algebra

The Auslander-Reiten Quiver of the 2-Kronecker Algebra

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İrem Yıldız, Dokuz Eylül University.
Date: 10th of January, 2023, Tuesday, Time: 13:30 – 15:30.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206 (Online-Sakai-Graduate Meetings).

Abstract: The Auslander-Reiten quiver of a finite dimensional k-algebra A is the quiver whose vertices are the isoclasses of indecomposable modules over A, and the number of arrows between two vertices [M],[N] is given by the k-dimension of Irr_{A}(M,N).
In this talk, we will review some basic facts about almost split sequences and explain briefly how to construct them. Our aim is to construct the Auslander-Reiten quiver of the path algebra associated to the 2-Kronecker quiver.

Torsion and Torsion-Free Classes Arising Through Objects Of Finite Type in a Grothendieck Category

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Sinem Odabaşı, Universidad de Murcia.
Date: 3th of January, 2023, Tuesday, Time: 13.00 - 14.30.
Place: Online-Join Zoom Meeting (https://zoom.us/j/99412179515?pwd=bUlLdStFaVpjMHhWb21LaFlRMWI1Zz09) 
Meeting ID: 994 1217 9515 Passcode: 553266

Abstract: In a module category, the class FPn(R) of the so-called finitely n-presented left R-modules induces two crucial classes In(R) and Fn(Rop) of left FPn-injective and right FPn-flat Rmodules, respectively. It is known that certain homological properties of In(R) and Fn(Rop) and closure properties of FPn(R) are closely related, and these properties determine ring theoretic properties of R. The authors in [1] introduce the class FPn(G) of objects of type FPn in a Grothendieck category G as a generalization of the class FPn(R), and study certain aspects of the associated class In(G) of FPn-injective objects in G. In this talk, further homological aspects of the class In(G) will be presented. We show that the projective dimension of the class FPn(G) controls how far is In(G) from being a torsion class. Under mild conditions on G, which permit us to have ‘the external tensor product functor’, we introduce the class Fn(G) of FPn-flat objects in G. We will exhibit a close interaction between the classes In(G) and Fn(G). Further applications will be presented in Ab-valued functor categories showing n-coherency in terms of the domain category This is a joint work with Daniel Bravo, Carlos Parra and Marco Perez; see [2].

References

[1] Bravo, D., Gillespie, J. & Perez, M. A. (2019). Locally typeFPn and n-coherent categories. arXiv:1908.10987

[2] Bravo, D., Odabas¸ı, S., Parra, C & Perez, M. A. (2022). Torsion and Torsion-free classes from objects of finite type in Grothendieck categories. Journal of Algebra, 608, 412-444. DOI: 10.1016/j.jalgebra.2022.05.029.

On coGalois Groups

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Canan Özeren, Dokuz Eylül University.
Date: 26th of December, 2022, Tuesday, Time: 13:30 – 15:30.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.

Abstract:Torsion-free covers exist for abelian groups (see [1]).
The coGalois group of a torsion-free cover \phi: T \rightarrow A of an abelian group is defined in [2] as the group of f: T \rightarrow T s.t. \phi f= \phi and is denoted by G(\phi).
The abelian groups for which the coGalois group is trivial characterized in [3].
The notion of coGalois group can be defined in any category where we have a covering class.
In [4], coGalois groups are studied in the category of representation of quiver q_2 : \cdot \rightarrow \cdot .
We talk about the necessary conditions, give in [4], the coGalois group, associated to torsion free-cover, for an object in (q_2, Z-mod).

References

[1] E. Enochs: Torsion-free covering modules. (1963)

[2] E. Enochs, J. R. García Rozas and L. Oyonarte: Compact coGalois groups. (2000).

[3] E. Enochs and J. Rada: Abelian groups which have trivial absolute \ coGalois group. (2005).

[4] Paul Hill, Abelian group pairs having a trivial coGalois Group. (2006).

Coxeter Functors and Gabriel’s Theorem III

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İrem Yıldız, Dokuz Eylül University.
Date: 20th of December, 2022, Tuesday, Time: 13:30 – 15:30.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.

Abstract: In this talk, we will introduce the basics of representation of quivers. We will define some functors which annihilate simples and carry indecomposables to indecomposables, which will help to prove Gabriel’s Theorem that states which types of quivers have finitely many isomorphisms classes of indecomposable representations.

References

[1] Some lecture notes in Bielefeld university from Henning Krause . 

[2] Article from I. N. Bernstein, I. M. Gel’fand, V. A. Ponomarev named “Coxeter functor and Gabriel ‘s theorem”.

Coxeter Functors and Gabriel’s Theorem II

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İrem Yıldız, Dokuz Eylül University.
Date: 13th of December, 2022, Tuesday, Time: 13:30 – 15:30.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.

Abstract: In this talk, we will introduce the basics of representation of quivers. We will define some functors which annihilate simples and carry indecomposables to indecomposables, which will help to prove Gabriel’s Theorem that states which types of quivers have finitely many isomorphisms classes of indecomposable representations.

References

[1] Some lecture notes in Bielefeld university from Henning Krause . 

[2] Article from I. N. Bernstein, I. M. Gel’fand, V. A. Ponomarev named “Coxeter functor and Gabriel ‘s theorem”.

Coxeter Functors and Gabriel’s Theorem

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İrem Yıldız, Dokuz Eylül University.
Date: 6th of December, 2022, Tuesday, Time: 13:30 – 15:30.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.

Abstract: In this talk, we will introduce the basics of representation of quivers. We will define some functors which annihilate simples and carry indecomposables to indecomposables, which will help to prove Gabriel’s Theorem that states which types of quivers have finitely many isomorphisms classes of indecomposable representations.

References

[1] Some lecture notes in Bielefeld university from Henning Krause . 

[2] Article from I. N. Bernstein, I. M. Gel’fand, V. A. Ponomarev named “Coxeter functor and Gabriel ‘s theorem”.

On Higher Commutators

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M. Pınar Eroğlu, Dokuz Eylül University.
Date: 25th of October, 2022, Tuesday, Time: 13:30 - 15:30.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.
Abstract:  A general result on higher commutators due to Herstein I.N. states that if A is a noncommutative simple algebra over a field of characteristic not 2, then the only higher commutators of A are A and [A,A]. For a more general approach, the question that arises naturally is: Which assumptions should be required to obtain a result similar to above in the case where A is an arbitrary noncommutative unital algebra? In this talk, we characterize higher commutators of unital algebras discussing the question above.

Almost Split Sequences

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Fatma Kaynarca, Afyon Kocatepe University.
Date: 15th of November, 2022, Tuesday, Time: 13.00 – 14.00.
Place: Online-Sakai-Graduate Meetings

Abstract: Almost split sequences arose from an attempt to understand the morphisms lying in the radical of a module category are minimal non-split short exact sequences. This sequences were introduced by Maurice Auslander and Idun Reitenin 1974-1975 and have become a central tool in the theory of representations of finite dimensional algebras. We start our discussion in seminar with a short description of the radical of a module category. Then we will define and study irreducible morphisms, almost split morphisms, minimal morphisms, almost split sequences and also give some characterizations of these notions.