Searcharxiv⌕ Search

arXiv subjects

Marcelo Lanzilotta

Publications and source records attributed to Marcelo Lanzilotta.

30 records · Page 2Linked to original sources
arXiv2015

Filtering subcategories of modules of an artinian algebra

Let $A$ be an artinian algebra, and let $\mathcal{C}$ be a subcategory of mod$A$ that is closed under extensions. When $\mathcal{C}$ is closed under kernels of epimorphisms (or closed under cokernels of monomorphisms), we describe the smallest class of modules that filters $\mathcal{C}$. As a consequence, we obtain sufficient conditions for the finitistic dimension of an algebra over a field to be finite. We also apply our results to the torsion pairs. In particular, when a torsion pair is induced by a tilting module, we show that the smallest classes of modules that filter the torsion and torsion-free classes are completely compatible with the quasi-equivalences of Brenner and Butler.

math.RT↗
arXiv2014

The Phi-dimension: A new homological measure

K. Igusa and G. Todorov 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 into 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 either of the bi-functors $\mathrm{Ext}^{i}_{A}(-, -)$ or Tor's bi-functors $\mathrm{Tor}^{A}_{i}(-,-).$ 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 as follows: For an artin algebra $A,$ a tilting $A$-module $T$ and the endomorphism algebra $B=\mathrm{End}_A(T)^{op},$ we have that $\mathrm{Fidim}\,(A)-\mathrm{pd}\,T\leq \mathrm{Fidim}\,(B)\leq \mathrm{Fidim}\,(A)+\mathrm{pd}\,T.$

math.RT↗
arXiv2013

Constructing nearly Frobenius algebras

In the first part we study nearly Frobenius algebras. The concept of nearly Frobenius algebras is a generalization of the concept of Frobenius algebras. Nearly Frobenius algebras do not have traces, nor they are self-dual. We prove that the known constructions: direct sums, tensor, quotient of nearly Frobenius algebras admit natural nearly Frobenius structures. In the second part we study algebras associated to some families of quivers and the nearly Frobenius structures that they admit. As a main theorem, we prove that an indecomposable algebra associated to a bound quiver $(Q,I)$ with no monomial relations admits a non trivial nearly Frobenius structure if and only if the quiver is $\overrightarrow{\mb{A}_n}$ and I=0. We also present an algorithm that determines the number of independent nearly Frobenius structures for Gentle algebras without oriented cycles.

math.RA↗
arXiv2013

Split-by-nilpotent extensions algebras and stratifying systems

Let $Γ$ and $Λ$ be artin algebras such that $Γ$ is a split-by-nilpotent extension of $Λ$ by a two sided ideal $I$ of $Γ.$ Consider the so-called change of rings functors $G:={}_ΓΓ_Λ\otimes_Λ-$ and $F:={}_ΛΛ_Γ\otimes_Γ-.$ In this paper, we find the necessary and sufficient conditions under which a stratifying system $(Θ,\leq)$ in $\moduΛ$ can be lifted to a stratifying system $(GΘ,\leq)$ in $\modu\,(Γ).$ Furthermore, by using the functors $F$ and $G,$ we study the relationship between their filtered categories of modules and some connections with their corresponding standardly stratified algebras are stated. Finally, a sufficient condition is given for stratifying systems in $\modu\,(Γ)$ in such a way that they can be restricted, through the functor $F,$ to stratifying systems in $\modu\,(Λ).$

math.RT↗
arXiv2011

Algebras Determined by Their Supports

In this paper, we introduce and study a class of algebras which we call ada algebras. An artin algebra is ada if every indecomposable projective and every indecomposable injective module lies in the union of the left and the right parts of the module category. We describe the Auslander-Reiten components of an ada algebra, showing in particular that its representation theory is entirely contained in that of its left and right supports, which are both tilted algebras. Also, we prove that an ada algebra over an algebraically closed field is simply connected if and only if its first Hochschild cohomology group vanishes.

math.RT↗
arXiv2011

An approach to the finitistic dimension conjecture

Let $R$ be a finite dimensional $k$-algebra over an algebraically closed field $k$ and $\mathrm{mod} R$ be the category of all finitely generated left $R$-modules. For a given full subcategory $\mathcal{X}$ of $\mathrm{mod} R,$ we denote by $\pfd \mathcal{X}$ the projective finitistic dimension of $\mathcal{X}.$ That is, $\pfd \mathcal{X}:=\mathrm{sup} \{\pd X : X\in\mathcal{X} \text{and} \pd X<\infty\}.$ \ It was conjectured by H. Bass in the 60's that the projective finitistic dimension $\pfd (R):=\pfd (\mathrm{mod} R)$ has to be finite. Since then, much work has been done toward the proof of this conjecture. Recently, K. Igusa and J. Todorov defined a function $Ψ:\mathrm{mod} R\to \Bbb{N},$ which turned out to be useful to prove that $\pfd (R)$ is finite for some classes of algebras. In order to have a different approach to the finitistic dimension conjecture, we propose to consider a class of full subcategories of $\mathrm{mod} R$ instead of a class of algebras, namely to take the class of categories $\F(θ)$ of $θ$-filtered $R$-modules for all stratifying systems $(θ,\leq)$ in $\mathrm{mod} R.$

math.RT↗
arXiv2011

Self-injective right artinian rings and Igusa Todorov functions

We show that a right artinian ring $R$ is right self-injective if and only if $ψ(M)=0$ (or equivalently $ϕ(M)=0$) for all finitely generated right $R$-modules $M$, where $ψ, ϕ: \mod R \to \mathbb N$ are functions defined by Igusa and Todorov. In particular, an artin algebra $Λ$ is self-injective if and only if $ϕ(M)=0$ for all finitely generated right $Λ$-modules $M$.

math.RT↗
arXiv2010

Toupie algebra, some examples of laura algebras

We consider the bound quiver algebras whose ordinary quiver is that of a canonical algebra. We determine which of those algebras are hereditary, tilted, quasitilted, weakly shod or laura algebras.

math.RT↗
arXiv2009

Combinatorial classification of piecewise hereditary algebras

We use the characteristic polynomial of the Coxeter matrix of an algebra to complete the combinatorial classification of piecewise hereditary algebras which Happel gave in terms of the trace of the Coxeter matrix. We also give a cohomological interpretation of the coefficients (other than the trace) of the characteristic polynomial of the Coxeter matrix of any finite dimensional algebra with finite global dimension.

math.RT↗
arXiv2008

Skew group algebras of piecewise hereditary algebras are piecewise hereditary

We show that the main results of Happel-Rickard-Schofield (1988) and Happel-Reiten-Smalo (1996) on piecewise hereditary algebras are coherent with the notion of group action on an algebra. Then, we take advantage of this compatibility and show that if G is a finite group acting on a piecewise hereditary algebra A over an algebraically closed field whose characteristic does not divide the order of G, then the resulting skew group algebra A[G] is also piecewise hereditary.

math.RA↗