arXiv subjects

Fahimeh Sadat Fotouhi

Publications and source records attributed to Fahimeh Sadat Fotouhi.

7 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↗

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

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↗