The Phi-dimension: A new homological measure

TL;DR

This paper introduces the phi-dimension, a new homological measure for artin algebras, characterizes it via Ext and Tor functors, and proves its invariance under derived equivalences, with applications to finitistic dimension bounds.

Contribution

It defines the phi-dimension for artin algebras, characterizes it using Ext and Tor, and proves its invariance under derived equivalences, extending classical results.

Findings

Finiteness of phi-dimension is invariant under derived equivalences.

Provides bounds for finitistic dimension related to tilting modules.

Characterizes phi-dimension using Ext and Tor bi-functors.

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 $\mathrm{Fidim}\,(A)-\mathrm{pd}\,T\leq \mathrm{Fidim}\,(B)\leq \mathrm{Fidim}\,(A)+\mathrm{pd}\,T.$