
Mathematics 2013
The Phidimension: A new homological measureAbstract: 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 IgusaTodorov 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 IgusaTodorov function $\phi,$ we characterise the $\phi$dimension of $A$ in terms either of the bifunctors $\mathrm{Ext}^{i}_{A}(, )$ or Tor's bifunctors $\mathrm{Tor}^{A}_{i}(,).$ 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 $\mathrm{Fidim}\,(A)\mathrm{pd}\,T\leq \mathrm{Fidim}\,(B)\leq \mathrm{Fidim}\,(A)+\mathrm{pd}\,T.$
