arXiv subjects

Shokrollah Salarian

Publications and source records attributed to Shokrollah Salarian.

15 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↗

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↗

arXiv2018

Orders of bounded and strongly unbounded lattice type

Brauer and Thrall conjectured that a finite-dimensional algebra over a field of bounded representation type is actually of finite representation type and a finite-dimensional algebra (over an infinite field) of infinite representation type has strongly unbounded representation type. These conjectures, now theorems, are our motivation for studying (generalized) orders of bounded and strongly unbounded lattice type. To each lattice over an order we assign a numerical invariant, $\underline{\h}$-length, measuring Hom modulo projectives. We show that an order of bounded lattice type is actually of finite lattice type, and if there are infinitely many non-isomorphic indecomposable lattices of the same $\underline{\h}$-length, then the order has strongly unbounded lattice type. For a hypersurface $R=k[[x_0,...,x_d]]/(f)$, we show that $R$ is of bounded (respectively, strongly unbounded) lattice type if and only if the double branched cover $R^{\sharp}$ of $R$ is of bounded (respectively, strongly unbounded) lattice type. This is an analog of a result of Knörrer and Buchweitz-Greuel-Schreyer for rings of finite mCM type. Consequently, it is proved that $R$ has strongly unbounded lattice type whenever $k$ is infinite.

math.RT↗

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↗

arXiv2014

Gorenstein Projective Modules Over Triangular Matrix Rings

We study totally acyclic complexes of projective modules over triangular matrix rings and then use it to classify Gorenstein projective modules over such rings. We also use this classification to obtain some information concerning Cohen-Macaulay finite and virtually Gorenstein triangular matrix artin algebras.

math.RT↗

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.

math.AC↗

arXiv2009

Totally acyclic complexes over noetherian schemes

We define a notion of total acyclicity for complexes of flat quasi-coherent sheaves over a semi-separated noetherian scheme, generalising complete flat resolutions over a ring. By studying these complexes as objects of the pure derived category of flat sheaves we extend several results about totally acyclic complexes of projective modules to schemes; for example, we prove that a scheme is Gorenstein if and only if every acyclic complex of flat sheaves is totally acyclic. Our formalism also removes the need for a dualising complex in several known results for rings, including Jorgensen's proof of the existence of Gorenstein projective precovers.

math.AG↗

arXiv2005

Characterizing local rings via homological dimensions and regular sequences

Let (R,m) be a Noetherian local ring of depth d and C a semidualizing R-complex. Let M be a finite R-module and t an integer between 0 and d. If G_C-dimension of M/IM is finite for all ideals I generated by an R-regular sequence of length at most d-t then either G_C-dimension of M is at most t or C is a dualizing complex. Analogous results for other homological dimensions are also given.

math.AC↗