On irreducible morphisms and Auslander-Reiten triangles in the stable category of modules over repetitive algebras

Hern\'an Giraldo; Jos\'e A. V\'elez-Marulanda; Yohny Calder\'on-Henao

REVIEW 2 major objections 6 minor 19 references

On irreducible morphisms and Auslander-Reiten triangles in the stable category of modules over repetitive algebras

T0 review · 2 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Irreducible morphisms in the stable category of modules over a repetitive algebra have exactly three canonical shapes, and every Auslander-Reiten triangle whose terms avoid projective summands is induced by an Auslander-Reiten sequence.

desk verdict Transfers the irreducible-shape classification to the stable category, but two load-bearing proof steps are invalid and need real revision. read the letter →

arxiv 1908.02912 v1 pith:JKREZD7C submitted 2019-08-08 math.RT

Yohny Calderón-Henao , Hernán Giraldo , José A. Vélez-Marulanda This is my paper · ORCID

classification math.RT MSC 16G1016G2020C20

keywords repetitivealgebrasstablecategoryAuslander-ReitentrianglesirreduciblemorphismssequencestriangulatedcategoriesFrobeniusgentle

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper pins down the possible shapes of irreducible morphisms and Auslander-Reiten triangles in the stable category of finitely generated modules over a repetitive algebra. It proves that, for objects with no projective direct summands, an irreducible stable morphism is always one of three types: all component maps are split monomorphisms, all are split epimorphisms, or exactly one component is irreducible. It then proves that every Auslander-Reiten triangle whose three terms avoid projective summands is induced by an ordinary Auslander-Reiten sequence of modules, possibly padded by a single indecomposable projective summand; when that summand is present, the two end terms are forced to be its radical and its top. These results turn questions about the triangulated stable category back into module-category problems, where Auslander-Reiten sequences are easier to construct and recognize.

What carries the argument

The machinery is the pair consisting of the stable category $\widehat{\Lambda}$-$\underline{\mathrm{mod}}$, a triangulated category whose translation is the first cosyzygy functor $\Omega^{-1}$, and the transfer theorem (Theorem 2.4) that compares irreducibility in the stable category with irreducibility in the module category $\widehat{\Lambda}$-$\mathrm{mod}$. The repetitive algebra $\widehat{\Lambda}$ is the doubly infinite matrix algebra built from $\Lambda$ and its injective cogenerator, and its module category is a Frobenius category in the sense that projective and injective objects coincide, so passing to the stable category produces the triangulated structure. The load-bearing transfer says that when $\hat M$ and $\hat M'$ have no projective direct summands, a morphism's stable class is split mono, split epi, or irreducible exactly when a representative is. The proof of Theorem 2.7 then starts from the Auslander-Reiten sequence beginning at $\hat M$ with middle term $\hat M'\oplus \hat Y$, uses the source-morphism property to cancel any non-projective summand of $\hat Y$, and concludes $\hat Y$ must be a projective $\hat P$; irreducibility of $\hat M\to \hat P$ forces $\hat M\simeq \operatorname{rad}\hat P$ and dually $\hat M''\simeq \hat P/\operatorname{soc}\hat P$.

What would settle it

Take the gentle algebra in Section 4 and inspect the displayed stable Auslander-Reiten quiver for an arrow whose source and target have no projective summands. The theorem predicts that this arrow is stably smonic, stably sepic, or stably sirreducible and that the surrounding triangle is induced by an Auslander-Reiten sequence with at most one indecomposable projective summand; an arrow of any other shape, or a triangle whose end terms are not $\operatorname{rad}\hat P$ and $\hat P/\operatorname{soc}\hat P$ when a projective summand appears, would disprove the claim.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is a structural trichotomy plus a lifting theorem. If $\hat h\colon \hat M\to \hat M'$ is an irreducible morphism in $\widehat{\Lambda}$-$\underline{\mathrm{mod}}$ with $\hat M$ and $\hat M'$ free of projective direct summands and at least one of them indecomposable, then $\hat h$ is stably smonic, stably sepic, or stably sirreducible, meaning its class in $\widehat{\Lambda}$-$\underline{\mathrm{mod}}$ has all split-monomorphism components, all split-epimorphism components, or exactly one irreducible component. In an Auslander-Reiten triangle $\hat M \xrightarrow{\hat h} \hat M' \xrightarrow{\hat h'} \hat M'' \to \Omega^{-1}\hat M$ with all three terms projective-summand-free, $\hat h$ and $\hat h'$ must pair as follows: smonic with sepic, sepic with sirreducible, and sirreducible with smonic or sirreducible. The enabling fact is Theorem 2.7: such a triangle is induced by an Auslander-Reiten sequence $0\to \hat M \to \hat M'\oplus \hat P \to \hat M''\to 0$ in $\widehat{\Lambda}$-$\mathrm{mod}$, where $\hat P$ is projective, and if $\hat P\neq 0$ then $\hat P$ is indecomposable with $\hat M\simeq \operatorname{rad}\hat P$ and $\hat M''\simeq \hat P/\operatorname{soc}\hat P$.

