Zübeyir Türkoğlu – Page 3 – Dokuz Eylül University Faculty of Science Department of Mathematics

On Rings Whose Finite Length Indecomposable Modules Are Completely Determined by Their Composition Factors II

March 7 2023 by Zübeyir Türkoğlu

Victor Blasco Jimenez, Dokuz Eylül University. Date: 8th of March, 2023, Wednesday, Time: 10.30 – 12.00. Place:Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206 (Online-Sakai-Graduate Meetings). Abstract: Today we will continue the discussion we started the other day. In particular, we will show that a commutative Artinian Ring $R$ of finite representation type satisfies $\mathfrak{X}$ , that is, finitely generated indecomposable $R$ -modules are completely determined by their composition factors.

On Rings Whose Finite Length Indecomposable Modules Are Completely Determined by Their Composition Factors

February 28 2023February 28 2023 by Zübeyir Türkoğlu

Victor Blasco Jimenez, Dokuz Eylül University.

Date: 1st of March, 2023, Wednesday, Time: 10.30 – 12.00.

Place:Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206 (Online-Sakai-Graduate Meetings).

Abstract: From the Fundamental Theorem of Abelian Groups we can deduce that every finite length indecomposable module over $\mathbb{Z}$ is uniquely determined by its composition factors set, meaning that if we have two indecomposable finite length abelian groups $N^1$ and $N^2$ with the same length and, if for any $0\subseteq N_{1}^j\subseteq ... \subseteq N_{r}^j=N^j$ composition series for $N^j$ , $j=1,2$ , we get $\{N_{i+1}^1/N_{i}^1\}_{i=0}^{r-1}=\{N_{i+1}^2/N_{i}^2\}_{i=0}^{r-1}$ , then we must have $N^1\cong N^2$ . In this series of talks , we will study this property about the finite length indecomposable abelian groups in a more general way. We will start by focusing on the class of commutative rings $R$ which satisfy it, showing that it contains the class of Dedekind Domains. If time permits, we will see that if $R$ is any unital ring (not necessarily commutative) satisfying this property, which we will call “Property $\mathfrak{X}$ “, and $I$ is an ideal of $R$ , then also $R/I$ satisfies it. This work is part of my ongoing master thesis “Some methods of Category Theory in the Representation Theory of Artin Algebras”.

The Auslander-Reiten Quiver of the 2-Kronecker Algebra III

February 21 2023February 21 2023 by Zübeyir Türkoğlu

İrem Yıldız, Dokuz Eylül University.
Date: 22nd of February, 2023, Wednesday, Time: 10.30 – 12.00.
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.

The Auslander-Reiten Quiver of the 2-Kronecker Algebra II

January 16 2023January 16 2023 by Zübeyir Türkoğlu

İrem Yıldız, Dokuz Eylül University.
Date: 17th 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).

The Auslander-Reiten Quiver of the 2-Kronecker Algebra

January 10 2023January 10 2023 by Zübeyir Türkoğlu

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).

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

January 9 2023January 9 2023 by Zübeyir Türkoğlu

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

December 26 2022December 26 2022 by Zübeyir Türkoğlu

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

January 10 2023December 14 2022 by Zübeyir Türkoğlu

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

December 13 2022December 13 2022 by Zübeyir Türkoğlu

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.

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

December 5 2022December 5 2022 by Zübeyir Türkoğlu

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.

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”.