$d\mathbb{Z}$-cluster tilting subcategories of singularity categories

For an exact category $\mathcal{E}$ with enough projectives and with a $d\mathbb{Z}$-cluster tilting subcategory, we show that the singularity category of $\mathcal{E}$ admits a $d\mathbb{Z}$-cluster tilting subcategory. To do this we introduce cluster tilting subcategories of left triangulated categories, and we show that there is a correspondence between cluster tilting subcategories of $\mathcal{E}$ and $\underline{\mathcal{E}}$. We also deduce that the Gorenstein projectives of $\mathcal{E}$ admit a $d\mathbb{Z}$-cluster tilting subcategory under some assumptions. Finally, we compute the $d\mathbb{Z}$-cluster tilting subcategory of the singularity category for a finite-dimensional algebra which is not Iwanaga-Gorenstein.


Introduction
Auslander-Reiten theory is a fundamental tool to describe the module category of finite-dimensional algebras, see [7] and [4,44,45]. A generalization of this theory, called higher Auslander-Reiten theory, was introduced by Iyama in [25] and further developed in [24,27]. In this case, the objects of study are module categories equipped with a d-cluster tilting subcategory. We refer to [2,16,21,22,23,29,30,31,32,33,35,37] for some other important papers. Also, see [26] for a survey of the theory and [36] for an introduction.
Let Λ be a finite-dimensional algebra and let mod Λ be the category of finitely generated right Λ-modules. Assume Λ has global dimension d. If M is a d-cluster tilting subcategory in mod Λ then the subcategory U = add{M [di] ∈ D b (mod Λ) | M ∈ M and i ∈ Z} is d-cluster tilting inside the bounded derived category D b (mod Λ) [27, Theorem 1.21] (this can be extended to τ d -finite algebras [27,Theorem 1.23] and cluster categories [1,Theorem 4.10] and [17,Theorem 2.2]). The subcategory U can be considered as a higher analogue of the derived category of a hereditary algebra. On the other hand, if Λ does not have global dimension d, then there is no known cluster tilting subcategory inside D b (mod Λ) in general. As shown in [34], the naive approach doesn't necessarily give a cluster tilting subcategory even when M is dZ-cluster tilting.
In this paper we consider instead the singularity category where K b (proj Λ) and K −,b (proj Λ) denote the bounded homotopy category of finitely generated projective modules and the right bounded homotopy category with bounded homology of finitely generated projective modules, respectively. The singularity category was introduced by Buchweitz in [11] as an useful invariant of the ring Λ. Via the equivalence K −,b (proj Λ) ∼ = D b (mod Λ) we get an equivalence and hence D sing (Λ) can be realized as a quotient of D b (mod Λ). We show that if mod Λ has a dZ-cluster tilting subcategory, then D sing (Λ) has a dZcluster tilting subcategory. In fact, we show this more generally for any exact category with enough projectives.
Theorem 1.2. Let E be an exact category with enough projectives P and with a dZ-cluster tilting subcategory M. Then the subcategory is a dZ-cluster tilting subcategory of K −,b (P)/K b (P).
Via the equivalence (1.1) this corresponds to the subcategory for some M ∈ M and i ∈ Z} in D b (mod Λ)/ perf Λ. Notice that this subcategory is closed under direct sums since any object in D b (mod Λ)/ perf Λ is isomorphic to a stalk complex.
There are many examples of d-cluster tilting subcategories of singularity categories of Iwanaga-Gorenstein rings, see [28], but Theorem 1.2 is the first result for non-Iwanaga-Gorenstein rings. For such rings the singularity category is more difficult to control since it is not enhanced by the Gorenstein projective modules. However, it is still possible to compute the subcategory explicitly, which we do in Example 8.4. To prove Theorem 1.2, we use the left triangulated structure of E, see Definition 3.1 and Theorem 3.2. More precisely, we introduce d-cluster tilting subcategories of left triangulated categories, and we show that there is a one-to-one correspondence between d and dZ-cluster tilting subcategories of E and E, see Theorem 6.1. Furthermore, we show that if a left triangulated category has a dZ-cluster tilting subcategory, then its stabilization has a dZ-cluster tilting subcategory. We then conclude using the fact that the stabilization of E is the singularity category, which was proved in [39].
We also obtain a corollary for Gorenstein projective modules, which we state below in the special case where E = mod Λ. Recall that Λ is Iwanaga-Gorenstein if it has finite selfinjective dimension, and that in this case the Gorenstein projectives are . Assume Λ is Iwanaga-Gorenstein, and let M be a dZ-cluster tilting subcategory of mod Λ. Then is a dZ-cluster tilting subcategory of GP(mod Λ).
We now describe the structure of the paper. In Section 2, 3 and 4 we recall the essential notions and results which we need. In Section 5 we introduce cluster tilting subcategories for left triangulated categories. We show that for a left triangulated category C with a dZ-cluster tilting subcategory, the stabilization ZC has a dZ-cluster tilting subcategory. We also investigate the (d + 2)-angulated structure of this subcategory. Our main result in Section 6 is Theorem 6.1, which gives a correspondence between d-cluster tilting subcategories of E and E when E is an exact category with enough projectives. In Section 7 we investigate the relationship with Gorenstein projectives. In Section 8 we compute the cluster tilting subcategory of the singularity category in two examples.