Load-bearing premise

The argument depends on the assumption that none of the three terms in the triangle contains a projective summand; if one does, the bridge between stable and module irreducibility breaks, so the classification could fail.

Editorial extensions

If this is right

Every Auslander-Reiten triangle in $\widehat{\Lambda}$-$\underline{\mathrm{mod}}$ whose terms avoid projective summands can be constructed from an ordinary Auslander-Reiten sequence in $\widehat{\Lambda}$-$\mathrm{mod}$, so such triangles can be studied through short exact sequences and projectives.
When the inducing sequence needs a nonzero projective summand $\hat P$, the triangle is completely pinned down by $\hat P$: it starts at $\operatorname{rad}\hat P$, ends at $\hat P/\operatorname{soc}\hat P$, and its middle term is the middle term of the corresponding Auslander-Reiten sequence plus $\hat P$.
The two middle maps of an Auslander-Reiten triangle have locked shapes: a smonic map forces its successor to be sepic, a sepic map forces its successor to be sirreducible, and a sirreducible map forces its successor to be smonic or sirreducible.
For algebras where the bounded derived category and $\widehat{\Lambda}$-$\underline{\mathrm{mod}}$ are equivalent as triangulated categories, the same trichotomy describes the shape of Auslander-Reiten triangles in the derived category.
The gentle algebra worked out in Section 4 shows that all four pairings allowed by Theorem 3.3 actually occur in a single stable Auslander-Reiten component of type $\mathbb{Z}A_\infty$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the transfer theorem is invoked only when no projective summands are present, the classification may need a separate bookkeeping for Auslander-Reiten triangles in which some term does contain a projective summand; a natural extension is to strip off the projective part and then apply the trichotomy to the remaining stable morphism.
The proof identifies the projective summand in the inducing sequence as the unique correction needed to turn a source morphism in the module category into one in the stable category, which suggests that any Frobenius category whose stable category is triangulated should admit a parallel 'one projective correction' description whenever a transfer theorem of the same kind holds.
For finite-dimensional algebras of infinite global dimension, Theorem 3.3 offers a way to read off the shape of derived-category Auslander-Reiten triangles directly from module-theoretic data, which could simplify explicit computations of stable Auslander-Reiten quivers in examples beyond gentle algebras.
A testable extension is to check whether the trichotomy survives when only one of the two objects is assumed free of projective summands; Lemma 2.3 gives partial statements in that direction, but the full classification may require an additional case.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the paper does what it says—transfers the shape classification of irreducible morphisms and AR triangles from Λ-hat-mod to its stable category—and the example is a genuine check. But the stress-test note is right: two steps in the load-bearing proofs are invalid, so this is not just a typo-fixing round.

What's new: Theorem 3.2 is an explicit transfer of Giraldo [7, Thm. 26] to the stable category using Theorem 2.4, and Theorem 3.3 generalizes [15, §4.1] from the bounded derived category to Λ-hat-mod with an indecomposability condition. The authors are upfront about the provenance. The Section 4 example is useful and appears to check all four cases of Theorem 3.3.

Now the soft spots. The first stress-test point lands. In the proof of Theorem 2.7, after showing α: M → P is an essential monomorphism, the authors assume that if P = P' ⊕ P'', then Im α = (Im α ∩ P') ⊕ (Im α ∩ P''). That equality is not valid for arbitrary essential submodules; the essentiality condition alone does not force it. The diagonal submodule in a direct sum is not essential, so it is not a counterexample in the essential case, but the equality still does not follow. The conclusion that P is indecomposable might be recoverable if one proves M is uniform or uses a different injective-hull argument, but the paper doesn't supply that. Since Theorem 2.7 underpins the reduction of Theorem 3.3 to the special AR sequences (2.9), this gap matters.

