Monomial Gorenstein algebras and the stably Calabi--Yau property

A celebrated result by Keller--Reiten says that $2$-Calabi--Yau tilted algebras are Gorenstein and stably $3$-Calabi--Yau. This note shows that the converse holds in the monomial case: a $1$-Gorenstein monomial algebra with a $3$-Calabi--Yau singularity category is $2$-Calabi--Yau tilted. We study the case of other Goresntein monomial algebras with stably Calabi--Yau singularity categories.


Introduction
A cluster-tilted algebra is the endomorphism algebra of a cluster-tilting object in a cluster category of a hereditary algebra, whereas 2-Calabi-Yau (2-CY) tilted algebras are obtained by replacing the cluster category by a 2-CY triangulated category. These algebras are 1-Iwanaga-Gorenstein (IG) [KR]. Cluster categories and cluster-tilted algebras were introduced in [BMRRT, CCS, BMR].
A large family of 2-CY tilted algebras are Jacobian algebras of a quiver with potential (QP) (Q, W ), [DWZ], so they are defined by a quiver with potential applyin cyclic derivatives (or a generalization called quiver with hyperpotential [Lad]). The 2-CY category C (Q,W ) was introduced by Amiot [Am] building on previous works by Keller and Ginzburg. Jacobian algebras include a well known class of gentle 2-CY tilted algebras whose QPs can be obtained from triangulations of unpunctured surfaces [ABCP]. This family includes the cluster-tilted algebras of type A andÃ. A related family of gentle algebras, that also have a geometric realization via unpunctured surfaces, are the m-cluster-tilted algebras of types A andÃ [T, Ba, Gub]. This family is a subfamily of gentle algebras defined by a bound quiver that can be realized as a dual graph obtained from (m + 2)-angulations as well. See the definition in Section 4.
This article provides a characterization of the monomial 2-CY tilted algebras mentioned above as Jacobian algebras, and also as 1-IG algebras with 3-CY stable category of Gorenstein projective objects (singularity category).
We also explore a related characterization over examples of gentle and monomial m-cluster tilted algebras. The main results are the following: Theorem 1 (Theorem 3.5). Let k be an algebraically closed field of characteristic zero and Λ = kQ/I a monomial k-algebra. The following are equivalent: (1) Λ is 2-CY tilted, (2) Λ is a 1-IG algebra with stably 3-CY singularity category, (3) Λ is Jacobian.
With the consideration of hyperpotentials instead of potentials, the condition char k = 0 can be removed.
Theorem 2 (Theorem 4.5). Let Λ T = Q T /I T be an algebra arising from a (m + 2)-angulation. Then N is Goresntein projective if and only if Ω m+1 τ N N over mod Λ T . In particular, this holds for m-cluster tilted algebras of type A andÃ.

Preliminaries
Throughout these notes, we consider k-algebras where k is an algebraically closed field of characteristic zero (if the contrary is not stated). Let Q = (Q 0 , Q 1 ) be a finite quiver, where Q 0 is the set of vertices and Q 1 the set of arrows. Let s, t : Q 1 → Q 0 be the functions that indicate the source and the target of each arrow, respectively. We will only consider finite dimensional basic k-algebras. Every finite dimensional Date: July 19, 2018. Key words: 2-Calabi-Yau tilted algebras, monomial algebras, Jacobian algebras, Gorenstein-projective modules. 2010 Mathematics Subject Classification: 16G20, 16E65, 18E30.
basic k-algebra Λ is isomorphic to a quotient kQ/I, where I is an admissible ideal. The pair (Q, I) is called a bound quiver [ASS].
Let Ω be the usual syzygy operator, τ the Auslander-Reiten (AR) translation, and D = Hom k (−, k). Denote by D b (A m [Be]. In this case each Λ-module either has infinite projective dimension or has projective dimension at most d. The stable subcategory of Gorenstein-projetive modules, also called singularity category, GP(Λ) is triangulated, with shift given by the formal inverse of Ω.
An algebra Λ = kQ/I is monomial if all the generators of I are paths α 1 . . . α t . Denote by F the minimal set of paths generating I. Following [LZ], when kQ/I is monomial and 1-IG the elements of F are very particular: F is a disjoint union m i=1 F i where the subsets F i consists of all sub-paths over a cyclic path The cycles c i are considered up to cyclic equivalence, and without arrow repetition. Denote by C(Λ) the set of cycles c 1 , . . . , c m that contain the paths in F 1 , . . . , F m respectively, by n i the length of the cycle c i and by r i the (common) length of paths in F i . We refer to the generators in F as zero-relations. By [LZ], two different cycles c i , c j in C(Λ) do not share arrows.

