Maximal $\tau_d$-rigid pairs

TL;DR

This paper extends the bijection between cluster tilting objects and support $ au$-tilting pairs to higher $(d+2)$-angulated categories, introducing maximal $ au_d$-rigid pairs as a higher analogue.

Contribution

It establishes a higher analogue of the known bijection in 2-Calabi--Yau categories, using maximal $ au_d$-rigid pairs in $(d+2)$-angulated categories.

Findings

Established a bijection between maximal $ au_d$-rigid pairs and support $ au_d$-tilting pairs.

Extended known results from 2-Calabi--Yau categories to higher $(d+2)$-angulated categories.

Introduced the concept of maximal $ au_d$-rigid pairs as a higher analogue.

Abstract

Let $\mathscr T$ be a $2$-Calabi--Yau triangulated category, $T$ a cluster tilting object with endomorphism algebra $\Gamma$. Consider the functor $\mathscr T( T,- ) : \mathscr T \rightarrow \mod \Gamma$. It induces a bijection from the isomorphism classes of cluster tilting objects to the isomorphism classes of support $\tau$-tilting pairs. This is due to Adachi, Iyama, and Reiten. The notion of $( d+2 )$-angulated categories is a higher analogue of triangulated categories. We show a higher analogue of the above result, based on the notion of maximal $\tau_d$-rigid pairs.

Equations

$$X_d\to \cdots \to X_0\to Y\to {\Sigma }^d{X}_d$$

$$\mathrm{add}X=\{Y\in \mathcal{T}\mid {\mathrm{Ext}}_{\mathcal{T}}^d(X,Y)=0\}.$$

$$\mathrm{add}X=\{Y\in \mathcal{T}\mid {\mathrm{Ext}}_{\mathcal{T}}^d(X\oplus Y,X\oplus Y)=0\}.$$

$$N\in\operatorname{add}M\Leftrightarrow\left\{\begin{array}[]{l}\operatorname{Hom}_{\Gamma}(M,\tau_{d}N)=0,\\[2.84526pt] \operatorname{Hom}_{\Gamma}(N,\tau_{d}M)=0,\\[2.84526pt] \operatorname{Hom}_{\Gamma}(P,N)=0.\end{array}\right.$$

$$Q\in \mathrm{add}P\iff {\mathrm{Hom}}_{\Gamma }(Q,M)=0.$$

$$\left\{\begin{array}[]{cc}\mbox{isomorphism classes of }\\ \mbox{maximal $d$-rigid objects in $\mathscr{T}$}\end{array}\right\}\rightarrow\left\{\begin{array}[]{cc}\mbox{isomorphism classes of}\\ \mbox{maximal $\tau_{d}$-rigid pairs in $\mathscr{D}$}\end{array}\right\}.$$

$$\left\{\begin{array}[]{cc}\mbox{isomorphism classes of }\\ \mbox{cluster tilting object in $\mathscr{T}$}\end{array}\right\}\rightarrow\left\{\begin{array}[]{cc}\mbox{isomorphism classes of}\\ \mbox{support $\tau$-tilting pairs in $\operatorname{mod}\Gamma$}\end{array}\right\}$$

$$\begin{array}[]{c}\mbox{$X$ is Oppermann--Thomas cluster tilting}\\ \Downarrow\\ \mbox{$X$ is $d$-self-perpendicular}\\ \Downarrow\\ \mbox{$X$ is maximal $d$-rigid}\\ \Downarrow\\ \mbox{$X$ is $d$-rigid.}\end{array}$$

$$X$$

$$\nu_{\Gamma}\mathscr{T}(T,T^{\prime})\cong\mathscr{T}\big{(}T,\Sigma^{2d}(T^{\prime})\big{)}$$

$$\mathscr{T}\big{(}T,\Sigma^{2d}(T^{\prime})\big{)}\cong\operatorname{D}\!\mathscr{T}(T^{\prime},T).$$

$$\operatorname{D}\!\mathscr{T}(T^{\prime},T)\cong\operatorname{D}\!\operatorname{Hom}_{\Gamma}\big{(}\mathscr{T}(T,T^{\prime}),\mathscr{T}(T,T)\big{)}=\operatorname{D}\!\operatorname{Hom}_{\Gamma}\big{(}\mathscr{T}(T,T^{\prime}),\Gamma\big{)}.$$

$$\operatorname{D}\!\operatorname{Hom}_{\Gamma}\big{(}\mathscr{T}(T,T^{\prime}),\Gamma\big{)}=\nu_{\Gamma}\mathscr{T}(T,T^{\prime}),$$

$$T_d\to \cdots \to T_0\to X\to {\Sigma }^d{T}_d$$

$$\mathcal{T}(T,T_d)\to \cdots \to \mathcal{T}(T,T_0)\to \mathcal{T}(T,X)\to 0$$

$${\Sigma }^{-d}X\to T_d\to \cdots \to T_0\to X$$

$$\mathcal{T}(T,{\Sigma }^{-d}X)\to \mathcal{T}(T,T_d)\to \cdots \to \mathcal{T}(T,T_0)\to \mathcal{T}(T,X)\to \mathcal{T}(T,{\Sigma }^d{T}_d)=0.$$

