Ring Theory Seminar
Speaker: Flávio U. Coelho (University of Sao Paulo)
Title: Mesh-comparable components of the AR-quiver
Date: 5/11/2025
Time: 10:00
Web: http://mat.uab.cat/web/ligat/
Abstract: Given a $k$-algebra $A$ (where $k$ is a field), one way of organizing the category mod$A$ of the finitely generated right $A$-modules is through the so-called Auslander-Reiten quiver $\Gamma$(mod$A)$. The vertices of such quiver correspond to the isoclasses of the indecomposable objects in mod$A$ and the arrows indicate the existence of irreducible morphisms between them. Recall that, for $A$-modules $X,Y$, $\mathrm{rad}_A(X,Y)$ denotes the set of non-isomorphisms $X \rightarrow Y$. Clearly, one can extend it to general modules as follows: $\mathrm{rad}_A(\oplus_{i=1}^n X_i,\ oplus_{j=1}^m Y_j) = \oplus_{i=1}^n\oplus_{j=1}^m \mathrm{rad}_A(X_i,Y_j)$. Using the fact that $\mathrm{rad}_A$ is an ideal of the category $\textrm{mod} A$, one can consider its powers, defined recursively by: $\mathrm{rad}_A^0 = \mathrm{Hom}_A, \mathrm{rad}_A^1= \mathrm{rad}_A, \mathrm{rad}_A^n = \mathrm{rad}_A^{n-1} \cdot \mathrm{rad}_A$, where the product $\cdot$ stands for composition of morphisms. We also define $\mathrm{rad}_A^{\infty} = \cap_{n \geq 0} \mathrm{rad}_A^n$. A morphism is called irreducible if it belongs to $\mathrm{rad}_A(X,Y) \setminus \mathrm{rad}_A^2(X,Y)$. Irreducible morphisms are of key importance, since, as shown by Auslander-Reiten theory, these morphisms generate any other morphism modulo $\mathrm{rad}^{\infty}$. Clearly, a composition of $n$ irreducible morphisms between indecomposable modules belongs to $\mathrm{rad}^n$ and one could wonder if it is also true that, provide it is non-zero, it does not belong to $\mathrm{rad}^{n+1}$. This is not true, not even for $n \geq 2$. Deciding in which cases there might be a non-zero composition of $n$ irreducible morphisms belonging to $\mathrm{rad}^{n+1}$ has become an interesting line of investigation. The purpose of this talk is to define mesh-comparable components using the so-called Riedtmann's functors and show that, in such components, one can choose for each of its arrow an irreducible morphism in such a way that a composite of $n\geq 2$ of them either belongs to rad$_A^{n} \setminus $ rad$_A^{n+1}$ or is zero. This is part of a joint work with Viktor Chust "Mesh-comparable components of the AR-quiver".