The Phi-dimension: A new homological measure

K. Igusa and G. Todorov introduced two functions $\phi$ and $\psi,$ which are natural and important homological measures generalising the notion of the projective dimension. These Igusa-Todorov functions have become into a powerful tool to understand better the finitistic dimension conjecture. In this paper, for an artin $R$-algebra $A$ and the Igusa-Todorov function $\phi,$ we characterise the $\phi$-dimension of $A$ in terms either of the bi-functors $\mathrm{Ext}^{i}_{A}(-, -)$ or Tor's bi-functors $\mathrm{Tor}^{A}_{i}(-,-).$ Furthermore, by using the first characterisation of the $\phi$-dimension, we show that the finiteness of the $\phi$-dimension of an artin algebra is invariant under derived equivalences. As an application of this result, we generalise the classical Bongartz's result as follows: For an artin algebra $A,$ a tilting $A$-module $T$ and the endomorphism algebra $B=\mathrm{End}_A(T)^{op},$ we have that $\mathrm{Fidim}\,(A)-\mathrm{pd}\,T\leq \mathrm{Fidim}\,(B)\leq \mathrm{Fidim}\,(A)+\mathrm{pd}\,T.$