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François Huard

Publications and source records attributed to François Huard.

6 recordsLinked 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↗
arXiv2015

Approximations of injective modules and finitistic dimension

Let $Λ$ be an artin algebra and let $\mathcal{P}^{<\infty}_Λ$ the category of finitely generated right $Λ$-modules of finite projective dimension. We show that $\mathcal{P}^{<\infty}_Λ$ is contravariantly finite in $\rm mod\,Λ$ if and only if the direct sum $E$ of the indecomposable Ext-injective modules in $\mathcal{P}^{<\infty}_Λ$ form a tilting module in $\rm mod\,Λ$. Moreover, we show that in this case $E$ coincides with the direct sum of the minimal right $\mathcal{P}^{<\infty}_Λ$-approximations of the indecomposable $Λ$-injective modules and that the projective dimension of $E$ equal to the finitistic dimension of $Λ$.

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↗