$${\tau }_d\mathcal{T}(T,X)\cong \mathcal{T}(T,{\Sigma }^{d}X).$$

$$\mathcal{T}(T,T_d)\to \cdots \to \mathcal{T}(T,T_0)\to \mathcal{T}(T,X)\to 0.$$

$${\tau }_d\mathcal{T}(T,X)\cong \mathcal{T}(T,{\Sigma }^{d}X).$$

$$D[\mathrm{add}T](X,Y)\cong {\mathrm{Hom}}_{\mathcal{T}/\mathrm{add}{\Sigma }^{d}T}(\overline{Y},\overline{{\Sigma }^{2d}X})$$

$$T_d\to \dots \to T_0\to Y\to {\Sigma }^d{T}_d,$$

$$\mathrm{Im}\Psi =[\mathrm{add}T](X,Y).$$

$$\cdots \to \mathcal{T}(X,T_0)\stackrel{\Psi }{\to }\mathcal{T}(X,Y)\to \mathcal{T}(X,{\Sigma }^d{T}_d)\to \cdots .$$

$$D[\mathrm{add}T](X,Y)\cong D\mathrm{Im}\Psi \cong \mathrm{Im}D\Psi \cong \mathcal{T}(Y,{\Sigma }^{2d}X)/[\mathrm{add}{\Sigma }^{d}T](Y,{\Sigma }^{2d}X)\cong {\mathrm{Hom}}_{\mathcal{T}/\mathrm{add}{\Sigma }^{d}T}(\overline{Y},\overline{{\Sigma }^{2d}X}).$$

$$0\to D{\mathrm{Hom}}_{\mathcal{T}/\mathrm{add}{\Sigma }^{d}T}(\overline{Y},\overline{{\Sigma }^{d}X})\to {\mathrm{Ext}}_{\mathcal{T}}^d(X,Y)\to {\mathrm{Hom}}_{\mathcal{T}/\mathrm{add}{\Sigma }^{d}T}(\overline{X},\overline{{\Sigma }^{d}Y})\to 0.$$

$$0\to [\mathrm{add}{\Sigma }^{d}T](X,{\Sigma }^{d}Y)\to \mathcal{T}(X,{\Sigma }^{d}Y)\to {\mathrm{Hom}}_{\mathcal{T}/\mathrm{add}{\Sigma }^{d}T}(\overline{X},\overline{{\Sigma }^{d}Y})\to 0.$$

$$[\mathrm{add}T]({\Sigma }^{-d}X,Y)\cong D{\mathrm{Hom}}_{\mathcal{T}/\mathrm{add}{\Sigma }^{d}T}(\overline{Y},\overline{{\Sigma }^{2d}{\Sigma }^{-d}X})\cong D{\mathrm{Hom}}_{\mathcal{T}/\mathrm{add}{\Sigma }^{d}T}(\overline{Y},\overline{{\Sigma }^{d}X}).$$

$$0\rightarrow\operatorname{D}\!\operatorname{Hom}_{\Gamma}\big{(}\mathscr{T}(T,Y),\tau_{d}\mathscr{T}(T,X)\big{)}\rightarrow\operatorname{Ext}_{\mathscr{T}}^{d}(X,Y)\rightarrow\operatorname{Hom}_{\Gamma}\big{(}\mathscr{T}(T,X),\tau_{d}\mathscr{T}(T,Y)\big{)}\rightarrow 0.$$

Full text

The notion of $(d+2)$-angulated categories is a higher analogue of triangulated categories. We show a higher analogue of the above result, based on the notion of maximal $\tau_{d}$-rigid pairs.

0. Introduction

In triangulated categories, the notions of cluster tilting objects (introduced in [4, p. 583]) and maximal rigid objects have recently been extensively investigated. They frequently coincide, by [22, thm. 2.6], and they are closely linked to the notion of support $\tau$-tilting pairs in abelian categories (introduced in [1, def. 0.3]). Indeed, there is often a bijection between the cluster tilting objects in a triangulated category and the support $\tau$-tilting pairs in a suitable (abelian) module category, see [1, thm. 4.1].

This paper investigates the analogous theory in $(d+2)$-angulated and $d$-abelian categories, which are the main objects of higher homological algebra, see [8, def. 2.1] and [15, def. 3.1]. Several key properties from the classic case do not carry over. For example, cluster tilting objects are maximal $d$-rigid, but the converse is rarely true. Moreover, the higher analogue of support $\tau$-rigid pairs permit a bijection to the maximal $d$-rigid objects, but not to the cluster tilting objects.

For further reading in higher homological algebra a number of references have been included in the bibliography, see [3], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21].

Let $k$ be an algebraically closed field, $d\geqslant 1$ an integer, $\mathscr{T}$ a $k$-linear $\operatorname{Hom}$-finite $(d+2)$-angulated category with split idempotents, see [8, def. 2.1]. Assume that $\mathscr{T}$ is $2d$-Calabi–Yau, see [21, def. 5.2], and let $\Sigma^{d}$ denote the $d$-suspension functor of $\mathscr{T}$.

