arXiv subjects

Rasool Hafezi

Publications and source records attributed to Rasool Hafezi.

At least 19 recordsLinked to original sources

arXiv2026

On simples in a cosilting heart and mutation of cosilting pairs

Let $A$ be a finite dimensional algebra. The lattice of torsion pairs in $\rm mod (A)$ is controlled by cosilting pairs, infinitely generated analogues of support $\tau^-$-tilting pairs. Then, edges in the Hasse quiver (i.e. minimal inclusions of torsion-free classes) correspond to irreducible mutations of cosilting pairs. An important difference with classical $\tau$-tilting theory is that not all indecomposable summands of a cosilting pair are mutable. So, it is very important to identify mutable indecomposable summands in a given cosilting pair. It is well-known that mutable summands correspond to injective envelopes of finitely presented simples in the HRS-tilted heart. Based on this correspondence, we first present a method for obtaining all simples in the HRS-tilted heart, and then give some necessary and sufficient conditions for an indecomposable summand of a given cosilting pair to be left mutable or right mutable.

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arXiv2026

Model structures arising from weak cotorsion pairs

Let $\mathcal{A}$ be an abelian category. Beligiannis and Reiten proved that there is a bijective correspondence between so-called projective model structures on $\mathcal{A}$ and hereditary cotorsion pairs in $\mathcal{A}$ with a contravariantly finite core. It is well-known that, tilting modules induce cotorsion pairs, so we may have a homotopicl interpretation of tilting modules. But a recent generalization of tilting modules, support $\tau$-tilting modules, induce weak cotorsion pairs. In this paper, we define weak projective model structures and prove that there is a bijective correspondence between weak projective model structures and left weak cotorsion pairs satisfying some mild conditions. This is a generalization of Beligiannis-Reiten correspondence from the perspective and philosophy of $\tau$-tilting theory. In particular, we prove that any support $\tau$-tilting module induce a model structure, and there is bijective correspondence between support $\tau$-tilting modules and a certain class of model structures.

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arXiv2026

Revising Auslander-Gruson-Jensen duality

For a ring $A$ there is a well-known duality between definable subcategories of right $A$-modules and definable subcategories of left $A$ modules. This is a consequence of Auslander-Gruson-Jensen duality $\rm mod\text{-}(mod\text{-}A)\rightarrow mod\text{-}(mod\text{-}A^{op})$. The existence of this duality arises from the fact that $\rm mod\text{-}(mod\text{-}A)$ is the free abelian category over the pre-additive category $A$ with a single object. In this note, first, we give a simple description of the free abelian category. This description clarifies Auslender-Gruson-Jensen duality and also the duality between definable subcategories of right $A$-modules and those of left $A$-modules.

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arXiv2025

Strongly flat modules via universal localization

In this paper, we investigate a non-commutative version of strongly flat modules, which is based on the concept of universal localization introduced by Cohn. We consider a set $σ$ consisting of maps of finitely generated projective $R$-modules, where $R$ is not necessarily a commutative ring. Let $R_σ$ denote the universal localization of $R$ with respect to $σ$. The class of $σ$-strongly flat modules is defined as the left class in the cotorsion pair generated by $R_σ$. We examine the homotopy category of $σ$-strongly flat modules and demonstrate that the thick subcategory $\mathscr{S}_σ$, consisting of acyclic complexes, wherein all syzygies are $σ$-strongly flat, forms a precovering class within this homotopy category. This implies that the quotient map from $\mathbb{K}({σ\mbox{-}\mathcal{SF}})$ to $\mathbb{K}({σ\mbox{-}\mathcal{SF}})/\mathscr{S}_σ$ always has a fully faithful right adjoint.

