arXiv subjects

Abdolnaser Bahlekeh

Publications and source records attributed to Abdolnaser Bahlekeh.

11 recordsLinked to original sources

arXiv2026

phantom stable categories of $n$-Frobenius categories are triangulated

Let $n$ be a non-negative integer. Motivated by the universal property of the stable category of Frobenius categories, the authors in \cite{bfss} generalized the stabilization of Frobenius categories to $n$-Frobenius categories, defining the phantom stable category. For an $n$-Frobenius category $\C$, this consists of a pair $(\C_{\p}, T)$, where $\C_{\p}$ is an additive category having the same objects as $\C$ and $T:\C\rt\C_{\p}$ an additive covariant functor that vanishes on $n$-$\Ext$-phantom morphisms and sends $n$-$\Ext$-invertible morphisms to isomorphisms, and $T$ has the universal property with respect to these conditions. The existence of the phantom stable category $(\C_{\p}, T)$ and its several interesting properties have appeared in \cite{bfss}. In this paper, we show that the syzygy functor $\syz$, constructed from $n$-projectives, from $\C$ to $\C_{\p}$ is not only an additive functor, but also it induces an auto-equivalence functor $\Syz$ on $\C_{\p}$. Then, as the main result, it is proved that phantom stable category $(\C_{\p}, T)$ is triangulated, with $\Syz$ serving as its shift functor.

math.RT↗

arXiv2026

On the natural transformations of extension functors

Assume that $\C$ is an exact category. This paper is concerned with the natural transformations between extension functors on $\C$. The first main result indicates that if $\C$ has enough projective objects, then for any pair of objects $M, N\in \C$ and any non-negative integer $n$, the group of all natural transformations from $\Ext^{n+1}_{\C}(N, -)$ to $\Ext^{n+1}_{\C}(M, -)$ is isomorphic to the quotient group $\Ext^n_{\C}(M, \syz^nN)/{\p}$, where $\p$ is the subgroup consisting of those extensions of length $n$ arising as a push-out along a morphism $P\rt\syz^nN$, with $P$ projective. This, together with the Auslander-Gruson-Jensen duality yields that if $\C$ is the category of all finitely presented left modules over an associative ring $R$, then the quotient group is isomorphic to the natural transformations from $\Tor_{n+1}^R(-, M)$ to $\Tor_{n+1}^R(-, N)$. The second main result proves that if $\C$ is an $n$-Frobenuis category, then the statement of the first result remains true, whenever projectives are replaced by $n$-projectives. This result is fruitful from the point of view that, $n$-Frobenius categories may not have projective objects. These results provide a far-reaching generalization of the Hilton-Rees theorem, in the sense that the case $n=0$, recover the Hilton-Rees theorem.

math.RT↗

arXiv2025

On the vanishing of Ext and Tor

This paper contains two theorems concerning the vanishing of natural transformations of (co)homology functors. Precisely, assume that $R$ is a right noetherian ring and $f: M\rt N$ is a morphism of finitely generated right $R$-modules. The first theorem proves that the natural transformation $\Ext^1(f, -)$ vanishes over the category of finitely generated right $R$-modules if and only if $\Tor_1(f, -)$ vanishes over the category of finitely generated left $R$-modules. As a corollary of this result, we establish that $\Ext^1(f, -)$ is epic if and only if $\Tor_1(f, -)$ is monic. The second theorem shows that if $R$ is left and right noetherian and $M, N$ are Gorenstein projective, then the natural transformations $\Tor_1(f, -)$, $\Ext^1(-, f)$ and $\Ext^1(f, -)$ vanish over the category of finitely generated Gorenstein projective modules, simultaneously. This, in particular, yields that over Gorenstein projective modules, the notions of phantom morphisms and $\Ext$-phantom morphisms coincide. Also, it is proved that if $R$ is $n$-Gorenstein, then for any integer $i>n$, the natural transformations $\Ext^{i}(f, -)$, $\Ext^{i}(-, f)$ and $\Tor_{i}(f, -)$ vanish over finitely generated modules, simultaneously. As an interesting consequence, we show that under the same assumptions, $\Ext^i(-, f)$ is epic (resp. monic) if and only if $\Ext^i(f, -)$ is monic (resp. epic) if and only if $\Tor_i(f, -)$ is epic (resp. monic).