**Cluster tilting and maximal $d$-rigid objects. ** An object $X\in\mathscr{T}$ is $d$-rigid if $\operatorname{Ext}^{d}_{\mathscr{T}}(X,X)=0$. We recall three important definitions.

Definition 0.1** ([21, def. 5.3]).**

An object $X\in\mathscr{T}$ is Oppermann–Thomas cluster tilting in $\mathscr{T}$ if:

1. (i)

$X$ is $d$-rigid. 2. (ii)

For any $Y\in\mathscr{T}$ there exists a $(d+2)$-angle

[TABLE]

with $X_{i}\in\operatorname{add}X$ for all $0\leq i\leq d$.

Definition 0.2**.**

An object $X\in\mathscr{T}$ is $d$-self-perpendicular in $\mathscr{T}$ if

[TABLE]

Definition 0.3**.**

An object $X\in\mathscr{T}$ is maximal $d$-rigid in $\mathscr{T}$ if

[TABLE]

Our first main result is:

*Theorem A. ** $X$ is Oppermann–Thomas cluster tilting $\Rightarrow$ $X$ is $d$-self-perpendicular $\Rightarrow$ $X$ is maximal $d$-rigid. *

We prove this in Theorem 1.1. Of equal importance is that the implications cannot be reversed in general, see Remark 1.2. In particular, when $d\geqslant 2$, the class of maximal $d$-rigid objects is typically strictly larger than the class of Oppermann–Thomas cluster tilting objects, in contrast to the classic case $d=1$ where the two classes usually coincide, see [22, thm. 2.6].

Maximal $\tau_{d}$-rigid pairs. Let $T\in\mathscr{T}$ be an Oppermann–Thomas cluster tilting object and let $\Gamma=\operatorname{End}_{\mathscr{T}}(T)$. Recall the following result.

Theorem 0.4** ([14, thm. 0.6]).**

Consider the essential image $\mathscr{D}$ of the functor $\mathscr{T}(T,-):\mathscr{T}\rightarrow\operatorname{mod}\Gamma$. Then $\mathscr{D}$ is a $d$-cluster tilting subcategory of $\operatorname{mod}\Gamma$. There is a commutative diagram, as shown below, where the vertical arrow is the quotient functor and the diagonal arrow is an equivalence of categories:

\mathscr{T}$$\mathscr{T}/\operatorname{add}\Sigma^{d}T.\mathscr{D}$$\scriptstyle\overline{(-)}$$\scriptstyle{\mathscr{T}(T,-)}$$\sim

The category $\mathscr{D}$ is a $d$-abelian category by [15, thm. 3.16]. It has a $d$-Auslander–Reiten translation $\tau_{d}$, which is a higher analogue of the classic Auslander–Reiten translation $\tau$, see [12, sec. 1.4.1]. A module $M\in\mathscr{D}$ is called $\tau_{d}$-rigid if $\operatorname{Hom}_{\Gamma}(M,\tau_{d}M)=0$.

Remark 0.5**.**

The classic $\operatorname{add}$-$\operatorname{proj}$-correspondence holds, as $\mathscr{T}(T,-)$ restricts to an equivalence $\operatorname{add}T\rightarrow\operatorname{proj}\Gamma$ . The functor also restricts to an equivalence $\operatorname{add}ST\rightarrow\operatorname{inj}\Gamma$. [14, lem. 2.1]

It is natural to ask if $\mathscr{D}$ permits a higher analogue of the $\tau$-tilting theory of [1]. We will not answer this question, but will instead introduce the following definitions inspired by it.

Definition 0.6**.**

A pair $(M,P)$ with $M\in\mathscr{D}$ and $P\in\operatorname{proj}\Gamma$ is called a $\tau_{d}$-rigid pair in $\mathscr{D}$ if $M$ is $\tau_{d}$-rigid and $\operatorname{Hom}_{\Gamma}(P,M)=0$.

Definition 0.7**.**

A pair $(M,P)$ with $M\in\mathscr{D}$ and $P\in\operatorname{proj}\Gamma$ is called a maximal $\tau_{d}$-rigid pair in $\mathscr{D}$ if it satisfies:

1. (i)

If $N\in\mathscr{D}$ then

[TABLE] 2. (ii)

If $Q\in\operatorname{proj}\Gamma$, then

[TABLE]

A maximal $\tau_{d}$-rigid pair is a $\tau_{d}$-rigid pair.

Our second main result is:

**Theorem B. ** If each indecomposable object of $\mathscr{T}$ is $d$-rigid, then there is a bijection

[TABLE]

We prove this in Section 3. If $d=1$, then $(M,P)$ is a maximal $\tau_{1}$-rigid pair if and only if it is a support $\tau$-tilting pair in the sense of [1, def. 0.3(b)], see [1, def. 0.3, prop. 2.3, and cor. 2.13]. Hence Theorem B is a higher analogue of the bijection

[TABLE]

which exists by [1, thm. 4.1] when $\mathscr{T}$ is triangulated, i.e. in the case $d=1$. However, when $d\geqslant 2$, we do not think of maximal $\tau_{d}$-rigid pairs as support $\tau_{d}$-tilting pairs. The reason is that by Theorem B, maximal $\tau_{d}$-rigid pairs are linked to maximal $d$-rigid objects in higher angulated categories. As remarked above, this class is typically strictly larger than the class of Oppermann–Thomas cluster tilting objects when $d\geqslant 2$.

