Irreducible Morphisms Between Modules over a Repetitive Algebras

Giraldo, Hernán

doi:10.1007/s10468-017-9733-9

Cited by 4 publications

(5 citation statements)

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“…Our main motivation is that the objects and the morphisms in Λ-mod are easier to understand than those in D b (Λ-mod). Our main results are Theorem 3.2, which extends [7,Thm. 26] to Λ-mod and Theorem 3.2, which provides a generalization of [15, §4.1].…”

Section: Introductionsupporting

confidence: 55%

“…Such task was approached in the more general setting of additive categories by M. J. Souto Salorio and R. Bautista in [4, §2]. More recently, the second author investigated in [7] the behavior of irreducible morphisms between objects in Λ-mod. This work was inspired by the second author's joint work with H. Merklen on the behavior of irreducible morphisms between objects in D b (Λ-mod) (see [8]).…”

Section: Introductionmentioning

confidence: 99%

“…In §2, we review the definition of Λ and some basic aspects concerning the categories Λ-mod and Λ-mod. We also review the results from [7] concerning irreducible morphisms in Λ-mod as well as the definition of Auslander-Reiten sequences in Λ-mod and Auslander-Reiten triangles in Λ-mod. We also prove Theorem 2.7 and some useful corollaries.…”

Section: Introductionmentioning

confidence: 99%

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

Calderón-Henao¹,

Giraldo²,

Vélez-Marulanda³

2019

Preprint

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Let k be an algebraically closed field, let Λ be a finite dimensional k-algebra, and let Λ be the repetitive algebra of Λ. For the stable category of finitely generated left Λ-modules Λ-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 Λ-mod. We use the fact (and prove) that every Auslander-Reiten triangle in Λ-mod is induced from an Auslander-Reiten sequence of finitely generated left Λ-modules.2010 Mathematics Subject Classification. 16G10 and 16G20 and 20C20.

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Section: Introductionsupporting

confidence: 55%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Calderón-Henao¹,

Giraldo²,

Vélez-Marulanda³

2019

Preprint

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“…I, §2], i.e., Λ-mod is an exact category in the sense of [25] which has enough projective as well as injective objects, and these classes of objects coincide. (ii) By [16,Prop. 6] we have that the indecomposable projective-injective Λ-modules can be represented as follows:…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…It follows from [16,Prop. 6] that there exists an orthogonal primitive idempotent element in Λ such that P = k ⊗ R Q is as in (2.3).…”

Section: Lifts and Deformations Of Modules Over Repetitive Algebrasmentioning

confidence: 99%

On a deformation theory of finite dimensional modules over repetitive algebras

Fonce-Camacho¹,

Giraldo²,

Rizzo³

et al. 2020

Preprint

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Let Λ be a basic finite dimensional algebra over an algebraically closed field k, and let Λ be the repetitive algebra of Λ. In this article, we prove that if V is a left Λ-module with finite dimension over k, then V has a well-defined versal deformation ring R( Λ, V ), which is a local complete Noetherian commutative k-algebra whose residue field is also isomorphic to k. We also prove that in this situation R( Λ, V ) is stable after taking syzygies and that R( Λ, V ) is universal provided that End Λ ( V ) = k. We apply the obtained results to finite dimensional modules over the repetitive algebra of the 2-Kronecker algebra, which provides an alternative approach to the deformation theory of objects in the bounded derived category of coherent sheaves over P 1 k .2010 Mathematics Subject Classification. 16G10 and 16G20 and 16G70. Key words and phrases. Repetitive algebras and (Uni)versal deformation rings and stable endomorphism rings and Frobenius categories.

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