The Igusa-Todorov $\phi$ function for truncated path algebras

Abstract

Given a truncated path algebra $A=\frac{\Bbbk Q}{J^k}$ we prove that $\fidim A = \fidim A^{\op}$. We also compute the $\phi$-dimension of $A$ in function of the $\phi$-dimension of $\frac{\Bbbk Q}{J^2}$ when $Q$ has no sources nor sinks. This allows us to bound the $\phi$-dimension for truncated path algebras. Finally, we characterize $A$ when its $\phi$-dimension is equal to $1$.