where ${\mathcal W} = \mathsf {mod}\Lambda $ . Write $U = M\sqcup P[1]$ , $V = N\sqcup Q[1]$ , and $\mathcal E_U(V) = L\sqcup R[1]$ . For readability, denote ${\mathcal T}_0 = \mathsf {Gen}(N\sqcup M)$ and

. Noting that $\mathcal J(U\sqcup V) \subseteq \mathcal J(U)$ , Lemmas 6.5 and 6.6 then imply that

Before, we proceed with proving the second main theorem, we note that Theorem 6.4 has some interesting consequences.

Proof. Let $U \in \mathcal C(\mathsf {mod}\Lambda )$ and $V \in \mathcal C({\mathcal W})$ such that ${\mathcal W} = \mathcal J(U)$ and $\mathcal V = \mathcal J_{\mathcal W}(V)$ . By Theorems 5.1 and 6.4, it follows that $U \sqcup \left (\mathcal E_U^{-1}(V)\right ) \in \mathcal C(\mathsf {mod}\Lambda )$ is support $\tau $ -rigid and satisfies

$$ \begin{align*}\mathcal J\left(U \sqcup \left(\mathcal E_U^{-1}(V)\right)\right) = \mathcal J_{\mathcal J(U)}\left(\mathcal E_U\circ\mathcal E_U^{-1}(V)\right) = \mathcal J_{{\mathcal W}}(V) = \mathcal V.\end{align*} $$

We conclude that $\mathcal V$ is a $\tau $ -perpendicular subcategory of $\mathsf {mod}\Lambda $ .

In many cases, the converse of Corollary 1.2 holds as well. For example, in the $\tau $ -tilting finite case, all wide subcategories are $\tau $ -perpendicular and in the hereditary case, $\tau $ -perpendicular subcategories and left-finite wide subcategories coincide. Each of these implies that if $\mathcal V$ and ${\mathcal W}$ are $\tau $ -perpendicular subcategories of $\mathsf {mod}\Lambda $ with $\mathcal V \subseteq {\mathcal W}$ , then $\mathcal V$ is a $\tau $ -perpendicular subcategory of ${\mathcal W}$ . We expect that this is the case in general; that is, we propose the following conjecture.

As another consequence of Theorem 6.4, we have the following.

We now proceed with the proof of the main theorem. In [Reference Buan and Marsh9], which deals with the $\tau $ -tilting finite case, one establishes associativity of the composition operation, by proving that

(4) $$ \begin{align} \mathcal E_{\mathcal E_U(V)}^{\mathcal J(U)} \circ \mathcal E_U = \mathcal E_{U\sqcup V} \end{align} $$

for any basic support $\tau $ -rigid $U\sqcup V \in \mathcal C(\mathsf {mod}\Lambda )$ . This is shown to be a consequence of the fact that

(5) $$ \begin{align} \mathcal J_{\mathcal W}(U\sqcup V) = \mathcal J_{\mathcal J_{{\mathcal W}}(U)}(\mathcal E^{\mathcal W}_U(V)). \end{align} $$

However, the proof that (5) implies (4) given in [Reference Buan and Marsh9, Sec. 5–9] does not use that $\Lambda $ is $\tau $ -tilting finite. We have shown in Theorem 6.4 that Equation (5) holds for $\tau $ -perpendicular subcategories in the general case, and hence, we obtain (4) for free. Note that in the $\tau $ -tilting finite case, in fact, all wide subcategories are $\tau $ -perpendicular.

We will here provide an alternative and much more efficient proof of why (5) implies (4) in the general case, which only builds on two short lemmas in [Reference Buan and Marsh9], namely, Lemmas 5.5 (via Lemma 6.5 in the present article) and 6.2. This is completed in Theorem 6.12, but we first prepare with an additional technical lemma.

Proof. The equivalences $(1\iff 2)$ , $(3\iff 4)$ , and $(4\iff 5)$ all follow from the definitions of the “ $\mathcal E$ -maps” given in Theorem 5.1.

For the equivalence $(2\iff 3)$ , we note that $L' \notin \mathsf {Gen}(M\sqcup N)$ if and only if

$$ \begin{align*}f_{(M^{\perp})}(L') \notin f_{(M^{\perp})}(\mathsf{Gen}(M\sqcup N)) = (\mathsf{Gen}(f_{(M^{\perp})}(\overline{N}))\cap \mathcal J(U)\end{align*} $$

by Lemma 6.5. This, together with the fact that if $L' \notin \mathsf {Gen}(M\sqcup N),$ then $L' \notin \mathsf {Gen} M$ , proves the equivalence of (2) and (3).

