Relative Igusa-Todorov Functions and Relative Homological Dimensions

Abstract

Abstract. We develope the theory of the E-relative Igusa-Todorov functions in an exact IT -context (C, E) (see Definition 2.1). In the case when C = mod (Λ) is the category of finitely generated left Λ-modules, for an artin algebra Λ, and E is the class of all exact sequences in C, we recover the usual Igusa-Todorov functions [33]. We use the setting of the exact structures and the Auslander-Solberg relative homological theory to generalise the original Igusa-Todorov's results. Furthermore, we introduce the E-relative IgusaTodorov dimension and also we obtain relationships with the relative global and relative finitistic dimensions and the Gorenstein homological dimensions.

Generalised Igusa-Todorov functions and Lat-Igusa-Todorov algebras

“…Another application of Theorem B is related with Gorenstein homological algebra. Indeed, the global Gorenstein projective dimension of an Artin algebra Λ is gl.Gpdim(Λ) := sup{Gpd(M ) : M ∈ mod (Λ)}, where Gpd(M ) is the Gorenstein projective dimension of M. The following result generalises [20,Theorem 4.7], for a complete version see Theorem 4.5.…”

“…This raises the question of exactly how can be changed this class P of modules in order to define new Igusa-Todorov functions and what is the relation between them. In order to see a different solution of this question, by using relative homological algebra, we recommend the reader to see [20]. In what follows, we will address this issues.…”

Pullback diagrams, syzygy finite classes and Igusa–Todorov algebras

“…These functions are nowadays known as the Igusa-Todorov functions, or IT-functions for short. For a further development of IT-functions, we recommend the reader to see in [2,6].…”

“…Proof. See the proofs given in [3,5,6,7]. ✷ Related with the IT-functions, there are the Φ-dimension and the Ψ-dimension which were introduced in [4] and defined as follows…”

A Counterexample to the $ϕ$-Dimension Conjecture

“…In particular, finiteness of either the φ-or ψ-dimension implies finiteness of the finitistic dimension. This fact, and the prevalence of the φ-and ψ-dimensions in recent literature on the finitistic dimension conjecture, has led to the so called φ-dimension conjecture and ψ-dimension conjecture, formally stated by Fernandes-Lanzilotta-Mendoza [FLM15] and Lanzilotta-Mendoza [LM17]. These conjectures state, respectively, that φdimΛ < ∞ and ψdimΛ < ∞ for all Artin algebras.…”