arXiv subjects

Javad Asadollahi

Publications and source records attributed to Javad Asadollahi.

At least 19 recordsLinked to original sources

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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arXiv2025

The Homotopy Category of Strongly flat modules

In this paper, we plan to build upon significant results by Amnon Neeman regarding the homotopy category of flat modules to study ${\mathbb{K}}({S\rm{SF}}\mbox{-}R)$, the homotopy category of $S$-strongly flat modules, where $S$ is a multiplicatively closed subset of a commutative ring $R$. The category ${\mathbb{K}}({S\rm{SF}}\mbox{-}R)$ is an intermediate triangulated category that includes ${\mathbb{K}}({\rm{Prj}\mbox{-}} R)$, the homotopy category of projective $R$-modules, which is always well generated by a result of Neeman, and is included in ${\mathbb{K}}({\rm{Flat}}\mbox{-} R)$, the homotopy category of flat $R$-modules, which is well generated if and only if $R$ is perfect, by a result of Štovíček. We analyze corresponding inclusion functors and the existence of their adjoints. In this way, we provide a new, fully faithful embedding of the homotopy category of projectives to the homotopy category of $S$-strongly flat modules. We introduce the notion of $S$-almost well generated triangulated categories. If $R$ is an $S$-almost perfect ring, ${\mathbb{K}}({\rm{Flat}}\mbox{-} R)$ is $S$-almost well generated. We show that the converse is true under certain conditions on the ring $R$. We hope that this approach provides insights into the largely mysterious class of $S$-strongly flat modules.

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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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arXiv2022

On $τ$-tilting subcategories

The main theme of this paper is to study $τ$-tilting subcategories in an abelian category $\mathscr{A}$ with enough projective objects. We introduce the notion of $τ$-cotorsion torsion triples and show a bijection between the collection of $τ$-cotorsion torsion triples in $\mathscr{A}$ and the collection of $τ$-tilting subcategories of $\mathscr{A}$, generalizing the bijection by Bauer, Botnan, Oppermann and Steen between the collection of cotorsion torsion triples and the collection of tilting subcategories of $\mathscr{A}$. General definitions and results are exemplified using persistent modules. If $\mathscr{A}={\rm{Mod\mbox{}}R}$, where $R$ is an unitary associative ring, we characterize all support $τ$-tilting, resp. all support $τ^-$-tilting, subcategories of ${\rm{Mod\mbox{}}R}$ in term of finendo quasitilting, resp. quasicotilting, modules. As a result, it will be shown that every silting module, respectively every cosilting module, induces a support $τ$-tilting, respectively support $τ^{-}$-tilting, subcategory of ${\rm{Mod\mbox{}}R}$. We also study the theory in ${\rm Rep}(Q, \mathscr{A})$, where $Q$ is a finite and acyclic quiver. In particular, we give an algorithm to construct support $τ$-tilting subcategories in ${\rm Rep}(Q, \mathscr{A})$ from certain support $τ$-tilting subcategories of $\mathscr{A}$ and present a systematic way to construct $(n+1)$-tilting subcategories in ${\rm Rep}(Q, \mathscr{A})$ from $n$-tilting subcategories in $\mathscr{A}$.

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arXiv2022

On higher torsion classes

Building on the embedding of an $n$-abelian category $\mathscr{M}$ into an abelian category $\mathcal{A}$ as an $n$-cluster-tilting subcategory of $\mathcal{A}$, in this paper we relate the $n$-torsion classes of $\mathscr{M}$ with the torsion classes of $\mathcal{A}$. Indeed, we show that every $n$-torsion class in $\mathscr{M}$ is given by the intersection of a torsion class in $\mathcal{A}$ with $\mathscr{M}$. Moreover, we show that every chain of $n$-torsion classes in the $n$-abelian category $\mathscr{M}$ induces a Harder-Narasimhan filtration for every object of $\mathscr{M}$. We use the relation between $\mathscr{M}$ and $\mathcal{A}$ to show that every Harder-Narasimhan filtration induced by a chain of $n$-torsion classes in $\mathscr{M}$ can be induced by a chain of torsion classes in $\mathcal{A}$. Furthermore, we show that $n$-torsion classes are preserved by Galois covering functors, thus we provide a way to systematically construct new (chains of) $n$-torsion classes.