Exact categories
Here we define exact categories, following the conventions in [12].
1. An exact category E is an additive category equipped with a distinguished class of sequences where f is the kernel of g and g is the cokernel of f . The morphisms f are called admissible monomorphisms, and the morphisms g are called admissible epimorphisms. The following axioms need to be satisfied: (E0) For all object E in E the identity morphism 1 E : E → E is an admissible monomorphism; (E0 op ) For all object E in E the identity morphism 1 E : E → E is an admissible epimorphism; (E1) The composite of two admissible monomorphism is an admissible monomorphism; (E1 op ) The composite of two admissible epimorphisms is an admissible epimorphism; (E2) The pushout of an admissible monomorphism exists and yields an admissible monomorphisms. In other words, given an admissible monomorphism f : E 0 → E 1 and a morphism g : E 0 → E 2 there exists a pushout diagram where k is an admissible monomorphism; (E2 op ) The pullback of an admissible epimorphism exists and yields an admissible epimorphism. In other words, given an admissible epimorphism f : E 1 → E 0 and a morphism g : If E is an exact category, then the opposite category E op becomes an exact category in a natural way. If F ⊂ E is a full subcategory of E which is closed under extensions, then the class of sequences 0 → F 1 − → F 2 − → F 3 → 0 in F which are exact in E makes F into an exact category. We say that F is an exact subcategory of E.
An object P in E is projective if for any admissible epimorphism E 1 → E 0 the induced map E(P, E 1 ) → E(P, E 0 ) is an epimorphism. We let P denote the subcategory of E consisting of the projective objects. The exact category E has enough projectives if for any object E in E there exists an admissible epimorphism P → E with P ∈ P. In this case we let E = E/P denote the stable category of E modulo projectives. For any object E or morphism f in E we denote the corresponding object or morphism in E by E or f . Since E has enough projectives, f = 0 for a morphism f : E → E ′ if and only if f factors through an admissible epimorphism P → E ′ with P projective.
Using this, it follows by the same argument as in the proof of [19, Theorem 2.2] that for any two objects E 0 and E 1 in E there exists an isomorphism E 1 ∼ = E 2 in E if and only if there exist projective objects P, Q ∈ E and an isomorphism Next we define the syzygy functor: . Let E be an exact category with enough projectives. For each object E ∈ E choose an admissible epimorphism p E : P E → E with P E projective. The syzygy functor Ω : E → E is defined as follows: For an object E ∈ E we set ΩE = Ker p E . For a morphism f : Note that Ω(f ) is independent of the choice of morphisms g and h, and up to isomorphism the syzygy functor is independent of the choice of the admissible epimorphisms p E , see Section 3 in [19] for details. By abuse of notation, for E ∈ E and i ≥ 0 we let Ω i E denote a choice of an object in E satisfying Ω i E ∼ = Ω i E.
An object I in E is injective if it is projective in E op , and E has enough injectives if E op has enough projectives. The exact category E is called Frobenius if E has enough projectives and enough injectives, and if the projective and injective objects coincide. In this case, the stable category E becomes a triangulated category where the suspension functor is the quasiinverse of Ω. We refer to [18, Section I.2] for more details.
We end this section with the following lemma, which we need later.
Lemma 2.4. Let E be an exact category with enough projectives, and let E be an object in E. The following statements are equivalent: Proof. This is well known, see [5,Proposition 2.43]. See also [41,Lemma 9] for a direct proof which can be translated verbatim to an exact category.

