Igusa–Todorov functions for Artin algebras

Abstract

In this paper we study the behaviour of the Igusa-Todorov functions for Artin algebras A with finite injective dimension, and Gorenstein algebras as a particular case. We show that the φ-dimension and ψ-dimension are finite in both cases. Also we prove that monomial, gentle and cluster tilted algebras have finite φ-dimension and finite ψ-dimension.

Igusa-Todorov function on path rings

“…In [11] it was proved that if A is of Ω n -finite representation type for some n then φdim(A) and ψdim(A) are both finite. In this article, as a consequence of the above theorem, we give an example of a family of algebras of Ω ∞ -infinite representation type with finite φ-dimension and ψ-dimension.…”

“…• ψdim(A) = sup{ψ(M ) such that M ∈ modA} The following results give properties of the Igusa-Todorov functions for an Artin algebra A with id(A) < ∞. For the proof see [11].…”

“…In [11] was proved that if A is of Ω n -finite representation type for some n then φdim(A) and ψdim(A) are both finite. A natural question to ask is which is the behaviour of these dimensions in the case that A is of Ω ∞ -infinite representation type.…”

Recollements and homological dimensions

“…Moreover, the φ-dimension can be used to describe selfinjective algebras: an algebra A is selfinjective if and only if φdim(A) = 0 [12]. Recently, various works were dedicated to study and generalise the properties of Igusa-Todorov function and the φdimension [7,12,13,18,25]. In particular, the φ-dimension of an algebra A was characterised in terms of the bi-functors Ext i A (−, −) and Tor A i (−, −) in [7].…”

Singularities of dual varieties and phi dimension of Nakayama algebras

“…For the small values of ϕ-dimension we have the following theorem where the first result appears in both [LM18][ES17] for general artin algebras, but for the Nakayama algebras we give an alternative short proof.…”