A counterexample to the $\phi$-dimension conjecture

TL;DR

This paper provides a counterexample to the long-standing {} dimension conjecture, challenging previous assumptions and impacting ongoing research in algebraic representation theory.

Contribution

It introduces the first known counterexample to the {} dimension conjecture, revealing limitations of the existing upper bound hypothesis.

Findings

Counterexample disproves the {} dimension conjecture.

Implications suggest the need to revise or refine the conjecture.

Discussion on how this affects the finitistic dimension conjecture.

Abstract

In 2005, the second author and Todorov introduced an upper bound on the finitistic dimension of an Artin algebra, now known as the {\phi}-dimension. The {\phi}-dimension conjecture states that this upper bound is always finite, a fact that would imply the finitistic dimension conjecture. In this paper, we present a counterexample to the {\phi}-dimension conjecture and explain where it comes from. We also discuss implications for further research and the finitistic dimension conjecture.