Note that [19] makes an approach to higher support tilting theory.

This paper is organised as follows: Section 1 proves Theorem A, Section 2 investigates the precise relation between $\operatorname{Hom}$ spaces in $\mathscr{T}$ and $\mathscr{D}$, Section 3 proves Theorem B, and Section 4 gives an example.

Setup 0.8**.**

Throughout the paper we use the following notation:

$k$**: **

An algebraically closed field.

$\operatorname{D}$**: **

The duality functor $\operatorname{Hom}_{k}(-,k)$.

$\mathscr{T}$**: **

A $k$-linear, $\operatorname{Hom}$-finite, $(d+2)$-angulated category with split idempotents. We assume that $\mathscr{T}$ is $2d$-Calabi–Yau, that is $\mathscr{T}(X,Y)\cong\operatorname{D}\!\mathscr{T}(Y,\Sigma^{2d}X)$ naturally in $X,Y\in\mathscr{T}$.

$\Sigma^{d}$**: **

The $d$-suspension functor on $\mathscr{T}$.

$T$**: **

An Oppermann–Thomas cluster tilting object in $\mathscr{T}$.

$\overline{(-)}$**: **

The canonical functor $\mathscr{T}\rightarrow\mathscr{T}/\operatorname{add}\Sigma^{d}T$, whose target is the naive quotient category of $\mathscr{T}$ modulo the morphisms which factor through an object in $\operatorname{add}\Sigma^{d}T$.

$\Gamma$**: **

The endomorphism ring $\operatorname{End}_{\mathscr{T}}(T)$.

$\nu_{\Gamma}$**: **

The Nakayama functor on $\operatorname{mod}\Gamma$.

$\tau_{d}$**: **

The $d$-Auslander–Reiten translation on $\operatorname{mod}\Gamma$.

$\mathscr{D}$**: **

The essential image of the functor $\mathscr{T}(T,-):\mathscr{T}\rightarrow\operatorname{mod}\Gamma$.

1. Proof of Theorem A

Theorem 1.1**.**

Let $X\in\mathscr{T}$ be given.

1. (i)

There are implications

[TABLE] 2. (ii)

If each indecomposable object in $\mathscr{T}$ is $d$-rigid, then

[TABLE]

Proof.

(i), the first implication: Suppose $X$ is Oppermann–Thomas cluster tilting. We must prove the equality in Definition 0.2, and the inclusion $\subseteq$ is clear. For the inclusion $\supseteq$, suppose $\operatorname{Ext}^{d}_{\mathscr{T}}(X,Y)=0$. Then each morphism $X_{0}\rightarrow\Sigma^{d}Y$ with $X_{0}\in\operatorname{add}X$ is zero. This applies in particular to the $(d+2)$-angle $X_{d}\rightarrow\cdots\rightarrow X_{0}\rightarrow\Sigma^{d}Y\rightarrow\Sigma^{d}X_{d}$ with $X_{i}\in\operatorname{add}X$, which exists since $X$ is Oppermann–Thomas cluster tilting. But then the morphism $\Sigma^{d}Y\rightarrow\Sigma^{d}X_{d}$ is a split monomorphism, and applying $\Sigma^{-d}$ gives a split monomorphism $Y\rightarrow X_{d}$ proving $Y\in\operatorname{add}X$.

(i), the second implication: Suppose that $X$ is $d$-self-perpendicular. We must prove the equality in Definition 0.3, and the inclusion $\subseteq$ is clear. For the inclusion $\supseteq$, suppose $\operatorname{Ext}^{d}_{\mathscr{T}}(X\oplus Y,X\oplus Y)=0$. Then in particular, $\operatorname{Ext}^{d}_{\mathscr{T}}(X,Y)=0$, whence $Y\in\operatorname{add}X$.

(i), the third implication: This is clear.

(ii): Suppose that each indecomposable object in $\mathscr{T}$ is $d$-rigid. Because of part (i), it is enough to prove the implication $\Leftarrow$ in (ii), so suppose that $X$ is maximal $d$-rigid. We must prove the equality in Definition 0.2, and $\subseteq$ is clear.

