Month: February 2023

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

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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

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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.