arXiv subjects

Ramin Ebrahimi

Publications and source records attributed to Ramin Ebrahimi.

13 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 $τ^-$-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 $τ$-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 $τ$-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 $τ$-tilting theory. In particular, we prove that any support $τ$-tilting module induce a model structure, and there is bijective correspondence between support $τ$-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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arXiv2024

On definable subcategories

Let $\mathcal{X}$ be a skeletally small additive category. Using the canonical equivalence between two different presentations of the free abelian category over $\mathcal{X}$, we give a new and simple characterization of definable subcategories of $\rm Mod\text{-}\mathcal{X}$, and in particular definable subcategories of modules over rings. In the end, we give a conceptual proof of Auslander-Gruson-Jensen duality, which makes the duality between definable subcategories of left and right module more transparent.

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arXiv2024

Semibrick-cosilting correspondence

Let $Λ$ be a finite dimensional algebra. In this paper we show that there is a natural bijection between cosilting modules in Mod$Λ$ and semibricks in Mod$Λ$ satisfying some condition. Also this bijection restricts to a bijection between all semibricks in mod$Λ$ and a certain subclass of cosilting modules. These bijections are generalizations of Asai's correspondence [7] between support $τ^-$-tilting modules and right finite semibricks.

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arXiv2023

The completion of $d$-abelian categories

Let $A$ be a finite-dimensional algebra, and $\mathfrak{M}$ be a $d$-cluster tilting subcategory of mod$A$. From the viewpoint of higher homological algebra, a natural question to ask is when $\mathfrak{M}$ induces a $d$-cluster tilting subcategory in Mod$A$. In this paper, we investigate this question in a more general form. Let $\mathcal{M}$ be a small $d$-abelian category of an abelian category $\mathcal{A}$. The completion of $\mathcal{M}$, denoted by Ind$(\mathcal{M})$, is defined as the universal completion of $\mathcal{M}$ with respect to filtered colimits. We explore Ind$(\mathcal{M})$ and demonstrate its equivalence to the full subcategory $\mathcal{L}_d(\mathcal{M})$ of Mod$\mathcal{M}$, comprising left $d$-exact functors. Notably, while Ind$(\mathcal{M})$ as a subcategory of $\frac{Mod\mathcal{M}}{Eff(\mathcal{M})}$, satisfies all properties of a $d$-cluster tilting subcategory except $d$-rigidity, it falls short of being a $d$-cluster tilting category. For a $d$-cluster tilting subcategory $\mathfrak{M}$ of mod$A$, $\overrightarrow{\mathfrak{M}}$, consists of all filtered colimits of objects from $\mathfrak{M}$, is a generating-cogenerating, functorially finite subcategory of Mod$A$. The question of whether $\mathfrak{M}$ is a $d$-rigid subcategory remains unanswered. However, if it is indeed $d$-rigid, it qualifies as a $d$-cluster tilting subcategory. In the case $d=2$, employing cotorsion theory, we establish that $\overrightarrow{\mathfrak{M}}$ is a $2$-cluster tilting subcategory if and only if $\mathfrak{M}$ is of finite type. Thus, the question regarding whether $\overrightarrow{\mathfrak{M}}$ is a $d$-cluster tilting subcategory of Mod$ A$ appears to be equivalent to the Iyama's qestion about the finiteness of $\mathfrak{M}$.

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arXiv2023

Higher Auslander's formula

Let $\mathcal{M}$ be a small $n$-abelian category. We show that the category of finitely presented functors $mod$-$\mathcal{M}$ modulo the subcategory of effaceable functors $mod_0$-$\mathcal{M}$ has an $n$-cluster tilting subcategory which is equivalent to $\mathcal{M}$. This gives a higher-dimensional version of Auslander's formula.

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arXiv2022

Yoneda extensions of abelian quotient categories

Let $\mathcal{A}$ be a essentially small abelian category and $\mathcal{C}$ be a Serre subcategory of $\mathcal{A}$. Consider the quotient functor $q:\mathcal{A}\rightarrow \mathcal{A}/\mathcal{C}$. For an object $A\in \mathcal{A}$ and a non-negative integer $k$ we investigate when the natural map $q_{X,A}^i: \rm Ext^i_{\mathcal{A}}(X,A)\rightarrow Ext^i_{\mathcal{A}/\mathcal{C}}(q(X),q(A))$ is invertible for every $X\in \mathcal{A}$ and every $i\in\{0,1,\cdots,k\}$. In the end we give an application of the main theorem.

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arXiv2022

$n\mathbb{Z}$-abelian and $n\mathbb{Z}$-exact categories

In this paper we introduce $n\mathbb{Z}$-abelian and $n\mathbb{Z}$-exact categories by axiomatising properties of $n\mathbb{Z}$-cluster tilting subcategories. We study this categories and show that every $n\mathbb{Z}$-cluster tilting subcategory of an abelian (resp., exact) category has a natural structure of an $n\mathbb{Z}$-abelian (resp., $n\mathbb{Z}$-exact) category. Also we show that every small $n\mathbb{Z}$-abelian category arise in this way, and discuss the problem for $n\mathbb{Z}$-exact categories.

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arXiv2021

Higher Auslander correspondence for exact categories

Inspired by the recent work of Henrard, Kvamme and van Roosmalen [17], we prove a categorified version of higher Auslander correspondence in the context of exact categories. We define n-Auslander exact categories and show that there is a bijection between the equivalence classes of n-cluster tilting subcategories of exact categories and the equivalence classes of n-Auslander exact categories.

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arXiv2021

Gabriel-Quillen embedding for $n$-exact categories

Our first aim is to provide an analog of the Gabriel-Quillen embedding theorem for $n$-exact categories. Also we give an example of an $n$-exact category that is not an $n$-cluster tilting subcategory, and we suggest two possible ways for realizing $n$-exact categories as $n$-cluster tilting subcategory.

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arXiv2020

Representation of n-abelian categories in abelian categories

Let $\mathcal{M}$ be a small $n$-abelian category. We show that the category of absolutely pure group valued functors over $\mathcal{M}$, denote by $\mathcal{L}_2(\mathcal{M},\mathcal{G})$, is an abelian category and $\mathcal{M}$ is equivalent to a full subcategory of $\mathcal{L}_2(\mathcal{M},\mathcal{G})$ in such a way that $n$-kernels and $n$-cokernels are precisely exact sequences of $\mathcal{L}_2(\mathcal{M},\mathcal{G})$ with terms in $\mathcal{M}$. This gives a higher-dimensional version of the Freyd-Mitchell embedding theorem for $n$-abelian categories.

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arXiv2020

Pure semisimple $n$-cluster tilting subcategories

From the viewpoint of higher homological algebra, we introduce pure semisimple $n$-abelian category, which is analogs of pure semisimple abelian category. Let $Λ$ be an Artin algebra and $\mathcal{M}$ be an $n$-cluster tilting subcategory of $Mod$-$Λ$. We show that $\mathcal{M}$ is pure semisimple if and only if each module in $\mathcal{M}$ is a direct sum of finitely generated modules. Let $\mathfrak{m}$ be an $n$-cluster tilting subcategory of $mod$-$Λ$. We show that $Add(\mathfrak{m})$ is an $n$-cluster tilting subcategory of $Mod$-$Λ$ if and only if $\mathfrak{m}$ has an additive generator if and only if $Mod(\mathfrak{m})$ is locally finite. This generalizes Auslander's classical results on pure semisimplicity of Artin algebras.

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