For the inclusion $\supseteq$, observe that $\{\,Y\in\mathscr{T}\mid\operatorname{Ext}^{d}_{\mathscr{T}}(X,Y)=0\,\}$ is closed under direct sums and summands by additivity of $\operatorname{Ext}$. Hence it is enough to suppose that $Y$ is an indecomposable object in this set and prove $Y\in\operatorname{add}X$. However, $\operatorname{Ext}^{d}_{\mathscr{T}}(X,Y)=0$ implies $\operatorname{Ext}^{d}_{\mathscr{T}}(Y,X)=0$ because $\mathscr{T}$ is $2d$-Calabi–Yau, and $\operatorname{Ext}^{d}_{\mathscr{T}}(Y,Y)=0$ by assumption. Finally, $X$ is $d$-rigid by part (i), so $\operatorname{Ext}^{d}_{\mathscr{T}}(X,X)=0$. Combining these equalities shows $\operatorname{Ext}^{d}_{\mathscr{T}}(X\oplus Y,X\oplus Y)=0$, and $Y\in\operatorname{add}X$ follows. ∎

Remark 1.2**.**

The implications in Theorem 1.1(i) cannot be reversed in general:

• –

An example of a $d$-self-perpendicular object $X$ which is not Oppermann–Thomas cluster tilting is given in Section 4. In fact, the objects in the last three rows of Figure 4 are such examples. The example was originally given in [21, p. 1735].

• –

An example of a maximal $d$-rigid object which is not $d$-self-perpendicular can be obtained by combining proposition 2.6 and corollary 2.7 in [5]. These results give a maximal $1$-rigid object which is not cluster tilting, but in the triangulated setting of [5], cluster tilting is equivalent to $1$-self-perpendicular, see [5, bottom of p. 963].

• –

Finally, an example of a $d$-rigid object which is not maximal $d$-rigid is the zero object, as soon as $\mathscr{T}$ has a non-zero $d$-rigid object.

We end the section by observing that Theorem 1.1(ii) can be applied to an important class of categories.

Proposition 1.3**.**

Let $\Lambda$ be a $d$-representation finite algebra, $\mathscr{O}_{\Lambda}$ the $(d+2)$-angulated cluster category associated to $\Lambda$ in [21, thm. 5.2]. Then each $X\in\mathscr{O}_{\Lambda}$ satisfies

[TABLE]

Proof.

Each indecomposable in $\mathscr{O}_{\Lambda}$ is $d$-rigid by [21, Lemma 5.41], so the equivalence follows from Theorem 1.1(ii). ∎

2. A dimension formula for $\operatorname{Ext}^{d}_{\mathscr{T}}$

Recall from Setup 0.8 that $T$ is a fixed Oppermann–Thomas cluster tilting object in $\mathscr{T}$, and that $\mathscr{T}$ is $2d$-Calabi–Yau, that is, $\mathscr{T}(X,Y)\cong\operatorname{D}\!\mathscr{T}(Y,\Sigma^{2d}X)$ naturally in $X,Y\in\mathscr{T}$.

Lemma 2.1**.**

There is a natural isomorphism

[TABLE]

for $T^{\prime}\in\operatorname{add}T$.

Proof.

By the 2$d$-Calabi-Yau property we have

[TABLE]

By [14, Lemma 2.2(i)],

[TABLE]

Finally, by definition we have

[TABLE]

see [2, def. III.2.8]. ∎

Lemma 2.2**.**

If $X\in\mathscr{T}$ has no non-zero direct summands in $\operatorname{add}\Sigma^{d}T$, then there exists a $(d+2)$-angle

[TABLE]

in $\mathscr{T}$ with the following properties: Each $T_{i}$ is in $\operatorname{add}T$, and applying the functor $\mathscr{T}(T,-)$ gives a complex

[TABLE]

which is the start of the augmented minimal projective resolution of $\mathscr{T}(T,X)$.

Proof.

Given $X$, there exists a $(d+2)$-angle

[TABLE]

with each $T_{i}$ in $\operatorname{add}T$ by Definition 0.1. Since $X$ has no non-zero direct summands in $\operatorname{add}\Sigma^{d}T$, the first morphism in the $(d+2)$-angle is in the radical of $\mathscr{T}$. By dropping trivial summands of the form $T^{\prime}\xrightarrow{\cong}T^{\prime}$, we can assume that so are the other morphisms except the last morphism.

By [8, prop. 2.5(a)], applying the functor $\mathscr{T}(T,-)$ gives an exact sequence

[TABLE]

By Theorem 0.4, applying the functor $\mathscr{T}(T,-)$ is, up to isomorphism, just to apply a quotient functor, and this preserves radical morphisms. So in the exact sequence each morphism, except possibly $\mathscr{T}(T,T_{0})\rightarrow\mathscr{T}(T,X)$, is in the radical of $\operatorname{mod}\Gamma$. This proves the claim of the lemma. ∎

Lemma 2.3**.**

If $X\in\mathscr{T}$ has no non-zero direct summands in $\operatorname{add}\Sigma^{d}T$, then there is a natural isomorphism

[TABLE]

Proof.

As $X$ has no non-zero direct summands in $\operatorname{add}\Sigma^{d}T$, we can consider the $(d+2)$-angle from Lemma 2.2. Apply $\mathscr{T}(T,-)$ to get the following part of an augmented minimal projective resolution in $\operatorname{mod}\Gamma$:

[TABLE]

Using the Nakayama functor and Lemma 2.1 we get the following commutative diagram.

