Let $A$ be an Artin algebra and $F$ a non-zero subfunctor of $\Ext_A^{ 1}(-,-)$. In this paper, we characterize the relative $\phi$-dimension of $A$ by the bi-functor $\Ext_F^1(-,-)$. Furthermore, we show that the finiteness of relative $\phi$-dimension of an Artin algebra is invariant under relative derived equivalence. More precisely, for an Artin algebra $A$, assume that $F$ has enough projectives and injectives, such that there exists $G\in \modcat{ A}$ such that $\add G=\mathcal { P}(F)$, where $\mathcal { P}(F)$ is the category of all $F$-projecitve $A$-modules. If $\cpx{ T}$ is a relative tilting complex over $A$ with term length $t(\cpx{ T})$ such that $B=\End(\cpx{ T})$, then we have $\phd_{ F}(A)-t(T^{ \bullet})\leq \phd(B)\leq\phd_{ F}(A)+t(T^{ \bullet})+2$.
Categories: math.RT