arXiv subjects

Alireza Nasr-Isfahani

Publications and source records attributed to Alireza Nasr-Isfahani.

At least 19 recordsLinked to original sources

arXiv2026

$r$ICE-closed subcategories induced by the morphism category of projective modules

Let $Λ$ be an Artin $R$-algebra, and ${\rm proj}\mbox{-}Λ$ denotes the category of all finitely generated projective $Λ$-modules. Define $\CP(Λ) := {\rm Mor}({\rm proj}\mbox{-}Λ)$. Due to the favorable homological properties of $\CP(Λ)$, we initially examine several noteworthy objects and subcategories of $\CP(Λ)$, subsequently relating these findings to $\mmod Λ$. Following our examination of Image-Cokernel-Extension closed (hereafter referred to as ICE-closed) subcategories of $\CP(Λ)$, among other bijections, we demonstrate a bijection between rigid objects in $\CP(Λ)$ and ICE-closed subcategories of $\CP(Λ)$ with enough Ext-projectives. In order to translate the concept of ICE-closed subcategory from $\CP(Λ)$ to $\mmod Λ$, it is necessary to introduce the framework of rICE-closed subcategories of $\mmod Λ$. We then establish a bijection between $τ$-rigid modules in $\mmod Λ$ and rICE-closed subcategories of $\mmod Λ$ that possess an rExt-progenerator. This is a generalization of a bijection given by Enomoto for hereditary algebras. Our morphism approach improves a bijection given by Buan and Zhou by introducing r-cotorsion-torsion triples. We conclude our paper with further applications for $τ$-tilting theory.

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arXiv2026

Semibricks and Brick-finite algebras

Let $Λ$ be a finite-dimensional algebra. Using the kappa order on the lattice of torsion classes with canonical join representations, we obtain an equivalent condition for $Λ$ to be brick-finite. We show that $Λ$ is brick-finite if and only if every widely generated torsion class in $\modd Λ$ has finitely many covers with respect to the kappa order. Furthermore, we prove that every semibrick in $\modd Λ$ is a finite set if and only if every chain of wide subcategories of $\modd Λ$ is eventually constant. We also show that $Λ$ is brick-finite if and only if every chain of widely generated torsion classes of $\modd Λ$ is eventually constant. Finally, we show that $Λ$ is brick-finite if and only if every cofinally closed monobrick is a cofinal closure of some semibrick.

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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.

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arXiv2024

The rigidity of filtered colimits of n-cluster tilting subcategories

Let $Λ$ be an artin algebra and $\mathcal{M}$ be an n-cluster tilting subcategory of $Λ$-mod with $n\ge 2$. From the viewpoint of higher homological algebra, a question that naturally arose in [17] is when $\mathcal{M}$ induces an n-cluster tilting subcategory of $Λ$-Mod. In this paper, we answer this question and explore its connection to Iyama's question on the finiteness of n-cluster tilting subcategories of $Λ$-mod. In fact, our theorem reformulates Iyama's question in terms of the vanishing of Ext; and highlights its relation with the rigidity of filtered colimits of $\mathcal{M}$. Also, we show that Add$(\mathcal{M})$ is an n-cluster tilting subcategory of $Λ$-Mod if and only if Add$(\mathcal{M})$ is a maximal n-rigid subcategory of $Λ$-Mod if and only if $\lbrace X\in Λ$-Mod$~|~ {\rm Ext}^i_Λ(\mathcal{M},X)=0 ~~~ {\rm for ~all}~ 0<i<n \rbrace \subseteq {\rm Add}(\mathcal{M})$ if and only if $\mathcal{M}$ is of finite type if and only if ${\rm Ext}_Λ^1({\underrightarrow{\lim}}\mathcal{M}, {\underrightarrow{\lim}}\mathcal{M})=0$. Moreover, we present several equivalent conditions for Iyama's question which shows the relation of Iyama's question with different subjects in representation theory such as purity and covering theory.

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arXiv2024

Semibrick-cosilting correspondence

Let $Λ$ be a finite dimensional algebra. In this paper we show that there is a natural bijection between cosilting modules in Mod$Λ$ and semibricks in Mod$Λ$ satisfying some condition. Also this bijection restricts to a bijection between all semibricks in mod$Λ$ and a certain subclass of cosilting modules. These bijections are generalizations of Asai's correspondence [7] between support $τ^-$-tilting modules and right finite semibricks.

