Related papers: Stability of Gorenstein objects in triangulated ca…
We develop some aspects of the theory of derivators, pointed derivators, and stable derivators. As a main result, we show that the values of a stable derivator can be canonically endowed with the structure of a triangulated category.…
We classify indecomposable non-projective Gorenstein-projective modules over a monomial algebra via the notion of perfect paths. We apply this classification to a quadratic monomial algebra and describe explicitly the stable category of its…
Let $(\mathcal{A,B})$ be a GP-admissible pair and $(\mathcal{Z,W})$ be a GI-admissible pair of classes of objects in an abelian category $\mathcal{C}$, and consider the class $\pi\mathcal{GP}_{(\omega,\mathcal{B},1)}$ of $1$-periodic…
We give conditions on when a triangular matrix ring is Gorenstein of a given selfinjective dimension. We apply the result to the category algebra of a finite EI category. In particular, we prove that for a finite EI category, its category…
Let $\mathcal{G}$ be a fusion category acting on a triangulated category $\mathcal{D}$, in the sense that $\mathcal{D}$ is a $\mathcal{G}$-module category. Our motivation example is fusion-weighted species, which is essentially Heng's…
We study some examples of Bridgeland-Douglas stability conditions on triangulated categories. From one side we give a complete description of the stability manifolds for smooth projective curves of positive genus. From the other side we…
Given a locally coherent Grothendieck category G, we prove that the homotopy category of complexes of injective objects (also known as the coderived category of G) is compactly generated triangulated. Moreover, the full subcategory of…
We analyze stable homology over associative rings and obtain results over Artin algebras and commutative noetherian rings. Our study develops similarly for these classes; for simplicity we only discuss the latter here. Stable homology is a…
We study the homotopy category of unbounded complexes with bounded homologies and its quotient category by the homotopy category of bounded complexes. We show the existence of a recollement of the above quotient category and it has the…
In the paper, we mainly connect the Gorenstein derived equivalence and stable functors of Gorenstein projective modules. Specially, we prove that a Gorenstein derived equivalence between CM-finite algebras A and B can induce a stable…
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…
Let $C$ be a semidualizing module. We first investigate the properties of finitely generated $G_C$-projective modules. Then, relative to $C$, we introduce and study the rings of Gorenstein (weak) global dimensions at most 1, which we call…
We introduce the notion of ST-pairs of triangulated subcategories, a prototypical example of which is the pair of the bound homotopy category and the bound derived category of a finite-dimensional algebra. For an ST-pair $(\C,\D)$, we…
We prove that if a positively-graded ring $R$ is Gorenstein and the associated torsion functor has finite cohomological dimension, then the corresponding noncommutative projective scheme ${\rm Tails}(R)$ is a Gorenstein category in the…
The present paper focuses on the study of the stable category of vector bundles for the weighted projective lines of weight triple. We find some important triangles in this category and use them to construct tilting objects with tubular…
The relation between the $n$-recollements of stable categories of Gorenstein projective modules and the virtual Gorensteinness of algebras are investigated. Let $A,B$, and $C$ be finite dimensional algebras. We prove that if the stable…
Let l be a commutative ring with unit. Garkusha constructed a functor from the category of l-algebras into a triangulated category D, that is a universal excisive and homotopy invariant homology theory. Later on, he provided different…
Let $\Lambda$ be an arbitrary monomial algebra. We investigate the stable category $\underline{\operatorname{Gproj}}^{\mathbb{Z}}\Lambda$ of graded Gorenstein-projective $\Lambda$-modules and the orbit category…
In this paper, we discuss a relationship between representation theory of graded self-injective algebras and that of algebras of finite global dimension. For a positively graded self-injective algebra $A$ such that $A_0$ has finite global…
We develop some aspects of the homological algebra of persistence modules, in both the one-parameter and multi-parameter settings, considered as either sheaves or graded modules. The two theories are different. We consider the graded module…