arXiv subjects

R. Hafezi

Publications and source records attributed to R. Hafezi.

4 recordsLinked to original sources

arXiv2024

Strong Purity and Phantom Morphisms

Let $R$ be a commutative ring and $S \subseteq R$ be a multiplicative subset. We introduce and study the concept of $S$-purity based on the notion of $S$-strongly flat modules. The class of $S$-pure injective modules will be studied. We demonstrate that this class is enveloping and explore its closedness under extension. The concept of purity is closely connected to the existence of phantom maps. So we will delve into the study of the $S$-phantom morphisms. We will establish that the $S$-phantom ideal is a precovering ideal and examine the situations where it becomes a covering ideal. Finally, in the last section, we will investigate an ideal version of the `Optimistic Conjecture', raised by Positselski and Slávik.

math.AC↗

arXiv2019

A functorial approach to categorical resolutions

Using a relative version of Auslander's formula, we give a functorial approach to show that the bounded derived category of every Artin algebra admits a categorical resolution. This, in particular, implies that the bounded derived categories of Artin algebras of finite global dimension determine bounded derived categories of all Artin algebras. Hence, this paper can be considered as a typical application of functor categories, introduced in representation theory by Auslander, to categorical resolutions.

math.RT↗

arXiv2017

Duality and Serre functor in homotopy categories

For a (right and left) coherent ring $A$, we show that there exists a duality between homotopy categories ${\mathbb{K}}^{\rm{b}}({\rm mod}{\mbox{-}}A^{\rm op})$ and ${\mathbb{K}}^{\rm{b}}({\rm mod}{\mbox{-}}A)$. If $A=Λ$ is an artin algebra of finite global dimension, this duality restricts to a duality between their subcategories of acyclic complexes, ${\mathbb{K}}^{\rm{b}}_{\rm ac}({\rm mod}{\mbox{-}}Λ^{\rm op})$ and ${\mathbb{K}}^{\rm{b}}_{\rm ac}({\rm mod}{\mbox{-}}Λ).$ As a result, it will be shown that, in this case, ${\mathbb{K}}_{\rm ac}^{\rm{b}}({\rm mod}{\mbox{-}}Λ)$ admits a Serre functor and hence has Auslander-Reiten triangles.

math.RT↗

arXiv2016

On relative derived categories

The paper is devoted to study some of the questions arises naturally in connection to the notion of relative derived categories. In particular, we study invariants of recollements involving relative derived categories, generalise two results of Happel by proving the existence of AR-triangles in Gorenstein derived categories, provide situations for which relative derived categories with respect to Gorenstein projective and Gorenstein injective modules are equivalent and finally study relations between the Gorenstein derived category of a quiver and its image under a reflection functor. Some interesting applications are provided.

math.RT↗