0\tau_{d}\mathscr{T}(T,X)$$\nu_{\Gamma}\mathscr{T}(T,T_{d})$$\cdots$$\nu_{\Gamma}\mathscr{T}(T,T_{0})[math]\mathscr{T}(T,\Sigma^{d}X)$$\mathscr{T}(T,\Sigma^{2d}T_{d})$$\cdots$$\mathscr{T}(T,\Sigma^{2d}T_{0})$$\sim$$\sim

The top sequence is exact by the definition of $\tau_{d}$, see [12, sec. 1.4.1]. The bottom sequence is exact because it is obtained by applying $\operatorname{Hom}_{\mathscr{T}}(T,-)$ to a $(d+2)$-angle in $\mathscr{T}$, see [8, prop. 2.5(a)]. The first term of the bottom sequence is actually $\mathscr{T}(T,\Sigma^{d}T_{0})$, but this is zero. Since we have $d\geq 1$, the diagram implies

[TABLE]

∎

We write $[\operatorname{add}T](X,Y)=\{\,f\in\mathscr{T}(X,Y)\mid f\text{ factors through an object of }\operatorname{add}T\,\}.$

Lemma 2.4**.**

There is a natural isomorphism

[TABLE]

for $X,Y\in\mathscr{T}$.

Proof.

Pick a $(d+2)$-angle in $\mathscr{T}$:

[TABLE]

with $T_{i}\in\operatorname{add}T$. Use $\mathscr{T}(X,-)$ to obtain the morphism $\Psi:\mathscr{T}(X,T_{0})\rightarrow\mathscr{T}(X,Y)$. This is a homomorphism of $k$-vector spaces, hence we can talk about the image of $\Psi$. We first note that any morphism $f$ in the image of $\Psi$ must factor through $\operatorname{add}T$. Now suppose $f\in\mathscr{T}(X,Y)$ factors through $T^{\prime}\in\operatorname{add}T$. We have the following commutative diagram, where the lower row is a part of the $(d+2)$-angle above:

\cdots$$T_{0}$$Y$$\Sigma^{d}T_{d}.\cdots$$T^{\prime}$$T^{\prime}[math]X$$1_{T^{\prime}}$$f

The dashed arrow exists by completing the commutative square to a morphism of $(d+2)$-angles. We conclude that $f\in\operatorname{Im}\Psi$. Hence

[TABLE]

We now return to the long exact sequence

[TABLE]

Using the duality functor $\operatorname{D}$ and Serre duality we get the following diagram with exact rows:

\operatorname{D}\!\mathscr{T}(X,\Sigma^{d}T_{d})$$\operatorname{D}\!\mathscr{T}(X,Y)$$\operatorname{D}\!\mathscr{T}(X,T_{0})$$\mathscr{T}(\Sigma^{d}T_{d},\Sigma^{2d}X)$$\mathscr{T}(Y,\Sigma^{2d}X)$$\mathscr{T}(T_{0},\Sigma^{2d}X)$$\scriptstyle{\operatorname{D}\!\Psi}$$\scriptstyle{\alpha^{\prime}}$$\scriptstyle{\beta^{\prime}}$$\sim$$\sim$$\sim$$[\operatorname{add}\Sigma^{d}T](Y,\Sigma^{2d}X)$$\mathscr{T}(Y,\Sigma^{2d}X)/[\operatorname{add}\Sigma^{d}T](Y,\Sigma^{2d}X)$$\scriptstyle{\alpha}$$\scriptstyle{\beta}

Analogous to the above discussion, the space $[\operatorname{add}\Sigma^{d}T](Y,\Sigma^{2d}X)$ is the image of the map $\alpha^{\prime}$. Hence $\alpha$ is the kernel of $\beta^{\prime}$ and $\operatorname{D}\!\Psi$ (by isomorphism). The morphism $\beta$ is by definition the cokernel of $\alpha$, and $\mathscr{T}(Y,\Sigma^{2d}X)/[\operatorname{add}\Sigma^{d}T](Y,\Sigma^{2d}X)$ is thus the image of $\operatorname{D}\!\Psi$. Thus we have

[TABLE]

∎

Lemma 2.5**.**

Suppose $X,Y\in\mathscr{T}$.Then we have a short exact sequence

[TABLE]

Proof.

By the definition of the quotient functor we have a short exact sequence

[TABLE]

We have $[\operatorname{add}\Sigma^{d}T](X,\Sigma^{d}Y)\cong[\operatorname{add}T](\Sigma^{-d}X,Y)$. By Lemma 2.4 we have

[TABLE]

We also know that $\mathscr{T}(X,\Sigma^{d}Y)\cong\operatorname{Ext}^{d}_{\mathscr{T}}(X,Y)$, so the conclusion follows. ∎

Lemma 2.6**.**

Suppose $X,Y\in\mathscr{T}$ have no non-zero direct summands in $\operatorname{add}\Sigma^{d}T$. Then we have a short exact sequence

[TABLE]

Proof.