The second point also lands. In Corollary 2.5, from h and h' irreducible and h − h' factoring through a projective, the proof claims the difference is irreducible because Irr is a k-vector space. That is not valid: the class of the difference in rad/rad^2 can be zero even when the morphism is nonzero, so the difference need not be irreducible. The corollary's conclusion—that the stable class determines the module representative—is used to define stably smonic/sepic/sirreducible and in the proof of Theorem 3.3. It may be true, but the given argument doesn't establish it.

Overall: the mathematics looks plausible and the results are modest, useful extensions. The proof gaps are not typos; they are invalid inference steps. The paper deserves a serious referee—not a desk reject—but the authors need to fix these proofs or supply alternative arguments.

Audience: representation theorists working on repetitive algebras, AR theory, and stable categories. I would engage with it after revision; right now, the proofs as written do not fully support the main theorems.

Referee Report

2 major / 6 minor

Summary. The paper studies the stable category of finitely generated modules over the repetitive algebra Λ̂ of a finite-dimensional algebra Λ. It claims a trichotomy for irreducible morphisms in Λ̂-mod (smonic, sepic, sirreducible) under a no-projective-direct-summands hypothesis, and it describes the shape of Auslander-Reiten triangles by proving that every AR triangle whose three terms have no projective summands is induced by an AR sequence in Λ̂-mod. The engine is Theorem 2.7, which asserts that the inducing AR sequence has middle term M′⊕P with P projective, and that if P≠0 then P is indecomposable and the sequence has the form 0→radP→M′⊕P→P/socP→0. Section 3 uses this to classify the stable irreducible morphisms appearing in AR triangles, and Section 4 gives a gentle-algebra example.

Significance. If correct, the results provide a concrete structural description of irreducible morphisms and AR triangles in the stable category of a repetitive algebra, connecting the module-theoretic AR sequences of Hughes–Waschbüsch with Happel’s triangulated stable category. The paper follows a plausible strategy based on standard Frobenius-category and AR-theory methods, and it includes a substantial example verifying the claimed classification. However, the central results currently rest on two proof steps that are invalid as written: an asserted submodule equality in the proof of Theorem 2.7, and a vector-space argument in Corollary 2.5. Both steps are load-bearing for Theorems 3.2 and 3.3, so the manuscript requires substantive revision before the claims can be regarded as established.

major comments (2)

[§2.3, Theorem 2.7] In the proof of Theorem 2.7, after establishing that α:M→P is an essential monomorphism, the authors assume P=P′⊕P″ and assert Imα=(Imα∩P′)⊕(Imα∩P″). This equality is false in general: a submodule of a direct sum need not be the direct sum of its intersections with the summands, with a diagonal submodule as a counterexample. This step is the only argument given to prove that P is indecomposable, and it is also used to reach the conclusions M≅radP and M″≅P/socP, which in turn are needed to reduce Theorem 3.3 to AR sequences of the special form (2.9). The theorem may still be true, since the isomorphism δ:I(M)→P obtained earlier, together with the standard fact that the injective hull of an indecomposable module is indecomposable, directly yields that P is indecomposable. The authors should replace the faulty equality with a correct proof.
[§2.2, Corollary 2.5] The proof of Corollary 2.5 contains two invalid steps. First, from h_k0,h′_k0 ∈ Irr(M,M′_k0) and the fact that Irr is a k-vector space, the authors conclude that v_k0∘u_k0∈Irr(M,M′_k0). This does not follow: elements of Irr are classes in rad/rad², and the difference of two nonzero classes can be zero, so the representative v_k0∘u_k0 may lie in rad² and need not be irreducible. Second, the assertion that each component h_k−h′_k factors through an indecomposable projective summand P_k is not justified; a morphism factoring through a projective module P need not factor through a single indecomposable summand of P. Corollary 2.5 is used to make Definition 3.1 well-defined and to justify replacing h′ by h″ in the proof of Theorem 3.3, so this gap directly affects Theorems 3.2 and 3.3. The authors need either a valid proof of Corollary 2.5 or a reformulation that avoids relying on it.