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arXiv2025

Covering techniques in higher Auslander-Reiten theory

This paper investigates the behavior of $n$-precluster tilting subcategories under the push-down functor in the context of Galois coverings of locally bounded categories. Building on higher Auslander-Reiten theory and covering techniques, we establish that for a locally support-finite category $\mathcal{C}$ with a free group action $G$ on its indecomposables, the push-down functor maps $G$-equivariant $n$-precluster tilting subcategories of ${\rm mod}\mbox{-}\mathcal{C}$ to $n$-precluster tilting subcategories of ${\rm mod}\mbox{-}(\mathcal{C}/G)$, and vice versa. These results provide a framework for studying $τ_n$-selfinjective algebras. We further prove that ${\rm mod}\mbox{-}\mathcal{C}$ is $n$-minimal Auslander-Gorenstein if and only if ${\rm mod}\mbox{-}(\mathcal{C}/G)$ is so, under square-free conditions on $\mathcal{C}/G$. Additionally, we analyze support $τ_n$-tilting pairs via the push-down functor, showing that locally $τ_n$-tilting finiteness is preserved under Galois coverings. Our work offers new insights into the interplay between higher homological algebra and covering theory in representation-finite contexts.

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arXiv2025

On the monomorphism category of large modules

Let $R$ be an associative ring with identity. This paper investigates the structure of the monomorphism category of large $R$-modules and establishes connections with the category of contravariant functors defined on finitely presented $R$-modules. Several equivalences and dualities will be presented. Our results highlight the role of pure-injective modules in studying the homological properties of functor categories.

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arXiv2024

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

Let $\Lambda$ be an Artin $R$-algebra, and ${\rm proj}\mbox{-}\Lambda$ denotes the category of all finitely generated projective $\Lambda$-modules. Define $\CP(\Lambda) := {\rm Mor}({\rm proj}\mbox{-}\Lambda)$. Due to the favorable homological properties of $\CP(\Lambda)$, we initially examine several noteworthy objects and subcategories of $\CP(\Lambda)$, subsequently relating these findings to $\mmod \Lambda$. Following our examination of Image-Cokernel-Extension closed (hereafter referred to as ICE-closed) subcategories of $\CP(\Lambda)$, among other bijections, we demonstrate a bijection between rigid objects in $\CP(\Lambda)$ and ICE-closed subcategories of $\CP(\Lambda)$ with enough Ext-projectives. In order to translate the concept of ICE-closed subcategory from $\CP(\Lambda)$ to $\mmod \Lambda$, it is necessary to introduce the framework of rICE-closed subcategories of $\mmod \Lambda$. We then establish a bijection between $\tau$-rigid modules in $\mmod \Lambda$ and rICE-closed subcategories of $\mmod \Lambda$ 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 $\tau$-tilting theory.

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arXiv2024

G-semisimple algebras

Let $Λ$ be an Artin algebra and ${\mathsf{mod}}\mbox{-} ({\underline{\mathsf{Gprj}}}\mbox{-}Λ)$ the category of finitely presented functors over the stable category ${\underline{\mathsf{Gprj}}}\mbox{-}Λ$ of finitely generated Gorenstein projective $Λ$-modules. This paper deals with those algebras $Λ$ in which ${\mathsf{mod}}\mbox{-} ({\underline{\mathsf{Gprj}}}\mbox{-}Λ)$ is a semisimple abelian category, and we call G-semisimple algebras. We study some basic properties of such algebras. In particular, it will be observed that the class of G-semisimple algebras contains important classes of algebras, including gentle algebras and more generally quadratic monomial algebras. Next, we construct an epivalence from the stable category of Gorenstein projective representations $\underline{\mathsf{Gprj}}(\mathcal{Q}, Λ)$ of a finite acyclic quiver $\mathcal{Q}$ to the category of representations ${\rm rep}(\mathcal{Q}, \underline{\mathsf{Gprj}}\mbox{-} Λ)$ over $\underline{\mathsf{Gprj}}\mbox{-} Λ)$, provided $Λ$ is a G-semisimple algebra over an algebraic closed field. Using this, we will show that the path algebra $Λ\mathcal{Q}$ of the G-semisimple algebra $Λ$ is Cohen-Macaulay finite if and only if $\mathcal{Q}$ is Dynkin. In the last part, we provide a complete classification of indecomposable Gorenstein projective representations within ${\mathsf{Gprj}}(A_n, Λ)$ of the linear quiver $A_n$ over a G-semisimple algebra $Λ$. We also determine almost split sequences in ${\mathsf{Gprj}}(A_n, Λ)$ with certain ending terms. We apply these results to obtain insights into the cardinality of the components of the stable Auslander-Reiten quiver ${\mathsf{Gprj}}(A_n, Λ)$.