Gentle algebras.
We recall the definition of gentle algebra. By [GR] it is known that gentle algebras are IG.
For each x 0 ∈ Q 0 there are at most two arrows with source x 0 , and at most two arrows with target x 0 , (g2) the ideal I is generated by paths of length 2, (g3) for each β ∈ Q 1 there is at most one arrow α ∈ Q 1 and at most one arrow γ such that αβ ∈ I and βγ ∈ I, and (g4) for each β ∈ Q 1 there is at most one arrow α and at most one arrow γ such that αβ / ∈ I and βγ / ∈ I.
An algebra Λ = kQ/I, where I is generated by paths and (Q, I) satisfies the two conditions (g1) and (g4), is called a string algebra. Thus every gentle algebra is a string algebra. A string in Λ is by definition a reduced walk w in Q avoiding the zero-relations, thus w is a sequence where the x i are vertices of Q and each α i is an arrow between the vertices x i and x i+1 in either direction such that there is no If the first and the last vertex of w coincide, then the string is cyclic. A band is a cyclic string b such that each power b n is a cyclic string but b is not a power of some string. The classification of indecomposable modules over a string algebra Λ = kQ/I is given by Butler-Ringel [BuRi] in terms of strings and bands in (Q, I). Each string w defines an indecomposable module M (w), called a string module, and each band b defines a family of indecomposable modules M (b, λ, n), called band modules, with parameters λ ∈ k and n ∈ N.
Definition 2.6. Let Λ = kQ/I be a gentle algebra.

α n is critical if it is formed by consecutive relations and α 1 is a gentle arrow.
When there is no gentle arrow, we set n(Λ) = 0. When there is a gentle arrow, let n(Λ) be the maximal length computed over all critical paths. This number is bounded, since Q is finite.
Theorem 2.7. [GR] Let Λ = kQ/I be a gentle algebra with n(Λ) the maximum length over all critical paths.
be a saturated cycle. Then the indecomposable projective module P (x i ) and the indecomposable injective module I(x i ) are string modules given by P ( Figure 1).
αi ui αi+1 ui+1 ui+2 Figure 1. Local situation for a saturated cycle. The path u i is the maximal path starting at the vertex x i and the path v i is the maximal path ending at x i .
Theorem 2.8. [Ka] Let Λ = kQ/I be a gentle algebra. Let where u i is the string starting at x i as in Figure 1, is Gorenstein-projective. Moreover, all indecomposable modules in GP(Λ) are obtained in such manner.

Monomial 2-CY tilted algebras are Jacobian
for all X, Y ∈ C. This means that the d-th power of [1] is a Serre functor for the category, and this is to say that there is a funtorial isomporphism (1) An object T is cluster-tilting if it is basic and addT = {X ∈ C : Hom C (X, T [1]) = 0.
(2) The endomorphism algebra End C (T ) of a cluster-tilting object is called a 2-CY tilted algebra.
Examples of 2-CY tilted algebras are the cluster-tilted algebras defined in [BMR]. A more general family is obtained from generalized cluster categories defined by Amiot, [Am]. Quivers with potential were introduced in [DWZ]. A potential W is a (possibly infinite) linear combination of cycles in Q, up to cyclic equivalence. Given an arrow α and a cycle c = α 1 . . . α l , the cyclic derivative ∂ α (c) is defined by where δ αα k is the Kronecker delta. This derivative can be applied to a sum of cycles extending by linearity. Notice that a cycle α 1 . . . α l may have arrow repetitions. Let R Q be the complete path algebra consisting of all (possibly infinite) linear combinations of paths in Q. Let (Q, W ) be a quiver with potential (QP), the Jacobian algebra is defined to be Jac(Q, W ) = R Q / ∂ α W, α ∈ Q 1 . When one considers the algebra R Q / ∂ α W, α ∈ Q 1 and the result is finite dimensional then it is a Jacobian algebra Jac(Q, W ) and there is no need to consider completions and ideal closures.