minor comments (6)

[Introduction] The introduction refers to 'Theorem 3.2' twice where the second mention should clearly be Theorem 3.3.
[§2.3, Theorem 2.7] There is a typo in the phrase 'Let ι_M : M → I(M) the the injective Λ̂-hull of M'; 'the the' should be 'be the'.
[§3, proof of Theorem 3.3] The sentence 'such that h′ is equal to h″ module an isomorphism' should read 'modulo an isomorphism'.
[§2.3, Corollary 2.8] In part (ii), the statement 'M′_i = 0 for all n≠i0,i0+1' uses the variable n where i is intended.
[§3, Lemma 3.5] The reference 'Remark rem1.1' in part (i) is broken; it should be 'Remark 2.1'.
[§4, Figure 1] The quiver and the Auslander-Reiten quiver in Section 4 are extremely difficult to read; the labels around Figure 1 appear garbled. A redrawn, clean version would greatly improve the exposition.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's main results are applications of separately established structural theorems and explicit proofs, not restatements of their own inputs.

full rationale

The derivation chain in the paper is linear and non-circular. Theorem 2.7, which states that every Auslander-Reiten triangle in the stable category, under the stated no-projective-summand hypothesis, is induced by an Auslander-Reiten sequence with a projective middle summand, is proved using Happel's triangulated structure, the Hughes-Waschbüsch existence of Auslander-Reiten sequences, and properties of Auslander-Reiten triangles from [9, Chap. I]; it is not assumed as its own conclusion. Theorem 3.2 is a direct consequence of the cited external classification [7, Thm. 26], Theorem 2.4 from [7], and Corollary 2.5, together with Definition 3.1, which simply names the three module-category shapes in the stable category. Definition 3.1 is a labeling convention, not a hidden assumption of Theorem 3.2. Theorem 3.3 reduces Auslander-Reiten triangles to Auslander-Reiten sequences via Theorem 2.7 and then verifies the three cases with Lemmas 3.4 and 3.5; none of those lemmas assumes the conclusion of Theorem 3.3. Self-citations occur ([7], [15]), but they are published structural results about irreducible morphisms and Auslander-Reiten triangle shapes, and the present argument does not require accepting the present paper's conclusion in order to justify them. No fitted parameter is renamed as a prediction, and no conclusion coincides by construction with an input. Any alleged gap in the proof of Theorem 2.7, such as the submodule-intersection step, would be a mathematical correctness issue, not circularity. The appropriate finding is therefore no significant circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new free parameters or entities. Its central claims rest on standard structure theory of repetitive algebras and stable categories (Happel, Hughes and Waschbüsch) and on the author's earlier theorem [7] relating irreducibility in the module category and the stable category. The main additional premise is the no-projective-direct-summands hypothesis, which is explicit in every main theorem.

assumptions (6)

standard math The stable category Λ̂-mod of finitely generated Λ̂-modules is a triangulated category with translation functor Ω^{-1} (Happel, cited in Section 2.1).
Used to define distinguished triangles and Auslander-Reiten triangles; taken from [9, Chap. I, Section 2.6].
standard math The category Λ̂-mod is Frobenius, with projective and injective objects coinciding, and the indecomposable projective-injective modules have the form (2.3) (Happel, [9, Chap. II, Section 2.2]).
Underpins the definition of the stable category and the cosyzygy functor Ω^{-1}.
standard math Λ̂-mod has Auslander-Reiten sequences starting and ending at every indecomposable module (Hughes-Waschbüsch [12, Section 2.5]).
Guarantees the existence of the AR sequence (2.7) used in the proof of Theorem 2.7.
standard math An irreducible morphism in Λ̂-mod between objects without projective direct summands is also irreducible in the stable category, and vice versa (Theorem 2.4, from [7, Prop. 41 and Thm. 42]).
The bridge between the abelian and stable settings; used in Theorems 2.7, 3.2, and 3.3.
domain assumption All modules considered are finitely generated and k is an algebraically closed field of arbitrary characteristic (Section 2).
The global hypothesis of the paper; used throughout but not proved.
ad hoc to paper The terms M, M', M'' of the Auslander-Reiten triangle have no projective direct summands (hypothesis in Theorem 2.7 and Theorem 3.3).
Load-bearing premise that allows Theorem 2.4 to transfer irreducibility between Λ̂-mod and Λ̂-mod; the theorems are not stated without it.

