Maximal $\tau_d$-rigid pairs

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.


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 τ -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 τ -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 τ -rigid pairs permit a bijection to the maximal d-rigid objects, but not to the cluster tilting objects.
Cluster tilting and maximal d-rigid objects. An object X ∈ T is d-rigid if Ext d T (X, X) = 0. We recall three important definitions.
Definition 0.1 ([21, def. 5.3]). An object X ∈ T is Oppermann-Thomas cluster tilting in T if: (ii) For any Y ∈ T there exists a (d + 2)-angle Our first main result is: 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 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 τ d -rigid pairs. Let T ∈ T be an Oppermann-Thomas cluster tilting object and let Γ = End T (T ). Recall the following result. Then D is a d-cluster tilting subcategory of mod Γ. 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: The category D is a d-abelian category by [15, thm. 3.16]. It has a d-Auslander-Reiten translation τ d , which is a higher analogue of the classic Auslander-Reiten translation τ , see [12, Remark 0.5. The classic add-proj-correspondence holds, as T (T, −) restricts to an equivalence add T → proj Γ . The functor also restricts to an equivalence add ST → inj Γ. [14, lem. 2.1] It is natural to ask if D permits a higher analogue of the τ -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 ∈ D and P ∈ proj Γ is called a τ d -rigid pair in D if M is τ d -rigid and Hom Γ (P, M) = 0.
Definition 0.7. A pair (M, P ) with M ∈ D and P ∈ proj Γ is called a maximal τ d -rigid pair in D if it satisfies: A maximal τ d -rigid pair is a τ d -rigid pair.
Our second main result is: we do not think of maximal τ d -rigid pairs as support τ d -tilting pairs. The reason is that by Theorem B, maximal τ 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 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 Hom spaces in T and 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. D: The duality functor Hom k (−, k). T : A k-linear, Hom-finite, (d + 2)-angulated category with split idempotents. We assume that T is 2d-

Proof of Theorem A
Proof. (i), the first implication: Suppose X is Oppermann-Thomas cluster tilting. We must prove the equality in Definition 0.2, and the inclusion ⊆ is clear. For the inclusion ⊇, suppose Ext d which exists since X is Oppermann-Thomas cluster tilting. But then the morphism Σ d Y → Σ d X d is a split monomorphism, and applying Σ −d gives a split monomorphism Y → X d proving Y ∈ 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 ⊆ is clear.
, the third implication: This is clear.
(ii): Suppose that each indecomposable object in T is d-rigid. Because of part (i), it is enough to prove the implication ⇐ in (ii), so suppose that X is maximal d-rigid. We must prove the equality in Definition 0.2, and ⊆ is clear.
is closed under direct sums and summands by additivity of Ext. Hence it is enough to suppose that Y is an indecomposable object in this set and prove Y ∈ add X. However, Ext d 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 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. Recall from Setup 0.8 that T is a fixed Oppermann-Thomas cluster tilting object in T , and that T is 2d-Calabi-Yau, that is, T (X, Y ) ∼ = DT (Y, Σ 2d X) naturally in X, Y ∈ T .
Lemma 2.1. There is a natural isomorphism Proof. By the 2d-Calabi-Yau property we have Finally, by definition we have Lemma 2.2. If X ∈ T has no non-zero direct summands in add Σ d T , then there exists a (d+2)-angle Proof. Given X, there exists a (d + 2)-angle Proof. As X has no non-zero direct summands in add Σ d T , we can consider the (d + 2)-angle from Lemma 2.2. Apply T (T, −) to get the following part of an augmented minimal projective resolution in mod Γ: Using the Nakayama functor and Lemma 2.1 we get the following commutative diagram.
Proof. Pick a (d + 2)-angle in T : with T i ∈ add T . Use T (X, −) to obtain the morphism Ψ : T (X, T 0 ) → T (X, Y ). This is a homomorphism of k-vector spaces, hence we can talk about the image of Ψ. We first note that any morphism f in the image of Ψ must factor through add T . Now suppose f ∈ T (X, Y ) factors through T ′ ∈ add T . We have the following commutative diagram, where the lower row is a part of the (d + 2)-angle above: The dashed arrow exists by completing the commutative square to a morphism of (d + 2)-angles. We conclude that f ∈ Im Ψ. Hence Im Ψ = [add T ](X, Y ).
We now return to the long exact sequence Using the duality functor D and Serre duality we get the following diagram with exact rows: Analogous to the above discussion, the space [add Σ d T ](Y, Σ 2d X) is the image of the map α ′ . Hence α is the kernel of β ′ and DΨ (by isomorphism). The morphism β is by definition the cokernel of α, Lemma 2.5. Suppose X, Y ∈ T .Then we have a short exact sequence Proof. By the definition of the quotient functor we have a short exact sequence We also know that T (X, Σ d Y ) ∼ = Ext d T (X, Y ), so the conclusion follows. Lemma 2.6. Suppose X, Y ∈ T have no non-zero direct summands in add Σ d T . Then we have a short exact sequence Proof. Consider the short exact sequence from Lemma 2.5. By Theorem 0.4 we know that