Amiot [Am] showed that Jacobian algebras are 2-CY tilted whenever they are finite dimensional by defining a generalized cluster category C (Q,W ) . In [Am2], the author asked whether all 2-CY tilted algebras are Jacobian, and warned that the answer might be negative in a general setting. The first part of this note is devoted to this question, provided the algebra is monomial. Now we recall some main properties of 2-CY tilted algebras. The first part of the next theorem is a celebrated result by Keller-Reiten, the second part is a characterization of GP modules that can be computed over the module category.
Theorem 3.2. [KR, GS] Let Λ be a 2-CY tilted algebra, (1) Λ is 1-IG and stably 3-CY. That is, the triangulated category GP(Λ) is 3-CY. ( Let Λ = kQ/I be a monomial 2-CY tilted algebra. Since it is a monomial 1-IG algebra by the result above, the singularity category GP(Λ) is described in Lemma 2.4. Using this, we can gather information on cycles and relations in (Q, I). Remember that the zero-relations of length r i in F lie in a cycle c i of length (without repetition) n i . Lemma 3.3. Let Λ = kQ/I be a monomial 1-IG algebra with a stably 3-CY singularity category, and F = F i the corresponding set of zero-relations. Then Proof. We want to know for which parameters x, n ∈ Z + we have that an orbit category D b (A x )/τ n is 3-CY. Recall that in the orbit category [1] and τ are induced from D b (A x ). So, at least, we need the functor τ −1 [2] to send objects of a certain orbit to objects in the same orbit. One can easily compute that If we go back to the notation in Lemma 2.4, the claim follows.
With the last lemma we have restricted the possible lengths of cycles and zero-relations in kQ/I, so now there is a constraint for the elements in F . In the following we show that a finite dimensional algebra described by a bound quiver (Q, I) satisfying this constraint has to be 2-CY tilted.
Then the zero-relations arise from a potential W = i∈ [1,m] c bi i . Proof. Since two different cycles in C(Λ) do not share arrows, each α ∈ Q 1 is either in one of the cycles or is not present in C(Λ). If α is not in any cycle then the associated partial derivation adds zero to the ideal I. If α is in the cycle c j ∈ C(Λ) of length n j (with no repetition), then ∂ α ( i c bi i ) = ∂ α (c bj j ) and this is an element of kQ given by b j u where u is a subpath of length r j = n j b j − 1 in the cycle c bj j without the arrow α. Our general hypothesis is that k is a field of characteristic zero, so these paths b j u can be replaced by u as generators of I. Doing this for all α ∈ Q 1 we cover exactly all the elements in F . Therefore Λ is a Jacobian algebra hence it is 2-CY tilted.
The two lemmas above sum up in the following theorem.
Theorem 3.5. Let k be an algebraically closed field of characteristic zero and Λ = kQ/I a monomial k-algebra. The following are equivalent: ( Λ is a 1-IG algebra with stably 3-CY singularity category, (3) Λ is Jacobian. Proof.
(2 ⇒ 3) By Lemmas 3.3 and 3.4, a 1-IG algebra Λ = kQ/I with a 3-CY singularity category is such that the ideal I arises from the cyclic derivatives of a potential, and this potential can be easily defined from I.
The previous result is written with the restriction: b j ∈ Z + is non-zero over k. To guarantee this we say k is of characteristic zero.
To solve the problem presented by intergration-differentiation of potientials over fields of positive characteristic, an alternative is proposed: using hyperpotentials. A hyperpotential on a quiver Q is a collection of elements (ρ α ) α∈Q1 over the complete algebra R Q such that: for α : i → j, ρ α is a (possibly infinite) linear combination of paths j i, and α∈Q1 [α, ρ α ] = 0. The Jacobian algebra of a hyperpotential is the quotient R Q / ρ α (see [Lad]). In view of this, if we admit hyperpotentials, the last result extends to algebraically closed fields of positive characteristic.
Example 3.6. The algebra kQ/I in Example 2.3 is Jacobian. The potential is W = αβγδ + λ 5 .
3.1. Gentle case. Consider the blocks type I, II and loop in Figure 2. They also contain the information: All the vertices in the blocks are outlet vertices. A bound quiver (Q, I) is gentle-block-decomposable if it can be obtained from a collection of disjoint blocks by the following procedure. Take a partial matching of the combined set of outlets.
(1) Matching an outlet to itself or to another outlet from the same block is not allowed.
(2) Matching two outlets corresponding to different blocks type loop is not allowed.