Left triangulated categories
Here we recall the notion of a left triangulated category. This was first considered in [38], [39] (where it would be called a co-suspended category), and later in [3], [8], [9]. A higher dimensional version has also been introduced in [42].
Let C be a category and Ω : C → C an endofunctor. A sequence of the form Composition of morphism is given in the canonical way.
Definition 3.1. A left triangulated category is an additive category C equipped with an additive endofunctor Ω : C → C and a class of Ω-sequences called triangles, which satisfy the following axioms: (T0) Any Ω-sequence which is isomorphic to a triangle is a triangle itself; where the rows are triangles and the square commutes, then there exists a morphism f : A → A ′ making the whole diagram commute; (T5) Given two morphisms w : B → C and h : where the middle row and the second column are also triangles.
If ΩC → A → B − → C is a triangle and D ∈ C, then applying Hom C (D, −) gives a long exact sequence Assume E is an exact category with enough projectives. Any diagram of the form 0 The following result follows from [9, Theorem 3.1].
Theorem 3.2. Let E be an exact category with enough projectives. Then the stable category E together with syzygy functor Ω : E → E and the class of all Ω-sequences isomorphic to a sequence ΩE 1 Now assume C is a category and Ω : C → C is an endofunctor. Let ZC be the stabilization of E [20]. Explicitly, the objects of ZC are pairs (C, n) where C is an object in C and n ∈ Z is an integer. The morphism space between two objects (C, m) and (C ′ , n) is given by where the colimit is taken over all k ∈ Z such that m + k ≥ 0, and n + k ≥ 0. Since this is a filtered colimit, it follows that any morphism (C, m) → (C ′ , n) has a representative Ω m+k (C) → Ω n+k (C ′ ) in C for some k. Composition (C, m) → (C ′ , n) → (C ′′ , p) in ZC is given by composing representatives given by Σ(C, n) = (C, n − 1) and Ω(C, n) = (C, n + 1) on objects and by the canonical identifications to get maps on the morphisms spaces. If C is a left triangulated category, then ZC also comes equipped with a class of sequences called standard triangles, which are induced from a sequence in C for some integer k ∈ Z, and where Here K b (P) and K −,b (P) denote the bounded homotopy category with components in P and the right bounded homotopy category with bounded homology and with components in P, respectively. The Verdier quotient is the singularity category of E. We refer to Section 4 in [40] for details on Verdier quotients and localization of triangulated categories.

Cluster tilting subcategories
Let E be an additive category and M a full additive subcategory of E. We recall the following notions. ( covariantly finite in E. We recall the definition of d and dZ-cluster tilting subcategories in the following. If E is triangulated with suspension functor Σ, then by Ext i Definition 4.1. Let E be an exact or a triangulated category, let M be a full subcategory of E, and let d > 0 be a positive integer. We say that M is d-cluster tilting in E if the following hold: We need following result later: (a) M is a d-cluster tilting subcategory of E; Proof. For any E ∈ E choose a right M-approximation f : M → E and an admissible epimorphism g : Then the morphism f g : M ⊕ M ′ → E is a right M-approximation and an admissible epimorphism. Similarly, one can construct a left M-approximation which is an admissible monomorphism for any E ∈ E. Using this, the proof of Proposition 2.2.2 in [25] goes through in exactly the same way for exact categories, so the claim holds.
The following result shows that condition (b 0 ) + (c 0 ) is equivalent to dcluster tilting without the assumption that M is functorially finite.
Let M ∈ M be arbitrary. Applying Hom E (−, M ) to the short exact sequence (Coker f 1 , M ). Therefore, the map is an epimorphism. Since M ∈ M was arbitrary, we get that f 1 : E → M 1 is a left M-approximation. Hence, M is covariantly finite. Furthermore, if we assume Ext i E (M ′ , E) = 0 for 1 ≤ i ≤ d − 1 and M ′ ∈ M, then the same argument as above with M replaced by E shows that the map is an epimorphism. Therefore, f 1 : E → M 1 is a split monomorphism. Since f 1 is also an admissible monomorphism, it follows that E is a summand of

