  Let $\Lambda$ be an artin algebra and $\mathfrak{A}$ a two-sided idempotent
ideal of $\Lambda$, that is, $\mathfrak{A}$ is the trace of a projective
$\Lambda$-module $P$ in $\Lambda$. We consider the categories of finitely
generated modules over the associated rings $\Lambda/\mathfrak{A}, \Lambda$ and
$\Gamma=\mathrm{End}_{\Lambda}(P)^{op}$ and study the relationship between
their homological properties via the Igusa-Todorov functions.