Identify (or glue) the vertices within each pair of the matching. After the gluing, having a pair of arrows connecting the same pair of vertices but going in opposite directions is not allowed. By the results in this section all gentle 2-CY tilted algebras can be built as kQ/I where (Q, I) is gentle-block-decomposable. Since r i = b i n i − 1 = 2, we have b i n i = 3, so either b i = 3 and n i = 1, or b i = 1 and n i = 3.
Example 3.7. Let Q be the quiver in Figure 3, and consider the potential Then Jac(Q, W ) is a gentle 2-CY tilted algebra and Q is a matching of two blocks of type I, three blocks of type II and a block of type loop. If kQ/I is a gentle 2-CY tilted algebra and Q has no loops, then kQ/I is a Jacobian algebra arising from an unpunctured surface, in the sense of [ABCP]. The classification of tame QP Jacobian algebras (which includes the algebras arising from surfaces and some monomial algebras) was made in [GLaS].

Other monomial algebras and their singularity categories
During this section we study GP for a family of gentle algebras and some examples of non-gentle mcluster tilted algebras. We will obtain an analogous property to Theorem 3.2 (2) by explicit computation. This property is deeply related to the CY dimension of GP. 1 An algebra arising from a (m + 2)-angulation T is defined as kQ T  In particular, when the surface is a disk (resp. an annulus) the gentle algebra obtained is m-cluster tilted of type A (resp.Ã) in the sense of Thomas [T], as it was studied in several works [CCS, ABCP, Ba, Mu, Tol, Gub].
The following properties were observed in [Mu] and [Gub] in the case of the disc and annulus, but it is easy to see that they hold when the algebra arises from a (m + 2)-angulation of a general surface.

Proposition 4.2. Let (Q T , I T ) be a bound quiver arising from a (m + 2)-angulation.
(1) Λ T = kQ T /I T is a gentle algebra.
(3) There can be at most m − 1 consecutive zero-relations not lying in a saturated cycle.
Immediately, we have the following observation.

Lemma 4.3. Let Λ T = kQ T /I T be an algebra arising from a (m + 2)-angulation. Then, Λ T is m-IG.
Proof. The case m = 1 follows from Theorem 3.2 (1), and also from [ABCP]. Let m ≥ 2. Since Λ T is gentle, we can apply Theorem 2.7. First assume that there is no gentle arrow in (Q T , I T ), then n(Λ T ) = 0, so d is zero or one and d ≤ m. The statement follows. Now, assume there are gentle arrows in (Q T , I T ), and let α 1 be one of them. It follows that α 1 is not part of a saturated cycle. Let α 1 . . . α r be a critical path. Since α 1 is not part of a saturated cycle, then none of the arrows α i for 1 ≤ i ≤ r is part of a saturated cycle. By Proposition 4.2 (3), the maximal number of consecutive zero-relations outside of a saturated cycle is m − 1. Therefore, r ≤ m, and by Theorem 2.7, Λ T is Gorenstein of dimension d ≤ m.
Most of the arguments in the following lemma can be found also in [Ka].
Lemma 4.4. Let Λ = kQ/I be a gentle d-IG algebra, d ≥ 1. Let x ∈ Q 0 , and let N be an indecomposable direct summand of radP (x). Then, Proof. If N is projective, we are in case (b). Let N be non projective. Let P (x) = M (u −1 α −1 βw) be the indecomposable projective and N = M (u) so that S(t(α)) = topM (u). We study the cases: is not projective, there exists an arrow δ 1 such that αδ 1 ∈ I. The arrow δ 1 is not part of a saturated cycle, since then α would be part of the saturated cycle. Let P (t(α)) = M (c −1 δ −1 1 u), then there is an exact sequence If the string module M (c) is not projective, then it satisfies the same conditions as M (u), so we can construct a new exact sequence Recursively, we obtain a path αδ 1 · · · δ n such that each quadratic factor belongs to I. This process has to finish after a finite number of steps, being the direct summand M (c n ) of P (t(δ n−1 )) a projective module. If there were not finite steps and M (c n ) was not projective, we would find new arrows δ n+1 , . . . and form a path αδ 1 · · · δ n · · · such that each quadratic factor is in I. The quiver Q is finite, so the only way to construct an infinite path αδ 1 · · · δ n · · · is by reaching a saturated cycle. By the gentleness, if one of the arrows δ i is in a saturated cycle, then all α, δ 1 , . . . , δ n are in the saturated cycle, contradicting the condition imposed on α. Therefore the procedure to find the short exact sequences stops. The cup product of these sequences is a projective resolution for M (u), which is finite, so proj.dimM (u) < ∞. To complete the previous lemma, observe that if Λ is selfinjective (that is, Λ is 0-IG) then every indecomposable module is Gorenstein-projective.
