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The ϕ-Dimension: A New Homological Measure

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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)op, we have that ϕ dim (A) − pd T ≤ ϕ dim (B) ≤ ϕ dim (A) + pd T.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal de Viçosa, CP 36570000, Viçosa, MG, Brasil

    Sônia Maria Fernandes

  2. Instituto de Matemática y Estadística Rafael Laguardia, Facultad de Ingeniería, Universidad de la República, J. Herrera y Reissig 565, CP 11300, Montevideo, Uruguay

    Marcelo Lanzilotta

  3. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, C.P. 04510, México, D.F., México

    Octavio Mendoza Hernández

Authors
  1. Sônia Maria Fernandes
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  2. Marcelo Lanzilotta
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  3. Octavio Mendoza Hernández
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Correspondence to Octavio Mendoza Hernández.

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Communicated by: Raymundo Bautista

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Fernandes, S.M., Lanzilotta, M. & Mendoza Hernández, O. The ϕ-Dimension: A New Homological Measure. Algebr Represent Theor 18, 463–476 (2015). https://doi.org/10.1007/s10468-014-9504-9

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  • DOI: https://doi.org/10.1007/s10468-014-9504-9

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