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arXiv2020

Higher Ideal Approximation Theory

Let ${\mathscr{C}}$ be an $n$-cluster tilting subcategory of an exact category $({\mathscr{A}}, {\mathscr{E}})$, where $n \geq 1$ is an integer. It is proved by Jasso that if $n> 1$, then ${\mathscr{C}}$ although is no longer exact, but has a nice structure known as $n$-exact structure. In this new structure conflations are called admissible $n$-exact sequences and are ${\mathscr{E}}$-acyclic complexes with $n+2$ terms in ${\mathscr{C}}$. Since their introduction by Iyama, cluster tilting subcategories has gained a lot of traction, due largely to their links and applications to many research areas, many of them unexpected. On the other hand, ideal approximation theory, that is a gentle generalization of the classical approximation theory and deals with morphisms and ideals instead of objects and subcategories, is an active area that has been the subject of several researches. Our aim in this paper is to introduce the so-called `ideal approximation theory' into `higher homological algebra'. To this end, we introduce some important notions in approximation theory into the theory of $n$-exact categories and prove some results. In particular, the higher version of the notions such as ideal cotorsion pairs, phantom ideals, Salce's Lemma and Wakamatsu's Lemma for ideals will be introduced and studied. Our results motivate the definitions and show that $n$-exact categories are the appropriate context for the study of `higher ideal approximation theory'.

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arXiv2020

Cotorsion Classes in Higher Homological Algebra

In this note, the notion of cotorsion classes is introduced into the higher homological algebra. Our results motivate the definition, showing that this notion of $n$-cotorsion classes satisfies usual properties one could expect. In particular, a higher version of Wakamatsu's Lemma is proved. Connections with wide subcategories are also studied.

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arXiv2020

On the Monomorphism Category of $n$-Cluster Tilting Subcategories

Let $\mathcal{M}$ be an $n$-cluster tilting subcategory of ${\rm mod}\mbox{-}Λ$, where $Λ$ is an artin algebra. Let $\mathcal{S}(\mathcal{M})$ denotes the full subcategory of $\mathcal{S}(Λ)$, the submodule category of $Λ$, consisting of all monomorphisms in $\mathcal{M}$. We construct two functors from $\mathcal{S}(\mathcal{M})$ to ${\rm mod}\mbox{-}\underline{\mathcal{M}}$, the category of finitely presented (coherent) additive contravariant functors on the stable category of $\mathcal{M}$. We show that these functors are full, dense and objective. So they induce equivalences from the quotient categories of the submodule category of $\mathcal{M}$ modulo their respective kernels. Moreover, they are related by a syzygy functor on the stable category of ${\rm mod}\mbox{-}\underline{\mathcal{M}}$. These functors can be considered as a higher version of the two functors studied by Ringel and Zhang [RZ] in the case $Λ=k[x]/{\langle x^n \rangle}$ and generalized later by Eiríksson [E] to self-injective artin algebras. Several applications will be provided.

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arXiv2019

$n\mathbb{Z}$-Gorenstein cluster tilting subcategories

Let $Λ$ be an artin algebra. In this paper, the notion of $n\mathbb{Z}$-Gorenstein cluster tilting subcategories will be introduced. It is shown that every $n\mathbb{Z}$-cluster tilting subcategory of ${\rm{mod}}{\mbox{-}}Λ$ is $n\mathbb{Z}$-Gorenstein if and only if $Λ$ is an Iwanaga-Gorenstein algebra. Moreover, it will be shown that an $n\mathbb{Z}$-Gorenstein cluster tilting subcategory of ${\rm{mod}}{\mbox{-}}Λ$ is an $n\mathbb{Z}$-cluster tilting subcategory of the exact category ${\rm{Gprj}}{\mbox{-}}Λ$, the subcategory of all Gorenstein projective objects of ${\rm{mod}}{\mbox{-}}Λ$. Some basic properties of $n\mathbb{Z}$-Gorenstein cluster tilting subcategories will be studied. In particular, we show that they are $n$-resolving, a higher version of resolving subcategories.

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arXiv2018

Auslander's Formula for contravariantly finite subcategories

A relative version of Auslander's formula with respect to a contravariantly finite subcategory will be given. Dual version will be treated. Several examples and applications will be provided. In particular, we show that under certain circumstances, if relative Auslander algebras of artin algebras $Λ$ and $Λ'$ are Morita equivalent, then $Λ$ and $Λ'$ are also Morita equivalent.

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arXiv2017

On relative Auslander algebras

Relative Auslander algebras were introduced and studied by Beligiannis. In this paper, we apply intermediate extension functors associated to certain recollements of functor categories to study them. In particular, we study the existence of tilting-cotilting modules over such algebras. As a consequence, it will be shown that two Gorenstein algebras of G-dimension 1 being of finite Cohen-Macaulay type are Morita equivalent if and only if their Cohen-Macaulay Auslander algebras are Morita equivalent.