Cluster tilting subcategories of left triangulated categories
In this section we define cluster tilting subcategories of left triangulated categories. We show that when the ambient category is triangulated, then this coincides with the classical definition. Finally, we show that if a left triangulated category C has a dZ-cluster tilting subcategory, then so does the stabilization ZC.
Let C be a left triangulated category. We call a sequence in C where an arrow C ′ o o ✤ ❴ C denotes a morphism C ′ ← Ω(C) in C, each oriented triangle is a triangle in C, each non-oriented triangle commute, and α d+2 is equal to the composite Note that here the symbol n.5 means n + 0.5. This definition of (d + 2)angle differs slightly from [32], where they do not include the morphism Ω d (C 1 ) → C d+2 in the definition.
Definition 5.1. Let C be a left triangulated category and d > 0 a positive integer. A full additive subcategory X of C is d-cluster tilting if it satisfies the following: (i) X is closed under direct summands in C; (ii) For all objects C in C there exist (d + 2)-angles (iv) If X and X ′ are in X , then If furthermore Ω d (X ) ⊂ X , then we say that X is dZ-cluster tilting in C.
We use the terminology d-cluster tilting since for an exact category E with enough projectives we then get a correspondence between d-cluster tilting subcategories of E and E, see Theorem 6.1.

Remark 5.2.
We see that condition (ii) in Definition 5.1 is similar to condition (ii) in Proposition 4.4, and it can be considered as a substitute of Definition 4.1 (ii) and of functorially finiteness. The fact that it behaves much better under stabilization is also a crucial property we need.
Next we show that Definition 5.1 and Definition 4.1 are equivalent when Ω is an automorphism, i.e. when C is triangulated. In the following we let Σ denote the quasi-inverse of Ω. Proof. By Remark 4.2 (iii) the claim for dZ-cluster tilting subcategories follows immediately from the claim for d-cluster tilting subcategories. Hence, we only prove the latter. Note that a d-cluster tilting subcategory in the sense of Definition 4.1 obviously satisfies (i), (iii) and (iv) in Definition 5.1, while axiom (ii) follows from [32, Corollary 3.3] and its dual. For the converse, assume X ⊂ C is d-cluster tilting in the sense of Definition 5.1. As usual, for subcategories Y ′ and Y of C, we denote by Y ′ * Y the subcategory consisting of all C ∈ C admitting a triangle Y ′ → C → Y → ΣY ′ with Y ′ ∈ Y ′ and Y ∈ Y. By Definition 5.1 (ii) we have C = Ω d−1 X * · · · * ΩX * X . For each C ∈ C, take a triangle with Y ∈ Ω d−1 X * · · · * ΩX and X ∈ X . Since Hom C (Y, X ) = 0 by Definition 5.1 (iv), g is a left X -approximation and hence X is covariantly finite. Moreover, if Hom C (Ω i X, C) = 0 for 1 ≤ i ≤ d − 1, then f = 0 and hence C ∈ X . Since Hom C (X , Σ i X) = 0 for 1 ≤ i ≤ d− 1 and X ∈ X by Definition 5.1 (iv), this shows that The fact that X is contravariantly finite and the equality is shown dually.
Recall that the stabilization ZC of a left triangulated category C is a triangulated category, see Theorem 3.4.
Definition 5.4. Let C be a left triangulated category and X a dZ-cluster tilting subcategory of C. Define dZX to be the full subcategory of ZC consisting of all objects isomorphic to objects of the form (X, dk) with X ∈ X and k ∈ Z.
Our goal is to show that dZX is dZ-cluster tilting in ZC.
Lemma 5.5. The subcategory dZX is closed under direct summands.
Proof. Two objects (C, n) and (C ′ , n ′ ) are isomorphic in ZC if and only if there exists an integer k such that Ω k+n (C) and Ω k+n ′ (C ′ ) are isomorphic in C. Hence, dZX consists of all objects (C, n) such that there exists an integer k with Ω dk+n (C) ∈ X . Now assume that (C 1 , n 1 ) ⊕ (C 2 , n 2 ) ∈ dZX .
Therefore there exists an integer k such that Since X is closed under direct summands by Definition 5.1 (i), we have that Ω dk+n 1 (C 1 ) ∈ X and Ω dk+n 2 (C 2 ) ∈ X and hence (C 1 , n 1 ) ∈ dZX and (C 2 , n 2 ) ∈ dZX .
This proves the claim.
Theorem 5.7. Let C be a left triangulated category and X a dZ-cluster tilting subcategory of C. Then dZX is a dZ-cluster tilting subcategory of the triangulated category ZC.
Proof. We show that dZX satisfies Definition 5.1. Note that axiom (i) and (iv) follows from Lemma 5.5 and Lemma 5.6 respectively, and axiom (iii) is clear since ZC is a triangulated category. It therefore only remains to show axiom (ii). Let (C, n) ∈ ZC be arbitrary. Choose k such that dk < n. Then we have an isomorphism (C, n) ∼ = (Ω n−dk (C), dk). Hence, we can assume for simplicity that n = dk. Now choose (d + 2)-angles Applying Axiom (T3) in Definition 3.1 repeatedly, we obtain (d + 2)-angles in ZC, which prove the claim.
Remark 5.8. Note that the proof of Theorem 5.7 does not use axiom (iii) in Definition 5.1. This axiom is needed to prove Theorem 6.1 (and in particular Lemma 6.4) to get a correspondence between dZ-cluster tilting subcategories in the exact category E and in the left triangulated category E The category dZX is also equivalent to the stabilization of X with respect to the functor Ω d : X → X . Hence, it can be computed without describing the categories C and ZC, which is more complicated to do in general.
Since dZX is a dZ-cluster tilting subcategory of ZC, it has the structure of a (d + 2)-angulated category [16,Theorem 4.1], where is the suspension functor (see (3.3)) applied d times. The (d + 2)-angles in the sense of [16] are all (d + 2)-angles in ZC in our sense

