The ϕ-Dimension: A New Homological Measure

Abstract

In Igusa and Todorov (2005) introduced two functions ϕ and ψ, which are natural and important homological measures generalising the notion of the projective dimension. These Igusa-Todorov functions have become a powerful tool to understand better the finitistic dimension conjecture. In this paper, for an artin R-algebra A and the Igusa-Todorov function ϕ, we characterise the ϕ-dimension of A in terms of the bi-functors $\text{Ext}^{i}_{A}(-, -)$ and in terms of Tor’s bi-functors $\text{Tor}^{A}_{i}(-,-).$ Furthermore, by using the first characterisation of the ϕ-dimension, we show that the finiteness of the ϕ-dimension of an artin algebra is invariant under derived equivalences. As an application of this result, we generalise the classical Bongartz’s result (Bongartz, Lect. Notes Math. 903, 26–38, (1981), Corollary 1) as follows: For an artin algebra A, a tilting A-module T and the endomorphism algebra B = End A (T) o p , we have that ϕ dim (A) − pd T ≤ ϕ dim (B) ≤ ϕ dim (A) + pd T.