math.RT↗

arXiv2024

Representation-theoretic properties of balanced big Cohen-Macaulay modules

Let $(R, \m, k)$ be a complete Cohen-Macaulay local ring. In this paper, we assign a numerical invariant, for any balanced big Cohen-Macaulay module, called $\uh$-length. Among other results, it is proved that, for a given balanced big Cohen-Macaulay $R$-module $M$ with an $\m$-primary cohomological annihilator, if there is a bound on the $\uh$-length of all modules appearing in $\CM$-support of $M$, then it is fully decomposable, i.e. it is a direct sum of finitely generated modules. While the first Brauer-Thrall conjecture fails in general by a counterexample of Dieterich dealing with multiplicities to measure the size of maximal Cohen-Macaulay modules, our formalism establishes the validity of the conjecture for complete Cohen-Macaulay local rings. In addition, the pure-semisimplicity of a subcategory of balanced big Cohen-Macaulay modules is settled. Namely, it is shown that $R$ is of finite $\CM$-type if and only if the category of all fully decomposable balanced big Cohen-Macaulay modules is closed under kernels of epimorphisms. Finally, we examine the mentioned results in the context of Cohen-Macaulay artin algebras admitting a dualizing bimodule $ω$, as defined by Auslander and Reiten. It will turn out that, $ω$-Gorenstein projective modules with bounded $\CM$-support are fully decomposable. In particular, a Cohen-Macaulay algebra $Λ$ is of finite $\CM$-type if and only if every $ω$-Gorenstein projective module is of finite $\CM$-type, which generalizes a result of Chen for Gorenstein algebras. Our main tool in the proof of results is Gabriel-Roiter (co)measure, an invariant assigned to modules of finite length, and defined by Gabriel and Ringel. This, in fact, provides an application of the Gabriel-Roiter (co)measure in the category of maximal Cohen-Macaulay modules.

math.AC↗

arXiv2024

Phantom stable category of $n$-Frobenius categories

Let $n$ be a non-negative integer. An exact category $\C$ is said to be an $n$-Frobenius category, provided that it has enough $n$-projectives and $n$-injectives and the $n$-projectives coincide with the $n$-injectives. It is proved that any abelian category with non-zero $n$-projective objects, admits a non-trivial $n$-Frobenius subcategory. In particular, we explore several examples of $n$-Frobenius categories. Also, as a far reaching generalization of the stabilization of a Frobenius category, we define and study phantom stable category of an $n$-Frobenius category $\C$. Precisely, assume that $\p\subseteq\Ext^n_{\C}$ is the subfunctor consisting of all conflations of length $n$ factoring through $n$-projective objects. A couple $(\C_{\p}, T)$, where $\C_{\p}$ is an additive category and $T$ is a covariant additive functor from $\C$ to $\C_{\p}$, is a phantom stable category of $\C$, provided that for any morphism $f$ in $\C$, $T(f)=0$, whenever $f$ is an $n$-$\Ext$-phantom morphism and $T(f)$ is an isomorphism in $\C_{\p}$, if $f$ acts as invertible on $\Ext^n/{\p}$, and $T$ has the universal property with respect to these conditions. The main focus of this paper is to show that the phantom stable category of an $n$-Frobenius category always exists. Some properties of phantom stable categories that reveal the efficiency of these categories are studied.

math.RT↗

arXiv2024

The stable category of monomorphisms between (Gorenstein) projective modules with applications

Let (S; n) be a commutative noetherian local ring and let w in n be non-zero divisor. This paper is concerned with the two categories of monomorphisms between finitely generated (Gorenstein) projective S-modules, such that their cokernels are annihilated by w. It is shown that these categories, which will be denoted by Mon(w;P) and Mon(w; G), are both Frobenius categories with the same projective objects. It is also proved that the stable category Mon(w;P) is triangle equivalent to the category of D-branes of type B, DB(w), which has been introduced by Kontsevich and studied by Orlov. Moreover, it will be observed that the stable categories Mon(w;P) and Mon(w; G) are closely related to the singularity category of the factor ring R = S/(w). Precisely, there is a fully faithful triangle functor from the stable category Mon(w; G) to Dsg(R), which is dense if and only if R (and so S) are Gorenstein rings. Particularly, it is proved that the density of the restriction of this functor to Mon(w;P), guarantees the regularity of the ring S.