Now suppose that (1)–(5) hold. For readability, denote $L":= f_{(M^{\perp })}(L')$ and denote ${\mathcal T} := f_{(M^{\perp })}(\mathsf {Gen}(M\sqcup N))$ . Recall from Lemma 6.5 that ${\mathcal T} = (\mathsf {Gen}(f_{(M^{\perp })}(N))) \cap \mathcal J(U)$ . In particular, we have ${\mathcal T} \subseteq \mathcal J(U)\subseteq \mathcal J(M)$ . Now, denote by

$$ \begin{align*}0\rightarrow t_{\mathcal T}(L") \rightarrow L" \rightarrow f_{({\mathcal T}^{\perp}\cap \mathcal J(U))}(L")\rightarrow 0,\end{align*} $$

the canonical exact sequence with respect to the torsion pair $({\mathcal T},{\mathcal T}^{\perp } \cap \mathcal J(U))$ in $\mathcal J(U)$ . Likewise, denote by

$$ \begin{align*}0\rightarrow t_{\mathcal T}(L") \rightarrow L" \rightarrow f_{({\mathcal T}^{\perp}\cap \mathcal J(M))}(L")\rightarrow 0,\end{align*} $$

the canonical exact sequence with respect to the torsion pair $({\mathcal T},{\mathcal T}^{\perp } \cap \mathcal J(M))$ in $\mathcal J(M)$ . Since both sequences start with $t_{\mathcal T}(L")$ , we see that $f_{({\mathcal T}^{\perp }\cap \mathcal J(U))}(L") = f_{({\mathcal T}^{\perp }\cap \mathcal J(M))}(L")$ .

We now observe that $\mathcal E_{\mathcal E_U(V)}^{\mathcal J(U)} \circ \mathcal E_{U}(L') = f_{({\mathcal T}^{\perp }\cap \mathcal J(U))}(L")$ and $\mathcal E_{(U\sqcup V)} = f_{((M\sqcup N)^{\perp })}(L')$ by construction. Finally, it is shown in [Reference Buan and Marsh9, Lem. 6.2] that $f_{({\mathcal T}^{\perp }\cap \mathcal J(M))}(L") = f_{((M\sqcup N)^{\perp })}(L')$ . We conclude that $\mathcal E_{\mathcal E_U(V)}^{\mathcal J(U)} \circ \mathcal E_{U}(L') = \mathcal E_{U\sqcup V}(L')$ as desired.

We are now prepared to verify Equation (4).

Theorem 6.12. Let ${\mathcal W}\subseteq \mathsf {mod}\Lambda $ be a $\tau $ -perpendicular subcategory of $\mathsf {mod}\Lambda ,$ and let $U\sqcup V \in \mathcal C({\mathcal W})$ be support $\tau $ -rigid and basic. Then

$$ \begin{align*}\mathcal E_{\mathcal E_U(V)}^{\mathcal J_{\mathcal W}(U)} \circ \mathcal E^{\mathcal W}_U = \mathcal E^{\mathcal W}_{U\sqcup V}.\end{align*} $$

We are now ready to complete the proof that the $\tau $ -cluster morphism category is indeed a category. This essentially follows from Theorem 6.12 identically as in [Reference Buan and Marsh9, Cor. 1.8], [Reference Igusa17, Sec. 1], and [Reference Igusa and Todorov18, Cor. 1.10].

Proof. It is straightforward to show that for any $\tau $ -perpendicular subcategory ${\mathcal W} \subseteq \mathsf {mod}\Lambda $ , the morphism $g_0^{\mathcal W}$ is the identity of ${\mathcal W}$ . Thus, we need only show that the composition law is associative. Consider

$$ \begin{align*}{\mathcal W}_1\xrightarrow{g_U^{{\mathcal W}_1}} {\mathcal W}_2 \xrightarrow{g_V^{{\mathcal W}_2}} {\mathcal W}_3 \xrightarrow{g_W^{{\mathcal W}_3}} {\mathcal W}_4\end{align*} $$

a sequence of three composable morphisms in $\mathfrak {W}$ . Then, by Theorem 6.12 and the additivity of the “ $\mathcal E$ -maps,” we have

$$ \begin{align*} \left(g_W^{{\mathcal W}_3}\circ g_{V}^{{\mathcal W}_2}\right)\circ g_U^{{\mathcal W}_1} &= g_{V\sqcup \left(\mathcal E_V^{{\mathcal W}_2}\right)^{-1}(W)}^{{\mathcal W}_2}\circ g_U^{{\mathcal W}_1}\\ &= g_{U\sqcup \left(\mathcal E_U^{{\mathcal W}_1}\right)^{-1}(V) \sqcup \left(\mathcal E_V^{{\mathcal W}_2}\circ \mathcal E_U^{{\mathcal W}_1}\right)^{-1}(W)}^{{\mathcal W}_1}\\ &= g_W^{{\mathcal W}_3} \circ g_{U\sqcup \left(\mathcal E_W^{{\mathcal W}_1}\right)^{-1}(V)}^{{\mathcal W}_1}\\ &= g_W^{{\mathcal W}_3}\circ\left( g_{V}^{{\mathcal W}_2}\circ g_W^{{\mathcal W}_1}\right).\\[-37pt] \end{align*} $$

For ${\mathcal W}\subseteq \mathsf {mod}\Lambda $ a $\tau $ -perpendicular subcategory, we can likewise define the $\tau $ -cluster morphism category $\mathfrak {W}({\mathcal W})$ in the usual way; that is, by identifying ${\mathcal W}$ with some module category. We then obtain the following.