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arXiv2024

2-categorical approach to unifying constructions of precoverings and its applications

Throughout this paper $G$ is a fixed group, and $k$ is a fixed field. All categories are assumed to be $k$-linear. First we give a systematic way to induce $G$-precoverings by adjoint functors using a 2-categorical machinery, which unifies many similar constructions of $G$-precoverings. Now let $\mathcal{C}$ be a skeletally small category with a $G$-action, $\mathcal{C}/G$ the orbit category of $\mathcal{C}$, $(P, ϕ) : \mathcal{C} \rightarrow \mathcal{C}/G$ the canonical $G$-covering, and $\mathrm{mod}\mbox{-} \mathcal{C}$, $\mathrm{mod}\mbox{-} (\mathcal{C}/G)$ the categories of finitely generated modules over $\mathcal{C}, \mathcal{C}/G$, respectively. Then it is well known that there exists a canonical G-precovering $(P., ϕ.) : \mathrm{mod}\mbox{-} \mathcal{C} \rightarrow \mathrm{mod}\mbox{-} (\mathcal{C}/G)$. By applying the machinery above to this $(P., ϕ.)$, new $G$-precoverings $(\mathrm{mod}\mbox{-} \mathcal{C}) / S \rightarrow (\mathrm{mod}\mbox{-} \mathcal{C}/G)/S'$ are induced between the factor categories or localizations of $\mathrm{mod}\mbox{-} \mathcal{C}$ and $\mathrm{mod}\mbox{-} \mathcal{C}/G$, respectively. This is further applied to the morphism category $\mathrm{H}(\mathrm{mod}\mbox{-} \mathcal{C})$ of $\mathrm{mod}\mbox{-} \mathcal{C}$ to have a $G$-precovering $\mathrm{fp}(\mathcal{K}) \rightarrow \mathrm{fp}(\mathcal{K}')$ between the categories of finitely presented modules over suitable subcategories $\mathcal{K}$ and $\mathcal{K}'$ of $\mathrm{mod}\mbox{-}\mathcal{C}$ and $ \mathrm{mod}\mbox{-} \mathcal{C}/G$, respectively.

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arXiv2023

Relative Higher Homology and Representation Theory

Higher homological algebra, basically done in the framework of an $n$-cluster tilting subcategory $\mathcal{M}$ of an abelian category $\mathcal{A}$, has been the topic of several recent researches. In this paper, we study a relative version, in the sense of Auslander-Solberg, of the higher homological algebra. To this end, we consider an additive sub-bifunctor $F$ of $\mathrm{Ext}^n_{\mathcal{M}}( -,-)$ as the basis of our relative theory. This, in turn, specifies a collection of $n$-exact sequences in $\mathcal{M}$, which allows us to delve into the relative higher homological algebra. Our results include a proof of the relative $n$-Auslander-Reiten duality formula, as well as an exploration of relative Grothendieck groups, among other results. As an application, we provide necessary and sufficient conditions for $\mathcal{M}$ to be of finite type.