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arXiv2024

Locally finitely presented Grothendieck categories and the pure semisimplicity conjecture

In this paper, we investigate locally finitely presented pure semisimple (hereditary) Grothendieck categories. We show that every locally finitely presented pure semisimple (resp., hereditary) Grothendieck category A is equivalent to the category of left modules over a left pure semisimple (resp., left hereditary) ring when Mod(fp(A)) is a QF-3 category and every representable functor in Mod(fp(A)) has finitely generated essential socle. In fact, we show that there exists a bijection between Morita equivalence classes of left pure semisimple (resp., left hereditary) rings $Λ$ and equivalence classes of locally finitely presented pure semisimple (resp., hereditary) Grothendieck categories A that Mod(fp(A)) is a QF-3 category and every representable functor in Mod(fp(A)) has finitely generated essential socle. To prove this result, we study left pure semisimple rings by using Auslander's ideas. We show that there exists, up to equivalence, a bijection between the class of left pure semisimple rings and the class of rings with nice homological properties. These results extend the Auslander and Ringel-Tachikawa correspondence to the class of left pure semisimple rings. As a consequence, we give several equivalent statements to the pure semisimplicity conjecture.

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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 $\mathcal{C}$.

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arXiv2023

The completion of $d$-abelian categories

Let $A$ be a finite-dimensional algebra, and $\mathfrak{M}$ be a $d$-cluster tilting subcategory of mod$A$. From the viewpoint of higher homological algebra, a natural question to ask is when $\mathfrak{M}$ induces a $d$-cluster tilting subcategory in Mod$A$. In this paper, we investigate this question in a more general form. Let $\mathcal{M}$ be a small $d$-abelian category of an abelian category $\mathcal{A}$. The completion of $\mathcal{M}$, denoted by Ind$(\mathcal{M})$, is defined as the universal completion of $\mathcal{M}$ with respect to filtered colimits. We explore Ind$(\mathcal{M})$ and demonstrate its equivalence to the full subcategory $\mathcal{L}_d(\mathcal{M})$ of Mod$\mathcal{M}$, comprising left $d$-exact functors. Notably, while Ind$(\mathcal{M})$ as a subcategory of $\frac{Mod\mathcal{M}}{Eff(\mathcal{M})}$, satisfies all properties of a $d$-cluster tilting subcategory except $d$-rigidity, it falls short of being a $d$-cluster tilting category. For a $d$-cluster tilting subcategory $\mathfrak{M}$ of mod$A$, $\overrightarrow{\mathfrak{M}}$, consists of all filtered colimits of objects from $\mathfrak{M}$, is a generating-cogenerating, functorially finite subcategory of Mod$A$. The question of whether $\mathfrak{M}$ is a $d$-rigid subcategory remains unanswered. However, if it is indeed $d$-rigid, it qualifies as a $d$-cluster tilting subcategory. In the case $d=2$, employing cotorsion theory, we establish that $\overrightarrow{\mathfrak{M}}$ is a $2$-cluster tilting subcategory if and only if $\mathfrak{M}$ is of finite type. Thus, the question regarding whether $\overrightarrow{\mathfrak{M}}$ is a $d$-cluster tilting subcategory of Mod$ A$ appears to be equivalent to the Iyama's qestion about the finiteness of $\mathfrak{M}$.

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arXiv2023

Higher Auslander's formula

Let $\mathcal{M}$ be a small $n$-abelian category. We show that the category of finitely presented functors $mod$-$\mathcal{M}$ modulo the subcategory of effaceable functors $mod_0$-$\mathcal{M}$ has an $n$-cluster tilting subcategory which is equivalent to $\mathcal{M}$. This gives a higher-dimensional version of Auslander's formula.

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arXiv2023

Morita equivalence and Morita duality for rings with local units and subcategory of projective unitary modules

We study Morita equivalence and Morita duality for rings with local units. We extend the Auslander's results on the theory of Morita equivalence and the Azumaya-Morita duality theorem to rings with local units. As a consequence, we give a version of Morita theorem and Azumaya-Morita duality theorem over rings with local units in terms of their full subcategory of finitely generated projective unitary modules and full subcategory of finitely generated injective unitary modules.