is a (d + 2)-angle if and only if it is induced from a sequence in C of the form
Proof. This follows immediately from the description of the triangles in ZC together with axiom (T3) in Definition 3.1.

d-cluster tilting in stable categories
Let E be an exact category with enough projectives. In this section we compare cluster tilting subcategories in the exact category E and those in the left triangulated category E. Our main goal is to prove the following theorem: In the special case when E is Frobenius (and hence E is triangulated) the theorem is easy and well-known.
We start by proving one direction of the theorem.
Proof of "only if" part of Theorem 6.1. If M is dZ-cluster tilting, then for Therefore Ω d (M) ⊆ M. Hence the "only if" part of Theorem 6.1 (ii) follows from the "only if" part of Theorem 6.1 (i) by Remark 4.2 (ii).
Assume M is d-cluster tilting in E. Since M is closed under direct summands, it follows using the description of the isomorphisms from (2.2) that M is closed under direct summands. Now for E ∈ E we can choose exact sequences By definition of the left triangulated structure of E, we get (d + 2)-angles This shows that part (ii) of Definition 5.1 holds for M. Also, by Lemma 2.4 the map is an isomorphism for any M ∈ M, E ∈ E and 1 ≤ i ≤ d − 1 since = 0 for all P ∈ P. Hence, part (iii) of Definition 5.1 also holds for M. Finally, to prove part (iv), we use the basic fact that if Ext 1 E (E, P) = 0, then with exact rows, where P 1 , · · · , P d are projective. If we call such a sequence in dZM a standard (d + 2)-angle, then a (d + 2)-angle of dZM is precisely a sequence which is isomorphic to a standard (d + 2)-angle.
We now want to show the "if" part of Theorem 6.1. To this end, we fix a full subcategory M of E and assume M is d-cluster tilting in the left triangulated category E. Our goal is to show that M is d-cluster tilting in E.
Proof. Let E ∈ E be arbitrary, and choose a (d + 2)-angle Hence, by definition of triangles in E (see Section 3) there exist a projective object P ∈ E and an exact sequence is an epimorphism. Hence, the inclusion is a split monomorphism. The inclusion is also a composite of two admissible monomorphism, and it is therefore admissible. Therefore, its cokernel exists, which we denote by M 1 . We can therefore write the sequence as for some morphisms g 1 , g 2 . It follows from [12,Corollary 2.18] that the sequence is exact. This proves one part of the lemma. The other part is proved dually.
Proof of "if" part of Theorem 6.1. Part (i) follows from Proposition 4.4, Lemma 6.4 and Lemma 6.5. Part (ii) follows from part (i) together with Remark 4.2 (ii).

