arXiv subjects

Gustavo Mata

Publications and source records attributed to Gustavo Mata.

12 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

Handle element on nearly Frobenius algebras

In this article the concept of handle element of Frobenius algebras, as in [16], will be extended to nearly Frobenius algebras. The main properties of this element will be analyzed in this case and many examples will be constructed. Also the Casimir and Schur elements of symmetric algebras will be considered ([8]) and generalized for Frobenius and nearly Frobenius algebras showing which results still hold in this framework and which ones fail.

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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

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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arXiv2020

Nearly Frobenius dimension on Frobenius algebras

This article is divided into two parts. In the first part we work over a field $\mathbb{k}$ and prove that the Frobenius space associated to a Frobenius algebra is generated as left A-module by the Frobenius coproduct. In particular, we prove that the Frobenius dimension coincides with the dimension of the algebra. In the second part we work with a commutative ring $k$. We introduce the concept of nearly Frobenius algebras in this context and construct solutions of the Yang-Baxter equation starting from elements in the Frobenius space. Also, we give a list of equivalent characterizations of nearly Frobenius algebras.

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arXiv2019

Nearly Frobenius theory and semisimplicity of bimodules

In the first part of this article we prove that one of the conditions required in the original definition of nearly Frobenius algebra, the coassociativity, is redundant. Also, we determine the Frobenius dimension of the product and tensor product of two nearly Frobenius algebras from the Frobenius dimension of each of them. We apply these results to semisimple algebras. In the second part we introduce the notion of normalized nearly Frobenius algebra. We prove a series of equivalences: the concept of normalized nearly Frobenius algebra is equivalent to the concept of separable algebra, equivalent to the fact that the algebra is projective as a bimodule on itself and, finally, equivalent to the category of bimodules is semisimple. Also, we relate these concepts with the property of semisimplicity of the category of modules over the algebra.

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arXiv2019

Nearly Frobenius structures in some families of algebras

In this article we continue with the study started in [1] of nearly Frobenius structures in some representative families of finite dimensional algebras, as the radical square zero algebras, string algebras and the toupie algebras. We prove that the radical square zero algebras with at least one path of length two are nearly Frobenius. As for the string algebras, in the ones that are not gentle, we can afirm that there is at least one non-trivial nearly Frobenius structure. Finally, in the case of the toupie algebras, we prove that the existence of monomial relations is a suficient condition to have non-trivial nearly Frobenius structure. Using the technics developed for the previous families of algebras we prove suficient conditions for the existence of non-trivial Frobenius structures in quotients of path algebras in general.

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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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arXiv2016

Igusa-Todorov functions for Artin algebras

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.

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