A Counterexample to the $φ$-Dimension Conjecture

Eric J. Hanson, Kiyoshi Igusa

Published 2019-11-01Version 1

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.