Gorenstein projectives
In this section we consider the subcategory of Gorenstein projective objects in E. These objects were first introduced in [5] for modules over a noetherian ring. We refer to [13] for a survey of the theory for Artin algebras, and to [14] for more general rings.
Let E be an exact category with enough projectives P. Recall that a long exact sequence P • = · · · → P 0 → P 1 → · · · of projective objects in E is called totally acyclic if the complex is acyclic for all projective objects Q in E. An object G ∈ E is called Gorenstein projective if there exists a totally acyclic complex P • with G = Z 0 (P • ) := Ker(P 0 → P 1 ).
We let GP(E) denote the subcategory of E consisting of all Gorenstein projective objects. The subcategory GP(E) is closed in E under extensions, direct summands, and kernels of admissible epimorphisms. In fact, the proof of [13, Proposition 2.1.7] also works for exact categories. In particular, Ω : E → E restricts to a functor Ω : GP(E) → GP(E).
The Gorenstein projective dimension of an object E ∈ E, denoted dim GP(E) E, is the smallest integer n such that Ω n (E) ∈ GP(E). We write dim GP(E) E = ∞ if no such integer exists.
Since GP(E) is an extension closed subcategory of E, it inherits an exact structure making the inclusion GP(E) → E into an exact functor. Under this exact structure GP(E) becomes a Frobenius exact category with projective/injective objects being the objects in P, see [13,Proposition 2.1.11]. Hence, GP(E) is a triangulated category. In particular, Ω : GP(E) → GP(E) is an autoequivalence, and the quasi-inverse of Ω is the suspension functor for the triangulated category. The triangles in GP(E) are precisely all triangles in the left triangulated category E with components in GP(E). In particular, we see that the canonical functor restrict to a functor of triangulated categories This functor is fully faithful since Ω is an autoequivalence on GP(E). The result below gives necessary and sufficient condition for it to be an equivalence. It was first shown in [11] for a noetherian ring.
In particular, the functor is an equivalence if and only if dim GP(E) E < ∞ for all E ∈ E.
Proof. If (E, n) ∼ = (G, 0) in ZE with G ∈ GP(E), then Ω k+n (E) ∼ = Ω k (G) in E for some k > 0. Since Ω k (G) ∈ GP(E) it follows that E has finite Gorenstein projective dimension. Conversely, if dim GP(E) E = k < ∞, then for any n ∈ Z there exist isomorphisms and since Ω k E ∈ GP(E), it follows that Ω n−k (Ω k E) ∈ GP(E). This proves the claim. We now relate this to the theory of cluster tilting subcategories.

− → ZE
and since dZM is a dZ-cluster tilting subcategory of ZE by Theorem 5.7, the preimage of dZM is a dZ cluster tilting subcategory of GP(E). Explicitly, the preimage consists of all objects G ∈ GP(E) such that Ω dk (G) ∈ M for some integer k ≥ 0. To show that this is equal to M ∩ GP(E), we only need to show that such a G is contained in M. Note first that since G ∈ GP(E), it follows that Ext i E (G, P ) = 0 for all P ∈ P. Hence, by a dimension shifting argument we get that for any E ∈ E and any j ≥ 0. Now let M ∈ M be arbitrary, and choose an integer k ≥ 0 such that Ω dk (G) ∈ M and Ω dk (M ) ∈ GP(E). It follows that where we use dimension shifting and the fact that Ω dk M ∈ M since M is dZ-cluster tilting. Since M ∈ M was arbitrary, it follows that G ∈ M, which proves the claim.

Examples
In this section we compute the singularity category for a higher Nakayama algebras of type A ∞ ∞ , see [35]. For higher Nakayama algebras of typeÃ we refer to Section 4.3 in [43].
Example 8.1. We follow the notation in [35]. Let k be a field. Let l = (· · · , l −1 , l 0 , l 1 , · · · ) be the Kupisch Series of type with period 4, and with relations making all squares commute, and such that the following composites are 0.
By [35,Theorem 3.16] we know that the category of finitely presented modules mod A (2) l has a 2Z-cluster tilting subcategory M l . Explicitly, M (2) l is the k-linear additive category given by the infinite periodic quiver / / 012 infinite projective dimension. This shows that algebra A (2) l ′ is not Iwanaga-Gorenstein.