math.RT↗

arXiv2024

The monomorphism category of Gorenstein projective modules and comparision with the category of matrix factorization

Let ($S, \mathfrak{n})$ be a commutative noetherian local ring and let $ω\in\mathfrak{n}$ be non-zero divisor. This paper is concerned with the category of monomorphisms between finitely generated Gorenstein projective S-modules, such that their cokernels are annihilated by $ω$. We will observe that this category, which will be denoted by Mon$(ω,\mathcal{G})$, is an exact category in the sense of Quillen. More generally, it is proved that Mon$(ω,\mathcal{G})$ is a Frobenius category. Surprisingly, it is shown that not only the category of matrix factorizations embeds into Mon$(ω,\mathcal{G})$, but also its stable category as well as the singularity category of the factor ring $R = S/(ω)$, can be realized as triangulated subcategories of the stable category of Mon$(ω,\mathcal{G})$.

math.RT↗

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 ${\mathsf{Gprj}}(A_n, Λ)$.

math.RT↗

arXiv2023

The homotopy category of monomorphisms between projective modules

Let $(S, \n)$ be a commutative noetherian local ring and $ω\in\n$ be non-zerodivisor. This paper deals with the behavior of the category $\mon(ω, \cp)$ consisting of all monomorphisms between finitely generated projective $S$-modules with cokernels annihilated by $ω$. We introduce a homotopy category $\HT\mon(ω, \cp)$, which is shown to be triangulated. It is proved that this homotopy category embeds into the singularity category of the factor ring $R=S/{(ω)}$. As an application, not only the existence of almost split sequences {ending at indecomposable non-projective objects of} $\mon(ω, \cp)$ is proven, but also the Auslander-Reiten translation, $τ_{\mon}(-)$, is completely recognized. Particularly, it will be observed that any non-projective object of $\mon(ω, \cp)$ with local endomorphism ring is invariant under the square of the Auslander-Reiten translation.

math.AC↗

arXiv2018

Specifying The Auslander transpose in submodule category and its applications

Let $(R, \m)$ be a $d$-dimensional commutative noetherian local ring. Let $\M$ denote the morphism category of finitely generated $R$-modules and let $\Sc$ be the submodule category of $\M$. In this paper, we specify the Auslander transpose in submodule category $\Sc$. It will turn out that the Auslander transpose in this category can be described explicitly within ${\rm mod}R$, the category of finitely generated $R$-modules. This result is exploited to study the linkage theory as well as the Auslander-Reiten theory in $\Sc$. Indeed, a characterization of horizontally linked morphisms in terms of module category is given. In addition, motivated by a result of Ringel and Schmidmeier, we show that the Auslander-Reiten translations in the subcategories $\HH$ and $\G$, consisting of all morphisms which are maximal Cohen-Macaulay $R$-modules and Gorenstein projective morphisms, respectively, may be computed within ${\rm mod}R$ via $\G$-covers. Corresponding result for subcategory of epimorphisms in $\HH$ is also obtained.

math.AC↗

arXiv2015

Annihilation of cohomology, generation of modules and finiteness of derived dimension

Let $(R,\m,k)$ be a commutative noetherian local ring of Krull dimension $d$. We prove that the cohomology annihilator $\ca(R)$ of $R$ is $\m$-primary if and only if for some $n\ge0$ the $n$-th syzygies in $\mod R$ are constructed from syzygies of $k$ by taking direct sums/summands and a fixed number of extensions. These conditions yield that $R$ is an isolated singularity such that the bounded derived category $\db(R)$ and the singularity category $\ds(R)$ have finite dimension, and the converse holds when $R$ is Gorenstein. We also show that the modules locally free on the punctured spectrum are constructed from syzygies of finite length modules by taking direct sums/summands and $d$ extensions. This result is exploited to investigate several ascent and descent problems between $R$ and its completion $\widehat R$.

math.AC↗