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- W2952318082 abstract "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 $mathrm{Fidim},(A)-mathrm{pd},Tleq mathrm{Fidim},(B)leq mathrm{Fidim},(A)+mathrm{pd},T.$" @default.
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- W2952318082 date "2013-04-02" @default.
- W2952318082 modified "2023-09-27" @default.
- W2952318082 title "The Phi-dimension: A new homological measure" @default.
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