how reviews work

0 comments

Cite this review

Pith. "Pith review of On irreducible morphisms and Auslander-Reiten triangles in the stable category of modules over repetitive algebras." pith.science (2026). https://pith.science/paper/JKREZD7C

@misc{pith2026190802912,
  author       = {Pith},
  title        = {Pith review of: On irreducible morphisms and Auslander-Reiten triangles in the stable category of modules over repetitive algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JKREZD7C}},
  note         = {Machine review of arXiv:1908.02912}
}

abstract

Let $\mathbf{k}$ be an algebraically closed field, let $\Lambda$ be a finite dimensional $\mathbf{k}$-algebra, and let $\widehat{\Lambda}$ be the repetitive algebra of $\Lambda$. For the stable category of finitely generated left $\widehat{\Lambda}$-modules $\widehat{\Lambda}$-\underline{mod}, we show that the irreducible morphisms fall into three canonical forms: (i) all the component morphisms are split monomorphisms; (ii) all of them are split epimorphisms; (iii) there is exactly one irreducible component. We next use this fact in order to describe the shape of the Auslander-Reiten triangles in $\widehat{\Lambda}$-\underline{mod}. We use the fact (and prove) that every Auslander-Reiten triangle in $\widehat{\Lambda}$-\underline{mod} is induced from an Auslander-Reiten sequence of finitely generated left $\widehat{\Lambda}$-modules.

Figures

Figures reproduced from arXiv: 1908.02912 by the authors.

**Figure 1.** The component of the stable Auslander-Reiten quiver of Λ containing the simple b Λ-modules corresponding to the vertices 1 b z, 2z and 3z with z ∈ Z. In the following, we check that Λ verifies all the possibilities in Theorem b 3.3. In the followsing z ∈ Z is a fixed both arbitrary integer. (i) Consider the Auslander-Reiten triangle (4.1) Mc[θ −1 z αbz] bh −→ Mc[αb] ⊕ Mc[θ −1 z ] bh 0 −→ Mc[12z ] bh 00 −−→ Ω −1Mc[θ … view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

[1]

Assem, I

I. Assem, I. Simson, and A. Skowro´ nski, Elements of the Representation Theory of Associative Algebras , vol. 1, London Mathematical Society Student Texts, no. 65, Cambridge University Press, 2006

work page 2006
[2]

Assem and A

I. Assem and A. Skowro´ nski,Iterated tilted algebras of type ~An, Math. Z. 195 (1987), 269–290

work page 1987
[3]

Barot and O

M. Barot and O. Mendoza, An explicit construction for the Happel functor , Colloq. Math. 428 (2015), 141–149

work page 2015
[4]

Bautista and M

R. Bautista and M. J. Souto Salorio, Irreducible morphisms in the bounded derived category , J. Pure Appl. Algebra 104 (2006), no. 1, 866–884

work page 2006
[5]

M. C. R. Butler and C. M. Ringel, Auslander-Reiten sequences with few middle terms and applications to string algebras , Comm. Algebra 15 (1987), 145–179. 14 CALDER ´ON-HENAO, GIRALDO, AND V ´ELEZ-MARULANDA

work page 1987
[6]

X. W. Chen and P. Zhang, Quotient triangulated categories, Manuscripta Math. 123 (2007), 167–183

work page 2007
[7]

Giraldo, Irreducible morphisms between modules over a repetitive algebras , Algebr

H. Giraldo, Irreducible morphisms between modules over a repetitive algebras , Algebr. Represent. Theor. 21 (2018), no. 4, 683–702

work page 2018
[8]

Giraldo and H

H. Giraldo and H. Merklen, Irreducible morphisms of categories of complexes , J. Algebra 321 (2009), no. 10, 2716–2736

work page 2009

Show all 19 references

[9]

Happel, Triangulated Categories in the Representation Theory of Finite Dimensional Algebras , London Mathematical Society Lecture Notes Series, no

D. Happel, Triangulated Categories in the Representation Theory of Finite Dimensional Algebras , London Mathematical Society Lecture Notes Series, no. 119, Cambridge University Press, 1988

1988
[10]

, Auslander-Reiten triangles in derived categories of ﬁnite-dimensional algebras, Proc. Amer. Math. Soc. 112 (1991), no. 3, 641–648