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arXiv2017

Auslander's Formula: Variations and Applications

According to the Auslander's formula one way of studying an abelian category ${\mathcal{C}}$ is to study ${\rm mod}\mbox{-}{\mathcal{C}}$, that has nicer homological properties than ${\mathcal{C}}$, and then translate the results back to ${\mathcal{C}}$. Recently Krause gave a derived version of this formula and thus renewed the subject. This paper contains a detailed study of various versions of Auslander formula including the versions for all modules and for unbounded derived categories. We apply them to include some results concerning recollements of triangulated categories.

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arXiv2016

Minimal Injective Resolutions and Auslander-Gorenstein Property for Path Algebras

Let $R$ be a ring and $\mathcal{Q}$ be a finite and acyclic quiver. We present an explicit formula for the injective envelopes and projective precovers in the category $\rm{Rep} (\mathcal{Q} ,R)$ of representations of $\mathcal{Q}$ by left $R$-modules. We also extend our formula to all terms of the minimal injective resolution of $R\mathcal{Q}$. Using such descriptions, we study the Auslander-Gorenstein property of path algebras. In particular, we prove that the path algebra $R\mathcal{Q}$ is $k$-Gorenstein if and only if $\mathcal{Q}=\overrightarrow{A_{n}}$ and $R$ is a $k$-Gorenstein ring, where $n$ is the number of vertices of $\mathcal{Q}$.

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arXiv2015

Derived equivalences of functor categories

Let $\Mod \CS$ denote the category of $\CS$-modules, where $\CS$ is a small category. In the first part of this paper, we provide a version of Rickard's theorem on derived equivalence of rings for $\Mod \CS$. This will have several interesting applications. In the second part, we apply our techniques to get some interesting recollements of derived categories in different levels. We specialize our results to path rings as well as graded rings.

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arXiv2014

Recollements of Cohen-Macaulay Auslander algebras and Gorenstein derived categories

Let $A$, $B$ and $C$ be associative rings with identity. Using a result of Koenig we show that if we have a $\mathbb{D}^{\rm{b}}({\rm{mod\mbox{-}}} )$ level recollement, writing $A$ in terms of $B$ and $C$, then we get a $\mathbb{D}^-({\rm{Mod\mbox{-}}} )$ level recollement of certain functor categories, induces from the module categories of $A$, $B$ and $C$. As an application, we generalise the main theorem of Pan [Sh. Pan, Derived equivalences for Cohen-Macaulay Auslander algebras, J. Pure Appl. Algebra, 216 (2012), 355-363] in terms of recollements of Gorenstein artin algebras. Moreover, we show that being Gorenstein as well as being of finite Cohen-Macaulay type, are invariants with respect to $\mathbb{D}^{\rm{b}}_{{{\mathcal{G}p}}}({\rm{mod\mbox{-}}})$ level recollements of virtually Gorenstein algebras, where $\mathbb{D}^{\rm{b}}_{{{\mathcal{G}p}}}$ denotes the Gorenstein derived category.

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arXiv2012

Homotopy category of projective complexes and complexes of Gorenstein projective modules

Let $R$ be a ring with identity and $\C(R)$ denote the category of complexes of $R$-modules. In this paper we study the homotopy categories arising from projective (resp. injective) complexes as well as Gorenstein projective (resp. Gorenstein injective) modules. We show that the homotopy category of projective complexes over $R$, denoted $\KPC$, is always well generated and is compactly generated provided $\KPR$ is so. Based on this result, it will be proved that the class of Gorenstein projective complexes is precovering, whenever $R$ is a commutative noetherian ring of finite Krull dimension. Furthermore, it turns out that over such rings the inclusion functor $ι: \K(\RGPrj)\hookrightarrow \KR$ has a right adjoint $ι_ρ$, where $\K(\RGPrj)$ is the homotopy category of Gorenstein projective $R$ modules. Similar, or rather dual, results for the injective (resp. Gorenstein injective) complexes will be provided. If $R$ has a dualising complex, a triangle-equivalence between homotopy categories of projective and of injective complexes will be provided. As an application, we obtain an equivalence between the triangulated categories $\K(\RGPrj)$ and $\K(\RGInj)$, that restricts to an equivalence between $\KPR$ and $\KIR$, whenever $R$ is commutative, noetherian and admits a dualising complex.

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