We conclude this section by generalizing the results of [Reference Buan and Marsh9, Sec. 10].

7 An example

In this section, we consider a pair of examples. As examples in the $\tau $ -tilting finite case can be found in [Reference Buan and Marsh9, Sec. 12], our examples are both $\tau $ -tilting infinite. This means the corresponding $\tau $ -cluster morphism categories are infinite as well.

We first consider the Kronecker path algebra $\Lambda _1 = K(1\rightrightarrows 2).$ For $i \in \mathbb {N}$ , we denote by $M_{(i,i+1)}$ and $M_{(i+1,i)}$ the unique (up to isomorphism) indecomposable $\Lambda _1$ -modules with dimension vectors $(i,i+1)$ and $(i+1,i)$ . We note that $P(1) = M_{(1,2)}$ , $S(1) = M_{(1,0)}$ , and $P(2) = S(2) = M_{(0,1)}$ .

Figure 1 is an illustration of the category $\mathfrak {W}(\Lambda _1)$ . The vertices are the $\tau $ -perpendicular subcategories of $\mathsf {mod}\Lambda $ (which in this case are precisely the functorially finite wide subcategories). An irreducible morphism $g_U^{\mathcal W}:{\mathcal W}\rightarrow \mathcal V$ (so that U is indecomposable and support $\tau $ -rigid in $\mathcal C({\mathcal W})$ and $\mathcal J_{\mathcal W}(U) = \mathcal V)$ is shown as an arrow ${\mathcal W}\rightarrow \mathcal V$ labeled by U. The wide subcategories $\mathsf {add}\{M_{(i,i+1)}\}$ generated by the preprojective modules all appear above the horizontal dashed line, with i increasing as one moves counter clockwise. Likewise, the wide subcategories $\mathsf {add}\{M_{(i+1,i)}\}$ generated by the preinjective modules all appear below the horizontal dashed line, with i increasing as one moves clockwise. The category is drawn so that every square commutes, and wide subcategories which appear more than once in the figure should be identified.

Figure 1

The category $\mathfrak {W}(\Lambda _1)$ for $\Lambda _1 = K(1\rightrightarrows 2)$ .

For our second example, we consider quiver $Q = 1\rightrightarrows 2 \rightarrow 3$ and the algebra $\Lambda _2 = KQ/\mathrm {rad}^2 KQ$ . Again for $i \in \mathbb {N}$ , we denote by $M_{(i,i+1,0)}$ and $M_{(i+1,i,0)}$ the unique (up to isomorphism) $\Lambda _2$ -modules with dimension vectors $(i,i+1,0)$ and $(i+1,i,0)$ . The irreducible morphisms in $\mathfrak {W}(\Lambda _2)$ with source $\mathsf {mod}\Lambda _2$ are shown in Figure 2. Similarly to before, a morphism $g_U^{\mathsf {mod}\Lambda _2}: \mathsf {mod}\Lambda _2 \rightarrow {\mathcal W}$ is labeled by U. Moreover, every module of the form $M_{(i,i+1,0)}$ or $M_{(i+1,i,0)}$ corresponds to some morphism with source $\mathsf {mod}\Lambda _2$ .

Figure 2

The irreducible morphisms in $\mathfrak {W}(\Lambda _2)$ with source $\mathsf {mod}\Lambda _2$ . Here, $\Lambda _2 = K(1\rightrightarrows 2\rightarrow 3)/\mathrm {rad^2}$ .

To complete the picture, we can utilize Proposition 6.14. The Serre subcategory $P(3)^{\perp }$ is equivalent to $\mathsf {mod}\Lambda _1$ , so there is a copy of $\mathfrak {W}(\Lambda _1)$ sitting inside of $\mathfrak {W}(\Lambda _2)$ which has $P(3)^{\perp }$ identified with $\mathsf {mod}\Lambda _1$ . The Serre subcategory $P(1)^{\perp } = \mathsf {add}\{P(2),S(2),P(3)\}$ is equivalent to the module category of the path algebra of type $A_2$ . Thus, $\mathfrak {W}(\Lambda _2)$ contains five irreducible morphisms which have source $\mathsf {add}\{P(2),S(2),P(3)\}$ and five morphisms $\mathsf {add}\{P(2),S(2),P(3)\} \rightarrow 0$ . The remaining subcategories shown are semisimple, so each is the source of four irreducible morphisms and four morphisms with target 0 in $\mathfrak {W}(\Lambda _2)$ .