arXiv subjects

Raziyeh Diyanatnezhad

Publications and source records attributed to Raziyeh Diyanatnezhad.

3 recordsLinked to original sources

arXiv2025

Coxeter matrices and homological quadratic forms of $\boldsymbol{n}$-hereditary algebras

We study the Coxeter matrices and the homological quadratic forms of $n$-hereditary algebras within the framework of higher dimensional Auslander--Reiten theory. Let $Λ$ be a finite dimensional $n$-hereditary algebra with the Coxeter matrix $Φ$ and the homological quadratic form $χ$. We prove that if $Λ$ is $n$-representation finite, then there exists a positive integer $d$ such that $Φ^d=1$. In the case $n$ is an odd number, we show that if there exists a positive integer $d$ such that $Φ^d=1$, then $Λ$ is $n$-representation finite. Let $\mathcal{C}^0$ be the subcategory of $\moddΛ$ which is a higher analogue of the module category in the context of higher dimensional Auslander--Reiten theory. We introduce a Grothendieck group $\mathrm{K}_0(\mathcal{C}^0)$ associated with $\mathcal{C}^0$ and show that it is isomorphic to the Grothendieck group of $Λ$. We further prove that if the restriction of $χ$ to $\mathrm{K}_0(\mcc^0)$ is positive definite, then $Λ$ is $n$-representation finite for odd $n$. To prove these results, we first show that indecomposable $n$-preprojective and $n$-preinjective modules are uniquely determined up to isomorphism by their dimension vectors for odd $n$. We also provide examples of $n$-representation finite algebras that the restriction of $χ$ to $\mathrm{K}_0(\mcc^0)$ is not positive definite.

math.RT↗

arXiv2023

The radical of functorially finite subcategories

Let $Λ$ be an artin algebra and $\mathcal{C}$ be a functorially finite subcategory of mod$Λ$ which contains $Λ$ or $DΛ$. We use the concept of the infinite radical of $\mathcal{C}$ and show that $\mathcal{C}$ has an additive generator if and only if rad$^\infty_{\mathcal{C}}$ vanishes. In this case we describe the morphisms in powers of the radical of $\mathcal{C}$ in terms of its irreducible morphisms. Moreover, under a mild assumption, we prove that $\mathcal{C}$ is of finite representation type if and only if any family of monomorphisms (epimorphisms) between indecomposable objects in $\mathcal{C}$ is noetherian (conoetherian). Also, by using injective envelopes, projective covers, left $\mathcal{C}$-approximations and right $\mathcal{C}$-approximations of simple $Λ$-modules, we give other criteria to describe whether $\mathcal{C}$ is of finite representation type. In addition, we give a nilpotency index of the radical of $\mathcal{C}$ which is independent from the maximal length of indecomposable $Λ$-modules in $\mathcal{C}$.

math.RT↗

arXiv2021

Relations for Grothendieck groups of $n$-cluster tilting subcategories

Let $Λ$ be an artin algebra and $\mathcal{M}$ be an n-cluster tilting subcategory of mod$Λ$. We show that $\mathcal{M}$ has an additive generator if and only if the n-almost split sequences form a basis for the relations for the Grothendieck group of $\mathcal{M}$ if and only if every effaceable functor $\mathcal{M}\rightarrow Ab$ has finite length. As a consequence we show that if mod$Λ$ has n-cluster tilting subcategory of finite type then the n-almost split sequences form a basis for the relations for the Grothendieck group of $Λ$.

math.RT↗