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arXiv2023

From Morphism Categories to Functor Categories

For a nice-enough category $\mathcal{C}$, we construct both the morphism category ${\rm H}(\mathcal{C})$ of $\mathcal{C}$ and the category ${\rm mod}\mbox{-}\mathcal{C}$ of all finitely presented contravariant additive functors over $\mathcal{C}$ with values in Abelian groups. The main theme of this paper, is to translate some representation-theoretic attributes back and forth from one category to the other. This process is done by using an appropriate functor between these two categories, an approach which seems quite promising in particular when we show that many of almost split sequences are preserved by this functor. We apply our results to the case of wide subcategories of module categories to obtain certain auto-equivalences over them. Another part of the paper deals with Auslander algebras arising from algebras of finite representation type. In fact, we apply our results to study the Auslander-Reiten translates of simple modules over such algebras. In the last parts, we try to recognize particular components in the stable Auslander-Reiten quiver of Auslander algebras arising from self-injective algebras of finite representation type.

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arXiv2023

The Auslander-Reiten theory of the morphism category of projective modules

We investigate the structure of certain almost split sequences in $\mathcal{P}(Λ)$, i.e., the category of morphisms between projective modules over an Artin algebra $Λ$. The category $\mathcal{P}(Λ)$ has very nice properties and is closely related to $τ$-tilting theory, $g$-vectors, and Auslander-Reiten theory. We provide explicit constructions of certain almost split sequences ending at or starting from certain objects. Applications, such as to $g$-vectors, are given. As a byproduct, we also show that there exists an injection from Morita equivalence classes of Artin algebras to equivalence classes of 0-Auslander exact categories.

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arXiv2023

On the stable Auslander-Reiten components of certain monomorphism categories

Let $Λ$ be an Artin algebra and let $\rm{Gprj}\mbox{-}Λ$ denote the class of all finitely generated Gorenstein projective $Λ$-modules. In this paper, we study the components of the stable Auslander-Reiten quiver of a certain subcategory of the monomorphism category $\mathcal{S}({\rm Gprj}\mbox{-}Λ)$ containing boundary vertices. We describe the shape of such components. It is shown that certain components are linked to the orbits of an auto-equivalence on the stable category $\underline{\rm{Gprj}}\mbox{-}Λ$. In particular, for the finite components, we show that under certain mild conditions their cardinalities are divisible by $3$. We see that this three-periodicity phenomenon reoccurs several times in the paper.

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arXiv2022

Determination of some almost split sequences in morphism categories

Almost split sequences lie in the heart of Auslander-Reiten theory. This paper deals with the structure of almost split sequences with certain ending terms in the morphism category of an Artin algebra $Λ$. Firstly we try to interpret the Auslander-Reiten translates of particular objects in the morphism category in terms of the Auslander-Reiten translations within the category of $Λ$-modules, and then use them to calculate almost split sequences. In classical representation theory of algebras, it is quite important to recognize the midterms of almost split sequences. As such, another part of the paper is devoted to discuss the midterm of certain almost split sequences in the morphism category of $Λ$. As an application, we restrict in the last part of the paper to self-injective algebras and present a structural theorem that illuminates a link between representation-finite morphism categories and Dynkin diagrams.

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arXiv2022

On Cohen-Macaulay Auslander algebras

Cohen-Macaulay Auslander algebras are the endomorphism algebras of representation generators of the subcategory of Gorenstein projective modules over $\rm{CM}$-finite algebras. In this paper, we study Cohen-Macaulay Auslander algebras over $1$-Gorenstein algebras and $Ω_{\mathcal{G}}$-algebras. $1$-Gorenstein algebras are those of algebras with global Gorenstein projective dimension at most one and $Ω_{\mathcal{G}}$-algebras are a class of algebras introduced in this paper, including some important class of algebras for example Gentle algebras and more generally quadratic monomial algebras. It will be shown how the results for Gorenstein projective representations of a quiver over an Artin algebra, including the submodule category introduced in [RS], or more generally, the (separated) monomorphism category defined in [LZh2] and [XZZ], can be applied to study the Cohen-Macaulay Auslander algebras.