Consider the short exact sequence from Lemma 2.5. By Theorem 0.4 we know that

[TABLE]

Applying Lemma 2.3 we have

[TABLE]

Similarly we can show \operatorname{Hom}_{\mathscr{T}/\operatorname{add}\Sigma^{d}T}(\overline{X},\overline{\Sigma^{d}Y})\cong\operatorname{Hom}_{\Gamma}\big{(}\mathscr{T}(T,X),\tau_{d}\mathscr{T}(T,Y)\big{)}. ∎

The map defined next will eventually induce the equivalence of Theorem B.

Definition 2.7**.**

For each $X\in\mathscr{T}$, pick an isomorphism $X\cong X^{\prime}\oplus X^{\prime\prime}$ such that $X^{\prime}$ has no non-zero direct summands in $\operatorname{add}\Sigma^{d}T$ and $X^{\prime\prime}\in\operatorname{add}\Sigma^{d}T$. Let

[TABLE]

This is a pair of $\Gamma$-modules where $\mathscr{T}(T,X^{\prime})$ is in $\mathscr{D}$ and $\mathscr{T}(T,\Sigma^{-d}X^{\prime\prime})$ is in $\operatorname{proj}\Gamma$.

Proposition 2.8**.**

Given $X,Y\in\mathscr{T}$, set $(M,P)=\Delta(X)$ and $(N,Q)=\Delta(Y)$, where $\Delta$ is the map in Definition 2.7. Then

[TABLE]

Proof.

By additivity of $\operatorname{Ext}$ we have

[TABLE]

As $T$ is $d$-rigid, we see that $\operatorname{Ext}^{d}_{\mathscr{T}}(X^{\prime\prime},Y^{\prime\prime})=0$, and hence we have

[TABLE]

From Lemma 2.6 we have the short exact sequence:

[TABLE]

which means that

[TABLE]

We see that

[TABLE]

The third isomorphism follows from [14, Lemma 2.2(i)] and the fact that $\Sigma^{-d}X^{\prime\prime}\in\operatorname{add}T$. Similarly,

[TABLE]

Thus we have

[TABLE]

Substituting (2.2), (2.3), and (2.4) into (2.1) gives the result. ∎

As a consequence we have:

Corollary 2.9**.**

Given $X,Y\in\mathscr{T}$, set $(M,P)=\Delta(X)$ and $(N,Q)=\Delta(Y)$. Then

[TABLE]

3. Proof of Theorem B

The following results use the map $\Delta$ from Definition 2.7.

Lemma 3.1**.**

Given $X,Y\in\mathscr{T}$, set $(M,P)=\Delta(X)$ and $(N,Q)=\Delta(Y)$. Then $Y\in\operatorname{add}X$ if and only if $N\in\operatorname{add}M\mbox{ and }Q\in\operatorname{add}P$.

Proof.

Let $X\cong X^{\prime}\oplus X^{\prime\prime}$ be the decomposition from Definition 2.7, where $X^{\prime}$ has no non-zero direct summands from $\operatorname{add}\Sigma^{d}T$ while $X^{\prime\prime}$ is in $\operatorname{add}\Sigma^{d}T$. We have (M,P)=\big{(}\mathscr{T}(T,X^{\prime}),\mathscr{T}(T,\Sigma^{-d}X^{\prime\prime})\big{)}. Similarly, (N,Q)=\big{(}\mathscr{T}(T,Y^{\prime}),\mathscr{T}(T,\Sigma^{-d}Y^{\prime\prime})\big{)}.

The condition $Q\in\operatorname{add}P$ is equivalent to $Y^{\prime\prime}\in\operatorname{add}X^{\prime\prime}$ by the $\operatorname{add}$-$\operatorname{proj}$-correspondence, (see Remark 0.5). The condition $N\in\operatorname{add}M$ is equivalent to $Y^{\prime}\in\operatorname{add}X^{\prime}$ by Theorem 0.4 because $X^{\prime},Y^{\prime}$ have no non-zero direct summands in $\operatorname{add}\Sigma^{d}T$. The result follows. ∎

Lemma 3.2**.**

The category $\mathscr{T}$ is skeletally small. The map $\Delta$ induces a bijection

[TABLE]

where $\operatorname{iso}$ denotes the set of isomorphism classes of a skeletally small category.

Proof.

Let $\operatorname{Iso}$ denote the class of isomorphisms of a category. For a skeletally small category $\mathscr{C}$ we have that $\operatorname{Iso}\mathscr{C}=\operatorname{iso}\mathscr{C}$. Note that since a module category over a ring is skeletally small, we have that $\mathscr{D},\operatorname{proj}\Gamma\subseteq\operatorname{mod}\Gamma$ are skeletally small.

It is clear that $\Delta$ induces a well-defined map of the form

[TABLE]

To see that $\delta^{\prime}$ is injective, argue like the proof of Lemma 3.1, replacing membership of $\operatorname{add}$ with isomorphism.

It follows that $\mathscr{T}$ is skeletally small. We can thus replace $\delta^{\prime}$ with the map $\delta$ from (3.1).