We compute a minimal projective presentation of M (u i ).
Observe that Ω t M (u i ) = M (u i+t ), where t is an integer considered modulo m + 2. Applying the Nakayama functor we get The syzygy functor is additive, so Since Now, we only need to prove that Ω m M (w i+1 ) = 0. Observe that M (w i+1 ) is a direct summand of radP (y i+1 ).
We know by Lemma 4.3 that Λ T is Gorenstein of dimension d ≤ m. By Lemma 4.4 one of the following holds: If (1) holds, then Ω m M (w i+1 ) = 0 and we are done.
We assume (2) holds, so M (w i+1 ) ∈ GP(Λ T ). We prove that this leads to a contradiction. Let z i+1 be the vertex such that topM (w i+1 ) = S(z i+1 ). By the description in Theorem 2.8, the vertex z i+1 is the target of an arrow γ in a saturated (m + 2)-cycle and γw i+1 = 0.
(2a) If the arrow γ is y i+1 γ − → z i+1 , see the figure below (left), then there is an arrow a j in the saturated (m + 2)-cycle, such that a j γ ∈ I T . Then, a j v i+1 = 0, contradicting that I( (2b) If the arrow γ in a saturated cycle is such that s(γ) = y i+1 , see figure above (right), there is an arrow b j+2 following the saturated cycle such that γb j+2 ∈ I T . Thus, we have γw i+1 = 0 and by gentleness, a j+1 b j+2 / ∈ I T . But recall that M (w i+1 ) is a submodule of radP (y i+1 ), so b j+2 has to be the first arrow in the string w i+1 . This contradicts that M (w i+1 ) is a submodule of radP (y i+1 ). To sum up, M (w i+1 ) is not a trivial Gorenstein projective module and just (1) holds.
As a corollary, we obtain the next result that generalizes the properties known for cluster-tilted algebras: Theorem 3.2 (1) and (2). From a more general point of view, m-cluster tilted algebras are (m + 1)-CY tilted algebras. It is known, and expected, that Corollary 4.6 does not hold in general for d-CY tilted algebras by the next reasons: (1) In [KR] there is an example (due to Iyama) of a d-CY tilted algebra that is not IG.
Still, there are results in this direction due to Keller and Reiten [KR2], and Beligiannis [Be2], showing that a d-CY tilted algebra is IG under certain conditions. In both cases, it is required that addT is corigid to some degree u, meaning that Hom C (T, T [−t]) = 0 for all 1 ≤ t ≤ u. Over m-cluster tilting categories of type A orÃ, the full subcategory defined by a cluster tilting object add T is (m + 1)-cluster tilting. In the next example we show that Corollary 4.6 is independent of these results, by giving an example of a non-corigid cluster-tilting object in the 2-cluster category of type A 4 . Example 4.7. Let C 2 Q be the 2-cluster category, where Q is of type A 4 and let T be as in Figure 5. The object T is not corigid since Hom(T 3 , T 1 [−1]) Hom (T 3 [1], T 1 ) = 0. By Corollary 4.6 the 2-cluster tilted algebra End(T ) is 2-IG. In fact, in this example the algebra, given by the quiver bellow bounded by βα = 0, is of global dimension two. In the next example we see that there are m-cluster-tilted algebras not arising from (m + 2)-angulations, defined by non-corigid cluster tilting objects, for which the conclusion of Corollary 4.6 holds.
