arXiv subjects

Marcos Barrios

Publications and source records attributed to Marcos Barrios.

8 recordsLinked to original sources

arXiv2024

Delooping levels

In [8] V. G\'elinas introduced a homological invariant, called {\it delooping level} (dell), that bounds the finitistic dimension. In this article, we introduce another homological invariant (Dell) related to the delooping level for an Artin algebra. We compare this new tool with other dimensions as the finitistic dimension or the $\phi$-dimension (where $\phi$ is the first Igusa-Todorov function), and we also generalize Theorem 4.3. from [9] to truncated path algebras (Theorem 4.18). Finally, we show that for a monomial algebra $A$ the difference dell($A$) - Findim($A$) can be arbitrarily large (Example 4.22).

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arXiv2023

A survey on Igusa-Todorov functions

In this survey, we review the fundamental properties of the Igusa-Todorov functions, the $ϕ$-dimension, the $ψ$-dimension and their generalizations.

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arXiv2023

On algebras of $Ω^n$-finite and $Ω^{\infty}$-infinite representation type

Co-Gorenstein algebras were introduced by A. Beligiannis in \cite{B}. In \cite{KM}, the authors propose the following conjecture (Co-GC): if $Ω^n (\mod A)$ is extension closed for all $n \leq 1$, then $A$ is right Co-Gorenstein, and they prove that the Generalized Nakayama Conjecture implies the Co-GC, also that the Co-GC implies the Nakayama Conjecture. In this article we characterize the subcategory $Ω^{\infty}(\mod A)$ for algebras of $Ω^{n}$-finite representation type. As a consequence, we characterize when a truncated path algebra is a Co-Gorenstein algebra in terms of its associated quiver. We also study the behaviour of Artin algebras of $Ω^{\infty}$-infinite representation type. Finally, it is presented an example of a non Gorenstein algebra of $Ω^{\infty}$-infinite representation type and an example of a finite dimensional algebra with infinite $ϕ$-dimension.

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arXiv2022

Continuous homomorphisms on piecewise absolutely continuous maps of $\mathbb{S}^{1}$

Let $IET(\mathbb{S}^{1})$ be the group of interval exchange transformation of $\mathbb{S}^{1}$ and $\mathcal{AC}_{+}(\mathbb{S}^{1})$ be the group of absolutely continuous preserving orientation bijection with inverse absolutely continuous. We denote by $\mathcal{ACI}$ the group generated by $IET(\mathbb{S}^{1})$ and $\mathcal{AC}_{+}(\mathbb{S}^{1})$. Given a suitable distance on $\mathcal{ACI}$, we classify all continuous homomorphisms $ρ:\mathbb{R} \to \mathcal{ACI}$. More precisely, $ρ$ is conjugated to a continuous homomorphism $\hatρ:\mathbb{R} \to \mathcal{AC}_{+}(\uplus_{i}(\mathbb{S}^{1})_{i})$.

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arXiv2022

The Igusa-Todorov $ϕ$-dimension on Morita context algebras

In this article we prove that, under certain hypotheses, Morita context algebras that have zero bimodule morphisms have finite $ϕ$-dimension. We also study the behaviour of the $ϕ$-dimension for an algebra and its opposite. In particular we show that the $ϕ$-dimension of an Artin algebra is not symmetric, i.e. there exists a finite dimensional algebra $A$ such that $ϕ\dim (A) \not = ϕ\dim (A^{op})$.

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arXiv2021

On Lat-Igusa-Todorov algebras

Lat-Igusa-Todorov algebras are a natural generalization of Igusa-Todorov algebras. They are defined using the generalized Igusa-Todorov functions given in \cite{BLMV} and also verify the finitistic dimension conjecture. In this article we give new ways to construct examples of Lat-Igusa-Todorov algebras. On the other hand we show an example of a family of algebras that are not Lat-Igusa-Todorov.

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arXiv2018

Gaps for the Igusa-Todorov function

For a finite dimensional algebra $A$ with $0 < ϕdim (A) = m < \infty$ we prove that there always exist modules $M$ and $N$ such that $ϕ(M) = m-1$ and $ϕ(N) = 1$. On the other hand, we see an example of an algebra that not every value between $1$ and its $ϕ$-dimension is reached by the $ϕ$ function. We call that values gaps and we prove that the algebras with gaps verifies the finitistic conjecture.

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arXiv2018

The Igusa-Todorov $ϕ$ function for truncated path algebras

Given a truncated path algebra $A=\frac{\Bbbk Q}{J^k}$ we prove that $\fidim A = \fidim A^{\op}$. We also compute the $ϕ$-dimension of $A$ in function of the $ϕ$-dimension of $\frac{\Bbbk Q}{J^2}$ when $Q$ has no sources nor sinks. This allows us to bound the $ϕ$-dimension for truncated path algebras. Finally, we characterize $A$ when its $ϕ$-dimension is equal to $1$.

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