To see that $\delta$ is surjective, let $(M,P)$ be a pair with $M\in\mathscr{D}$ and $P\in\operatorname{proj}\Gamma$. By Theorem 0.4 there is an object $X^{\prime}\in\mathscr{T}$ with no non-zero direct summands in $\operatorname{add}\Sigma^{d}T$ such that $M\cong\mathscr{T}(T,X^{\prime})$. By the add-proj correspondence, see Remark 0.5, there is an object $X^{\prime\prime}\in\operatorname{add}\Sigma^{d}T$ such that $P\cong\mathscr{T}(T,\Sigma^{-d}X^{\prime\prime})$. Setting $X=X^{\prime}\oplus X^{\prime\prime}$ gives $(M,P)\cong\Delta(X)$. ∎

Lemma 3.3**.**

If $X\in\mathscr{T}$ is $d$-self-perpendicular, then $(M,P)=\Delta(X)$ is a maximal $\tau_{d}$-rigid pair.

Proof.

Let $N\in\mathscr{D}$ and $Q\in\operatorname{proj}\Gamma$ be given. By Lemma 3.2, there is an object $Y\in\mathscr{T}$ such that $(N,Q)\cong\Delta(Y)$. Then

[TABLE]

where the equivalences, respectively, are by Lemma 3.1, Definition 0.2, and Corollary 2.9.

The conditions of Definition 0.7 are recovered by setting $Q=0$ respectively $N=0$ ∎

Lemma 3.4**.**

Let $X\in\mathscr{T}$ be given. If $(M,P)=\Delta(X)$ is a maximal $\tau_{d}$-rigid pair, then $X$ is $d$-self-perpendicular.

Proof.

Let $Y\in\mathscr{T}$ be given and set $(N,Q)\cong\Delta(Y)$. Then

[TABLE]

where the equivalences, respectively, are by Corollary 2.9, Definition 0.7, and Lemma 3.1. ∎

Theorem 3.5**.**

Recall that the map $\Delta$ from Definition 2.7 induces the bijection $\delta:\operatorname{iso}\mathscr{T}\rightarrow\operatorname{iso}\mathscr{D}\times\operatorname{iso}\operatorname{proj}\Gamma$ from Lemma 3.2.

1. (i)

$\delta$* restricts to a bijection*

[TABLE] 2. (ii)

$\delta$* restricts further to a bijection*

[TABLE]

Proof.

(i): Consider $X\in\mathscr{T}$ and set $(M,P)=\Delta(X)$. Then

[TABLE]

by Corollary 2.9, so the result follows.

(ii): See Lemmas 3.3 and 3.4. ∎

*Proof *(of Theorem B from the introduction). Combine Theorems 3.5(ii) and 1.1(ii). $\Box$

4. An example

In this section we let $d=3$ and $\mathscr{T}=\mathscr{O}_{A_{2}^{3}}$. This is the $5$-angulated (higher) cluster category of type $A_{2}$, see [21, def. 5.2, sec. 6, and sec. 8]. The indecomposable objects can be identified with the elements of the set

[TABLE]

see [21, sec. 8]. The AR quiver of $\mathscr{T}$ is shown in Figure 1.

By [21, thm. 5.5 and sec. 8], the object

[TABLE]

is Oppermann–Thomas cluster tilting.

If $X,Y\in\mathscr{T}$ are indecomposable objects, then

[TABLE]

see [21, prop. 6.1 and def. 6.9]. It follows that $\Gamma=\operatorname{End}_{\mathscr{T}}(T)=kQ/I$, where

[TABLE]

and $I$ is the ideal generated by all compositions of two consecutive arrows. The action of the functor $\mathscr{T}(T,-):\mathscr{T}\rightarrow\operatorname{mod}\Gamma$ on indecomposable objects is shown in Figure 2, where $P(q)$ and $I(q)$ denote the indecomposable projective and injective modules associated to the vertex $q\in Q$.

Note that the essential image of $\mathscr{T}(T,-)$ is

[TABLE]

This is a $3$-cluster tilting subcategory of $\operatorname{mod}\Gamma$ and hence it is $3$-abelian.

The $3$-suspension functor $\Sigma^{3}$ acts on the AR quiver by moving four steps clockwise. Combined with our knowledge of $\operatorname{Hom}$, this shows that if $X$ is a fixed indecomposable object in $\mathscr{T}$, then the indecomposable objects $Y$ with $\operatorname{Ext}^{3}_{\mathscr{T}}(X,Y)\neq 0$ are precisely the two objects furthest from $X$ in the AR quiver, see Figure 3.

Based on this, we can compute all basic $3$-self-perpendicular objects in $\mathscr{T}$, and by Proposition 1.3 they coincide with the basic maximal $3$-rigid objects in $\mathscr{T}$. For each such object $X$, there is a maximal $\tau_{3}$-rigid pair \Delta(X)=\big{(}\mathscr{T}(T,X^{\prime}),\mathscr{T}(T,\Sigma^{-3}X^{\prime\prime})\big{)} by Theorem B. See Figure 4.

Note that the first nine objects in Figure 4 are Oppermann–Thomas cluster tilting, but the three last objects are not.

Acknowledgement. This work was supported by EPSRC grant EP/P016014/1 “Higher Dimensional Homological Algebra”. Karin M. Jacobsen is grateful for the hospitality of Newcastle University during her visit in October 2018.