The ϕ-Dimension: A New Homological Measure

• Published in: Algebras and representation theory 2015-04, Vol.18 (2), p.463-476
• Main Authors: Fernandes, Sônia Maria, Lanzilotta, Marcelo, Mendoza Hernández, Octavio
• Format: Article
• Language: English
• Subjects: Associative Rings and Algebras; Commutative Rings and Algebras; Mathematics; Mathematics and Statistics; Non-associative Rings and Algebras
• ISSN: 1386-923X; 1572-9079
• DOI: 10.1007/s10468-014-9504-9

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 Ext A i ( − , − ) and in terms of Tor’s bi-functors Tor i A ( − , − ) . 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 .