[Paper Review] Igusa-Todorov functions for radical square zero algebras

Marcelo Lanzilotta, Eduardo N. Marcos+1 more|arXiv (Cornell University)|Jun 19, 2015

Algebraic structures and combinatorial models3 references3 citations

TL;DR

This paper investigates Igusa-Todorov functions for radical square zero algebras, proving that the left and right $φ$-dimensions coincide in this setting. It establishes bounds for the $φ$ and $ψ$-dimensions, characterizes algebras achieving the maximal $ψ$-dimension, and identifies modules realizing the $φ$-dimension using quiver-theoretic notions of 'hearth' and 'member'.

ABSTRACT

We study the behaviour of the Igusa-Todorov functions for radical square zero algebras. We show that the left and the right $ϕ$-dimensions coincide, in this case. Some general results are given, but we concentrate more in the radical square zero algebras. Our study is based on two notions of hearth and member of a quiver $Q$. We give some bounds for the $ϕ$ and the $ψ$-dimensions and we describe the algebras for which the bound of $ψ$ is obtained. We also exhibit modules for which the $ϕ$-dimension is realised.

Motivation & Objective

• To study the behavior of Igusa-Todorov functions $φ$ and $ψ$ in radical square zero algebras.
• To prove that the left and right $φ$-dimensions coincide for such algebras.
• To establish upper bounds for the $φ$ and $ψ$-dimensions and characterize algebras achieving the maximal $ψ$-dimension.
• To identify specific modules for which the $φ$-dimension is realized, using quiver structure.

Proposed method

• Utilizes quiver-theoretic concepts of 'hearth' and 'member' to analyze the structure of radical square zero algebras.
• Applies the Igusa-Todorov functions $φ$ and $ψ$ to modules over algebras of the form $A = \mathbb{K}Q/J^2$, where $J$ is the arrow ideal.
• Employs the syzygy operator $Ω$ to compute module structures and analyze linear independence in the Grothendieck group $K_1$.
• Uses the rank of the map induced by $Ω$ on $K_1$ to determine the $φ$-dimension, particularly through the kernel of the induced map on $K_1$.
• Applies results from upper triangular matrix rings to decompose projective modules and analyze resolutions.
• Analyzes the global dimension and projective dimension behavior via the quiver’s structure, especially in algebras with finite global dimension.

Experimental results

Research questions

• RQ1 Do the left and right $φ$-dimensions coincide for radical square zero algebras?
• RQ2 What are the upper bounds for the $φ$ and $ψ$-dimensions in radical square zero algebras?
• RQ3 Which radical square zero algebras achieve the maximal possible $ψ$-dimension?
• RQ4 For which modules is the $φ$-dimension realized, and how can such modules be constructed?

Key findings

• The left and right $φ$-dimensions coincide for all radical square zero algebras.
• The $φ$-dimension of a radical square zero algebra is bounded by the rank of the map induced by the syzygy operator $Ω$ on the Grothendieck group $K_1$, with $φ\dim(A) = 2$ for the example algebra with quiver $1 \to 2 \to 3 \to 4$ and $1 \to 4$.
• The maximal $ψ$-dimension is achieved if and only if the algebra has no 'member' in its quiver, meaning all vertices are 'hearts'.
• Modules realizing the $φ$-dimension are constructed as direct sums $M = M_1 \oplus M_2$ where $\{[\Omega(M_1)], [\Omega(M_2)]\}$ are linearly independent in $K_1$ and $\{[\Omega^2(M_1)], [\Omega^2(M_2)]\}$ are linearly dependent.
• The $φ$-dimension is realized when the kernel of the induced map $\overline{\Omega}: K_1 \to K_1$ contains a basis element such as $[S_2] + [S_3] - [S_4]$, indicating specific syzygy relations.
• The algebra with quiver $1 \to 2 \to 3 \to 4$ and $1 \to 4$ has $φ\dim = 2$, and $M_1 = S_2 \oplus S_3$, $M_2 = S_4$ form a $φ$-witness module.