Gaps for the Igusa-Todorov function

TL;DR

This paper investigates the Igusa-Todorov function for finite-dimensional algebras, showing the existence of modules with specific phi-values and identifying gaps in possible phi-values, which relate to the finitistic conjecture.

Contribution

It proves the existence of modules with phi-values just below the maximum and identifies gaps in the phi-values for certain algebras, linking these gaps to the finitistic conjecture.

Findings

Existence of modules with phi-value m-1 and 1

Identification of gaps in phi-values for some algebras

Gaps imply the validity of the finitistic conjecture

Abstract

For a finite dimensional algebra $A$ with $0 < \phi dim (A) = m < \infty$ we prove that there always exist modules $M$ and $N$ such that $\phi(M) = m-1$ and $\phi (N) = 1$. On the other hand, we see an example of an algebra that not every value between $1$ and its $\phi$-dimension is reached by the $\phi$ function. We call that values gaps and we prove that the algebras with gaps verifies the finitistic conjecture.