Example 4.8. Let C 2 Q be the 2-cluster category of type D 6 , and T = 6 i=1 T i the 2-cluster tilting object in Figure 6. The algebra Λ = End C 2 Q (T ) in Example 4.8 is given by the quiver in Figure 7 and the ideal I = λα, αβγ, βγδ, δλ . We see that Λ is 2-IG and has infinite global dimension. The indecomposable modules in GP(Λ) are 3, 6, 5 4 , 2 1 , exactly those such that Ω 3 τ N = N . As a final remark, we may ask the next question: Let Λ be an Iwanaga-Gorenstein algebra. Since GP(Λ) is a functorially finite subcategory of mod Λ, it has Auslander-Reiten sequences [A], inducing an Auslander-Reiten translation τ GP on GP(Λ), typically different from the Auslander-Reiten translation τ Λ on mod Λ. The goal of this appendix is to relate the objects τ Λ M and τ GP M of mod Λ when M is Gorenstein projective. Indeed, we will show that these objects eventually coincide after repeated application of the syzygy functor. While this fact may not be surprising to experts, and can be deduced quickly from results already in the literature (most easily from [A1]), we give here a very direct proof, which even exhibits a natural isomorphism of functors. Moreover, we explain how this result provides a new perspective on results of Garcia Elsener (Corollary 4.6 of the present paper) and Garcia Elsener-Schiffler [GES] characterising Gorenstein projective modules over certain Calabi-Yau tilted algebras Λ, by relating these characterisations directly to a Calabi-Yau property of GP(Λ).
We denote by Ω : mod Λ → mod Λ the syzygy functor, taking a module to the kernel of a projective cover, and by Σ : mod Λ → mod Λ its left adjoint, taking a module to the cokernel of a right proj Λ-approximation. The restrictions of these functors to GP(Λ) are mutually inverse, the restriction of Σ being the suspension functor on this triangulated category. Proof. Using the Yoneda embedding, it suffices to show that there is a natural isomorphism of functors on mod Λ, for any X ∈ GP(Λ). By adjunction and the Auslander-Reiten formula for GP(Λ), we obtain natural isomorphisms The validity of the second isomorphism depends on Lemma A.1, showing that Σ d takes values in GP(Λ). Alternatively, using the Auslander-Reiten formula in mod Λ, we get The two right-hand sides coincide, completing the proof. Proof. By a result of Keller and Reiten [KR], a 2-Calabi-Yau tilted algebra is Iwanaga-Gorenstein of Gorenstein dimension at most 1. Thus Ω 2 N ∈ GP(Λ) for any N ∈ mod Λ, and the 'if' part of the theorem follows immediately.
For the 'only if' direction, Keller and Reiten also prove [KR] that GP(Λ) is a 3-Calabi-Yau category, meaning that Σ 3 is a Serre functor. On the other hand, the Auslander-Reiten formula states that Στ GP is always a Serre functor on GP(Λ). Since Serre functors are unique up to natural isomorphism, the 3-Calabi-Yau property is equivalent to there being a natural isomorphism τ GP Σ 2 . Now by Theorem A.2, using again the Iwanaga-Gorenstein property of Λ, we have Ω 2 τ Λ M ∼ = Ω 2 τ GP M ∼ = Ω 2 Σ 2 M ∼ = M.
We close by observing that characterisations of Gorenstein projective Λ-modules as in Theorem A.3 are closely related to Calabi-Yau properties of GP(Λ) in wider generality. Proof. As in the proof of Theorem A.2, GP(Λ) is (m + 1)-Calabi-Yau if and only if there is a natural isomorphism τ GP Σ m on GP(Λ). Since Σ m has quasi-inverse Ω m on this category, this is equivalent to asking that Ω m τ GP Id GP(Λ) . But Theorem A.2 tells us that Ω m τ GP Ω m τ Λ in this setting, and the first part of the statement follows. The second part is then proved as in Theorem A.3, replacing 2 by m.
Note that the isomorphisms constructed in the proof of Theorem A.3 arise from a natural isomorphism in this way, using the 3-Calabi-Yau property of GP(Λ) [KR]. We also deduce that a similar Calabi-Yau property holds for certain gentle m-cluster-tilted algebras. Corollary A.5. Let Λ be an m-cluster-tilted algebra of type A orÃ. Then GP(Λ) is (m + 1)-Calabi-Yau.
Proof. By Corollary 4.6, Λ is Iwanaga-Gorenstein of Gorenstein dimension at most m, and M ∼ = Ω m τ Λ M for each M ∈ GP(Λ). Since Λ is a gentle algebra, meaning that GP(Λ) is semi-simple [Ka], there must be a collection of such isomorphisms that is natural in M -for example, by first choosing such an isomorphism for each M in a set of representatives for the isoclasses of indecomposable objects, and then extending to arbitrary objects by choosing direct sum decompositions. (We use here that the ground field k is algebraically closed, so that each indecomposable has endomorphism ring k.) The result then follows by Proposition A.4.