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arXiv2021

When stable Cohen-Macaulay Auslander algebra is semisimple

Let $\text{Gprj}\mbox{-}Λ$ denote the category of Gorenstein projective modules over an Artin algebra $Λ$ and the category $\text{mod}\mbox{-} (\underline{\text{Gprj}}\mbox{-}Λ)$ of finitely presented functors over the stable category $\underline{\text{Gprj}}\mbox{-}Λ$. In this paper, we study those algebras $Λ$ with $\text{mod}\mbox{-} (\underline{\text{Gprj}}\mbox{-}Λ)$ to be a semisimple abelian category, and called $Ω_{\mathcal{G}}$-algebras. The class of $Ω_{\mathcal{G}}$-algebras contains important classes of algebras, including gentle algebras. Over an $Ω_{\mathcal{G}}$-algebra $Λ$, the structure of the almost split sequences in the morphism categories $\text{H}(\text{Gprj}\mbox{-}Λ)$ and the monomorphism categories $\mathcal{S}(\text{Gprj}\mbox{-}Λ)$ of $\text{Gprj}\mbox{-}Λ$ is investigated. Among other applications, we provide some results for the Cohen-Macaulay Auslander algebras of $Ω_{\mathcal{G}}$-algebras.

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arXiv2021

Covering theory, (mono)morphism categories and stable Auslander algebras

Let $\mathcal{A}$ be a locally bounded $k$-category and $G$ a torsion-free group of $k$-linear automorphisms of $\mathcal{A}$ acting freely on the objects of $\mathcal{A},$ and $F:\mathcal{A}\rightarrow \mathcal{B}$ is a Galois functor. We extend naturally the push-down functor $F_λ$ to the functor $\rm{H}\rm{F}_λ:\rm{H}(\rm{mod}\mbox{-} \mathcal{A})\rightarrow \rm{H}(\rm{mod}\mbox{-} \mathcal{B})$, resp. $\mathcal{S} \rm{F}_λ:\mathcal{S}(\rm{mod}\mbox{-} \mathcal{A})\rightarrow \mathcal{S}(\rm{mod}\mbox{-} \mathcal{B})$, between the corresponding morphism categories, resp. monomorphism categories, of $\rm{mod}\mbox{-} \mathcal{A}$ and $\rm{mod}\mbox{-} \mathcal{B}$. Under some additional conditions, we show that $\rm{H}(\rm{mod}\mbox{-}\mathcal{A})$, resp. $\mathcal{S}( \rm{mod}\mbox{-}\mathcal{A})$, is locally bounded if and only if $\rm{H}(\rm{mod}\mbox{-} \mathcal{B})$, resp. $\mathcal{S}(\rm{mod}\mbox{-}\mathcal{B})$, is of finite representation type. As an application, we show that the stable Auslander algebra of a representation-finite selfinjective algebra $Λ$ is again representation-finite if and only if $Λ$ is of Dynkin type $\mathbb{A}_{n}$ with $n\leqslant 4$.

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arXiv2020

From subcategories to the entire module categories

In this paper we show that how the representation theory of subcategories (of the category of modules over an Artin algebra) can be connected to the representation theory of all modules over some algebra. The subcategories dealing with are some certain subcategories of the morphism categories (including submodule categories studied recently by Ringel and Schmidmeier) and of the Gorenstein projective modules over (relative) stable Auslander algebras. These two kinds of subcategories, as will be seen, are closely related to each other. To make such a connection, we will define a functor from each type of the subcategories to the category of modules over some Artin algebra. It is shown that to compute the almost split sequences in the subcategories it is enough to do the computation with help of the corresponding functors in the category of modules over some Artin algebra which is known and easier to work. Then as an application the most part of Auslander-Reiten quiver of the subcategories is obtained only by the Ausalander-Reiten quiver of an appropriate algebra and next adding the remaining vertices and arrows in an obvious way. As a special case, whenever $Λ$ is a Gorenstein Artin algebra of finite representation type, then the subcategories of Gorenstein projective modules over the $2 \times 2$ upper triangular matrix algebra over $Λ$ and the stable Auslander algebra of $Λ$ can be estimated by the category of modules over the stable Cohen-Macaulay Auslander algebra of $Λ$.

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