Related papers: Proper classes and Gorensteinness in extriangulate…
We prove that some subquotient categories of exact categories are abelian. This generalizes a result by Koenig-Zhu in the case of (algebraic) triangulated categories. As a particular case, if an exact category B with enough projectives and…
We study Gorenstein categories. We show that such a category has Tate cohomological functors and Avramov-Martsinkovsky exact sequences connecting the Gorenstein relative, the absolute and the Tate cohomological functors. We show that such a…
We give a new characterization of silting subcategories in the stable category of a Frobenius extriangulated category, generalizing the result of Di et al. (J. Algebra 525 (2019) 42-63) about the Auslander-Reiten type correspondence for…
It is becoming increasingly difficult for geometers and even physicists to avoid papers containing phrases like `triangulated category', not to mention derived functors. I will give some motivation for such things from algebraic geometry,…
In this paper we introduce compatible cleft extensions of abelian categories, and we prove that if $(\mathcal{B},\mathcal{A}, e,i,l)$ is a compatible cleft extension, then both the functor $l$ and the left adjoint of $i$ preserve Gorenstein…
We show certain standard constructions of the theory of Verdier triangulated categories to be valid in the Heller triangulated framework as well; viz. Karoubi hull, exactness of adjoints, localisation.
We show that the category of projective modules over a graded commutative ring admits a triangulation with respect to module suspension if and only if the ring is a finite product of graded fields and exterior algebras on one generator over…
In this article we introduce the notion of cubical $(\omega,p)$-categories, for $p \in \mathbb N \cup \{\omega\}$. We show that the equivalence between globular and groupoid $\omega$-categories proven by Al-Agl, Brown and Steiner induces an…
In this paper, we study the relationship of Gorenstein projective objects among three Abelian categories in a recollement. As an application, we introduce the relation of $n$-Gorenstein tilting modules (and Gorenstein syzygy modules) in…
Let $B \subseteq A$ be an extension of finite dimensional algebras. We provide a sufficient condition for the existence of triangle equivalences of singularity categories (resp. Gorenstein defect categories) between $A$ and $B$. This result…
In this paper, we introduce the cofibrant derived category of a group algebra $kG$ and study its relation to the derived category of $kG$. We also define the cofibrant singularity category of $kG$, whose triviality characterizes the…
We introduce the notion of composition series of triangulated categories, which generalizes full exceptional sequences. The lengths of composition series yield invariants for triangulated categories. We study composition series of derived…
Let $\mathscr{A}$ be an abelian category and $\mathscr{C}$ an additive full subcategory of $\mathscr{A}$. We provide a method to construct a proper $\mathscr{C}$-resolution (resp. coproper $\mathscr{C}$-coresolution) of one term in a short…
In the last few years, Lopez-Permouth and several collaborators have introduced a new approach in the study of the classical projectivity, injectivity and flatness of modules. This way, they introduced subprojectivity domains of modules as…
We define $n$-angulated categories by modifying the axioms of triangulated categories in a natural way. We show that Heller's parametrization of pre-triangulations extends to pre-$n$-angulations. We obtain a large class of examples of…
We introduce the notion of categorical absorption of singularities: an operation that removes from the derived category of a singular variety a small admissible subcategory responsible for singularity and leaves a smooth and proper…
We introduce extriangulated factorization systems in extriangulated categories and show that there exists a bijection between $s$-torsion pairs and extriangulated factorization systems. We also consider the gluing of $s$-torsion pairs and…
Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an ${\rm Ext}$-finite, Krull-Schmidt and $k$-linear $n$-exangulated category with $k$ a commutative artinian ring. In this note, we define two additive subcategories $\mathscr{C}_r$ and…
We propose definitions of regular and exact (virtual) double categories, proving a number of results which parallel many basic results in the theory of regular and exact categories. We show that any regular virtual double category admits a…
In this paper, we first provide an explicit procedure to glue together hereditary exact model structures for the recollement of exact categories. To that end, we use the notion of cotorsion pairs and we investigate the gluing of complete…