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arXiv2022

Finiteness and purity of subcategories of the module categories

In this paper, by using functor rings and functor categories, we study finiteness and purity of subcategories of the module categories. We give a characterisation of contravariantly finite resolving subcategories of the module category of finite representation type in terms of their functor rings. We also characterize contravariantly finite resolving subcategories of the module category $Λ${\rm -mod} that contain the Jacobson radical of $Λ$ of finite type, by their functor categories. We study the pure semisimplicity conjecture for a locally finitely presented category ${\underrightarrow{\lim}}\mathscr{X}$ when $\mathscr{X}$ is a covariantly finite subcategory of $Λ$-mod and every simple object in Mod$(\mathscr{X}^{\rm op})$ is finitely presented and give a characterization of covariantly finite subcategories of finite representation type in terms of decomposition properties of their closure under filtered colimits. As a consequence we study finiteness and purity of $n$-cluster tilting subcategories and the subcategory of the Gorenstein projective modules of the module categories. These results extend and unify some known results.

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arXiv2022

$n\mathbb{Z}$-abelian and $n\mathbb{Z}$-exact categories

In this paper we introduce $n\mathbb{Z}$-abelian and $n\mathbb{Z}$-exact categories by axiomatising properties of $n\mathbb{Z}$-cluster tilting subcategories. We study this categories and show that every $n\mathbb{Z}$-cluster tilting subcategory of an abelian (resp., exact) category has a natural structure of an $n\mathbb{Z}$-abelian (resp., $n\mathbb{Z}$-exact) category. Also we show that every small $n\mathbb{Z}$-abelian category arise in this way, and discuss the problem for $n\mathbb{Z}$-exact categories.

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arXiv2021

Higher Auslander correspondence for exact categories

Inspired by the recent work of Henrard, Kvamme and van Roosmalen [17], we prove a categorified version of higher Auslander correspondence in the context of exact categories. We define n-Auslander exact categories and show that there is a bijection between the equivalence classes of n-cluster tilting subcategories of exact categories and the equivalence classes of n-Auslander exact categories.

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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 $Λ$.

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arXiv2021

Auslander correspondence for kawada rings

We study the Auslander ring of a basic left Köthe ring $Λ$ and give a characterization of basic left Köthe rings in terms of their Auslander rings. We also study the functor category Mod$((Λ$-mod$)^{\rm op})$ and characterize basic left Köthe rings $Λ$ by using functor categories Mod$((Λ$-mod$)^{\rm op})$. As a consequence we show that there exists a bijection between the Morita equivalence classes of left Kawada rings and the Morita equivalence classes of Auslander generalized right QF-2 rings.

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arXiv2020

Cluster algebras generated by projective cluster variables

We introduce the notion of a lower bound cluster algebra generated by projective cluster variables as a polynomial ring over the initial cluster variables and the so-called projective cluster variables. We show that under an acyclicity assumption, the cluster algebra and the lower bound cluster algebra generated by projective cluster variables coincide. In this case we use our results to construct a basis for the cluster algebra. We also show that any coefficient-free cluster algebra of types $A_n$ or $\widetilde{A}_n$ is equal to the corresponding lower bound cluster algebra generated by projective cluster variables.

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arXiv2020

Representation of n-abelian categories in abelian categories

Let $\mathcal{M}$ be a small $n$-abelian category. We show that the category of absolutely pure group valued functors over $\mathcal{M}$, denote by $\mathcal{L}_2(\mathcal{M},\mathcal{G})$, is an abelian category and $\mathcal{M}$ is equivalent to a full subcategory of $\mathcal{L}_2(\mathcal{M},\mathcal{G})$ in such a way that $n$-kernels and $n$-cokernels are precisely exact sequences of $\mathcal{L}_2(\mathcal{M},\mathcal{G})$ with terms in $\mathcal{M}$. This gives a higher-dimensional version of the Freyd-Mitchell embedding theorem for $n$-abelian categories.

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arXiv2020

Coxeter diagrams and the Köthe's problem

A ring $Λ$ is called right Köthe if every right $Λ$-module is a direct sum of cyclic modules. In this paper, we give a characterization of basic hereditary right Köthe rings in terms of their Coxeter valued quivers. Also we give a characterization of basic right Köthe rings with radical square zero. Therefore we give a solution of the Köthe's problem in this cases.

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