1991
[11]

Happel, B

D. Happel, B. Keller, and I. Reiten, Bounded derived categories and repetitive algebras , J. Algebra 319 (2008), no. 4, 1611–1635

2008
[12]

Hughes and J

D. Hughes and J. Waschb¨ usch,Trivial extensions of tilted algebras , Proc. London Math. Soc. 46 (1983), 347–364

1983
[13]

Mac Lane, Homology, Die Grundlehren der mathematischen Wissenschaften, Bd

S. Mac Lane, Homology, Die Grundlehren der mathematischen Wissenschaften, Bd. 114, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-G¨ ottingen-Heidelberg, 1963

1963
[14]

Quillen, Higher algebraic K-theory I , Lecture Notes in Mathematics, no

D. Quillen, Higher algebraic K-theory I , Lecture Notes in Mathematics, no. 341, Springer-Verlag, 1973

1973
[15]

Ribeiro Alvares, S

E. Ribeiro Alvares, S. M. Fernandes, and H. Giraldo, Shapes of Auslander-Reiten triangles , Under revisions. Available in https://arxiv.org/abs/1610.08457

arXiv
[16]

C. M. Ringel, Tame algebras and integral quadratic forms , Lecture Notes in Mathematics, no. 1099, Springer-Verlag, 1984

1984
[17]

Schr¨ oer,On the quiver with relations of a repetitive algebra , Arch

J. Schr¨ oer,On the quiver with relations of a repetitive algebra , Arch. Math. 72 (1999), no. 6, 426–432

1999
[18]

Verdier, Cat´ egories d´ eriv´ ees, ´ etat 0, Cohomologie Etale: S´ eminaire de G´ eom´ etrie Alg´ ebrique du Bois-Marie SGA 4 1/2 (P

J. Verdier, Cat´ egories d´ eriv´ ees, ´ etat 0, Cohomologie Etale: S´ eminaire de G´ eom´ etrie Alg´ ebrique du Bois-Marie SGA 4 1/2 (P. Deligne, ed.), Lecture Notes in Mathematics, no. 569, Springer-Verlag, 1977, pp. 262–311

1977
[19]

Wald and J

B. Wald and J. Waschb¨ usch,Tame biserial algebras, J. Algebra 95 (1985), 480–500. Instituto de Matem´aticas, Universidad de Antioquia, Medell´ın, Antioquia, Colombia E-mail address: yohny.calderon@udea.edu.co Instituto de Matem´aticas, Universidad de Antioquia, Medell´ın, Ant...

1985

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Assem, I

I. Assem, I. Simson, and A. Skowro´ nski, Elements of the Representation Theory of Associative Algebras , vol. 1, London Mathematical Society Student Texts, no. 65, Cambridge University Press, 2006

work page 2006

[2] [2]

Assem and A

I. Assem and A. Skowro´ nski,Iterated tilted algebras of type ~An, Math. Z. 195 (1987), 269–290

work page 1987

[3] [3]

Barot and O

M. Barot and O. Mendoza, An explicit construction for the Happel functor , Colloq. Math. 428 (2015), 141–149

work page 2015

[4] [4]

Bautista and M

R. Bautista and M. J. Souto Salorio, Irreducible morphisms in the bounded derived category , J. Pure Appl. Algebra 104 (2006), no. 1, 866–884

work page 2006

[5] [5]

M. C. R. Butler and C. M. Ringel, Auslander-Reiten sequences with few middle terms and applications to string algebras , Comm. Algebra 15 (1987), 145–179. 14 CALDER ´ON-HENAO, GIRALDO, AND V ´ELEZ-MARULANDA

work page 1987

[6] [6]

X. W. Chen and P. Zhang, Quotient triangulated categories, Manuscripta Math. 123 (2007), 167–183

work page 2007

[7] [7]

Giraldo, Irreducible morphisms between modules over a repetitive algebras , Algebr

H. Giraldo, Irreducible morphisms between modules over a repetitive algebras , Algebr. Represent. Theor. 21 (2018), no. 4, 683–702

work page 2018

[8] [8]

Giraldo and H

H. Giraldo and H. Merklen, Irreducible morphisms of categories of complexes , J. Algebra 321 (2009), no. 10, 2716–2736

work page 2009