相关论文: A short proof of HRS-tilting

Let $\mathcal{A}$ be an abelian category and $\mathcal{B}$ be the Happel-Reiten-Smal{\o} tilt of $\mathcal{A}$ with respect to a torsion pair. We give necessary and sufficient conditions for the existence of a derived equivalence between…

For an abelian category $A$ equipped with a torsion pair, we give an explicit description for the abelian category $B$ introduced by Happel-Reiten-Smalo, and also for the category of chain complexes $Ch(B)$ and the derived category $D(B)$…

We generalize the tilting process by Happel, Reiten and Smal{\o} to the setting of finitely presented modules over right coherent rings. Moreover, we extend the characterization of quasi-tilted artin algebras as the almost hereditary ones…

Let $\mathcal{A}$ be an abelian category with a torsion pair $(\mathcal{T},\mathcal{F})$. Happel-Reiten-Smalo tilting provides a method to construct a new abelian category $\mathcal{B}$ with a torsion pair associated to…

Let $\Lambda$ be a quasi-tilted algebra. If $\Lambda$ is representation-finite, it was shown by Happel, Reiten, and Smal{\o} that $\Lambda$ is tilted. We provide a new, short proof of this result.

In extended hearts of bounded $t$-structures on a triangulated category, we provide a Happel-Reiten-Smalo tilting theorem and a characterization for $s$-torsion pairs. Applying these to $m$-extended module categories, we characterize…

Tilting theory in cluster categories of hereditary algebras has been developed in [BMRRT] and [BMR]. These results are generalized to cluster categories of hereditary abelian categories. Furthermore, for any tilting object $T$ in a…

We establish connections between silting and tilting objects in an abelian category $\mathcal{B}$ and those in a cleft extension $\mathcal{A}$ of $\mathcal{B}$, which provides a method for constructing more silting and tilting objects. Then…

We show that variants of the classical reflection functors from quiver representation theory exist in any abstract stable homotopy theory, making them available for example over arbitrary ground rings, for quasi-coherent modules on schemes,…

Suppose that $\mathcal{A}$ is an abelian category whose derived category $\mathcal{D}(\mathcal{A})$ has $Hom$ sets and arbitrary (small) coproducts, let $T$ be a (not necessarily classical) ($n$-)tilting object of $\mathcal{A}$ and let…

To a big n-tilting object in a complete, cocomplete abelian category A with an injective cogenerator we assign a big n-cotilting object in a complete, cocomplete abelian category B with a projective generator, and vice versa. Then we…

We generalize Brenner and Butler's Theorem as well as Happel's Theorem on the equivalences induced by a finitely generated tilting module over artin algebras, to the case of an infinitely generated tilting module over an arbitrary…

In this paper we revisit the problem of determining when the heart of a t-structure is a Grothendieck category, with special attention to the case of the Happel-Reiten-Smal{\o} (HSR) t-structure in the derived category of a Grothendieck…

In the theory of triangulated categories, we propose to replace hearts of $t$-structures by proper abelian subcategories, which may be plentiful even when hearts are not. For instance, this happens in negative cluster categories. In support…

We generalize the construction of reflection functors from classical representation theory of quivers to arbitrary small categories with freely attached sinks or sources. These reflection morphisms are shown to induce equivalences between…

We introduce a relative tilting theory in abelian categories and show that this work offers a unified framework of different previous notions of tilting, ranging from Auslander-Solberg relative tilting modules on Artin algebras to…

Let $\mathcal{H}$ be a hereditary abelian category over a field $k$ with finite dimensional $\operatorname{Hom}$ and $\operatorname{Ext}$ spaces. It is proved that the bounded derived category $\mathcal{D}^b(\mathcal{H})$ has a silting…

In this paper we continue the project of generalizing tilting theory to the category of contravariant functors $Mod(C)$, from a skeletally small preadditive category $C$ to the category of abelian groups. We introduced the notion of a a…

Building on the recent work of Adachi, Enomoto and Tsukamoto on a generalization of the Happel-Reiten-Smal{\o} tilting process, we study extended tilting objects in extriangulated categories with negative first extension. These objects…

In 2018, Chen, Han and Zhou introduced a criterion to determine whether the HRS-tilt at a given torsion pair induces derived equivalence. We showcase four applications of this criterion: to stable torsion pairs in arbitrary abelian…