Similarly we can show Hom
The map defined next will eventually induce the equivalence of Theorem B.
Definition 2.7. For each X ∈ T , pick an isomorphism X ∼ = X ′ ⊕ X ′′ such that X ′ has no non-zero direct summands in add Σ d T and X ′′ ∈ add Σ d T . Let This is a pair of Γ-modules where T (T, X ′ ) is in D and T (T, Σ −d X ′′ ) is in proj Γ.
Proof. By additivity of Ext we have From Lemma 2.6 we have the short exact sequence: We see that N). The third isomorphism follows from [14, Lemma 2.2(i)] and the fact that Σ −d X ′′ ∈ add T . Similarly,

Proof of Theorem B
The following results use the map ∆ from Definition 2.7. The condition Q ∈ add P is equivalent to Y ′′ ∈ add X ′′ by the add-proj-correspondence, (see Remark 0.5). The condition N ∈ add M is equivalent to Y ′ ∈ add X ′ by Theorem 0.4 because X ′ , Y ′ have no non-zero direct summands in add Σ d T . The result follows.
Lemma 3.2. The category T is skeletally small. The map ∆ induces a bijection δ : iso T → iso D × iso proj Γ, (3.1) where iso denotes the set of isomorphism classes of a skeletally small category.
Proof. Let Iso denote the class of isomorphisms of a category. For a skeletally small category C we have that Iso C = iso C . Note that since a module category over a ring is skeletally small, we have that D, proj Γ ⊆ mod Γ are skeletally small.
It is clear that ∆ induces a well-defined map of the form To see that δ ′ is injective, argue like the proof of Lemma 3.1, replacing membership of add with isomorphism.
It follows that T is skeletally small. We can thus replace δ ′ with the map δ from (3.1).
To see that δ is surjective, let (M, P ) be a pair with M ∈ D and P ∈ proj Γ. By Theorem 0.4 there is an object X ′ ∈ T with no non-zero direct summands in add Σ d T such that M ∼ = T (T, X ′ ). By the addproj correspondence, see Remark 0.5, there is an object Proof. Let N ∈ D and Q ∈ proj Γ be given. By Lemma 3.2, there is an object Y ∈ T such that (N, Q) ∼ = ∆(Y ). Then N ∈ add M and Q ∈ add P ⇔ Y ∈ add X Proof. Let Y ∈ T be given and set (N, Q) ∼ = ∆(Y ). Then where the equivalences, respectively, are by Corollary 2.9, Definition 0.7, and Lemma 3.1. Proof (of Theorem B from the introduction). Combine Theorems 3.5(ii) and 1.1(ii).

An example
In this section we let d = 3 and T = O A 3 2 . 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 and I is the ideal generated by all compositions of two consecutive arrows. The action of the functor T (T, −) : T → mod Γ 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 ∈ Q. Note that the This is a 3-cluster tilting subcategory of mod Γ and hence it is 3-abelian.
The 3-suspension functor Σ 3 acts on the AR quiver by moving four steps clockwise. Combined with our knowledge of Hom, this shows that if X is a fixed indecomposable object in T , then the indecomposable objects Y with Ext 3 T (X, Y ) = 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 T , and by Proposition 1.3 they coincide with the basic maximal 3-rigid objects in T . For each such object X, there is a maximal τ 3 -rigid pair ∆(X) = T (T, X ′ ), T (T, Σ −3 X ′′ ) 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.