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Jiaqun Wei

Publications and source records attributed to Jiaqun Wei.

51 records · Page 3Linked to original sources
arXiv2020

Tate-Vogel and relative cohomologies of complexes with respect to cotorsion pairs

We study Tate-Vogel and relative cohomologies of complexes by applying the model structure induced by a complete hereditary cotorsion pair ($\A$, $\B$) of modules. We show first that the class of complexes admitting a complete $\A$ resolution is exactly the class of complexes with finite Gorenstein $\A$ dimension. This lets us give general techniques for computing Tate-Vogel cohomoloies of complexes with finite Gorenstein $\A$ dimension. As a consequence, relative cohomology groups for complexes with finite Gorenstein $\A$ dimension are investigated. Finally, the relationships between Gorenstein $\A$ dimensions and $\A$ dimensions for complexes are given.

math.RA↗
arXiv2020

Idempotent completion of extriangulated categories

Extriangulated categories were introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. In this paper, we show that the idempotent completion of an extriangulated category admits a natural extriangulated structure. As applications, we prove that cotorsion pairs in an extriangulated category induce cotorsion pairs in its idempotent completion under certain condition, and the idempotent completion of a recollement of extriangulated categories is still a recollement.

math.CT↗
arXiv2019

Recollements induced by good silting objects

Let $U$ be a silting object in a derived category over a dg-algebra $A$, and let $B$ be the endomorphism dg-algebra of $U$. Under some appropriate hypotheses, we show that if $U$ is good, then there exist a dg-algebra $C$, a homological epimorphism $B\rightarrow C$ and a recollement among the (unbounded) derived categories $\mathbf{D}(C,d)$ of $C$, $\mathbf{D}(B,d)$ of $B$ and $\mathbf{D}(A,d)$ of $A$. In particular, the kernel of the left derived functor $-\otimes^{\mathbb{L}}_{B}U$ is triangle equivalent to the derived category $\mathbf{D}(C,d)$. Conversely, if $-\otimes^{\mathbb{L}}_{B}U$ admits a fully faithful left adjoint functor, then $U$ is good. Moreover, we establish a criterion for the existence of a recollement of the derived category of a dg-algebra relative to two derived categories of weak non-positive dg-algebras. Finally, some applications are given related to good cosilting objects, good 2-term silting complexes, good tilting complexes and modules, which recovers a recent result by Chen and Xi.

math.CT↗
arXiv2018

On triangle equivalences of stable categories

We apply the Auslander-Buchweitz approximation theory to show that the Iyama and Yoshino's subfactor triangulated category can be realized as a triangulated quotient. Applications of this realization go in three directions. Firstly, we recover both a result of Iyama and Yang and a result of the third author. Secondly, we extend the classical Buchweitz's triangle equivalence from Iwanaga-Gorenstein rings to Noetherian rings. Finally, we obtain the converse of Buchweitz's triangle equivalence and a result of Beligiannis, and give characterizations for Iwanaga-Gorenstein rings and Gorenstein algebras

math.RT↗
arXiv2016

Wakamatsu's equivalence revisited

For a certain Wakamatsu-tilting bimodule over two artin algebras $A$ and $B$, Wakamatsu constructed an explicit equivalence between the stable module categories over the trivial extension algebra of $A$ and that of $B$. We prove that Wakamatsu's functor is a triangle functor, thus a triangle equivalence.

math.RT↗
arXiv2016

Cosilting complexes and AIR-cotilting modules

We introduce and study the new concepts of cosilting complexes, cosilting modules and AIR-cotilting modules. We prove that the three concepts AIR-cotilting modules, cosilting modules and quasi-cotilting modules coincide with each other, in contrast with the dual fact that AIR-tilting modules, silting modules and quasi-tilting modules are different. Further, we show that there are bijections between the following four classes (1) equivalent classes of AIR-cotilting (resp., cosilting, quasi-cotilting) modules, (2) equivalent classes of 2-term cosilting complexes, (3) torsion-free cover classes and (4) torsion-free special precover classes. We also extend a classical result of Auslander and Reiten on the correspondence between certain contravariantly finite subcategories and cotilting modules to the case of cosilting complexes.

math.RA↗
arXiv2016

Quasi-cotilting modules and torsion-free classes

We prove that all quasi-cotilting modules are pure-injective and cofinendo. It follows that the class CogenM is always a covering class whenever M is a quasi-cotilting module. Some characterizations of quasi-cotilting modules are given. As a main result, we prove that there is a bijective correspondence between the equivalent classes of quasi-cotilting modules and torsion-free covering classes.

math.RA↗
arXiv2015

Relative singularity categories, Gorenstein objects and silting theory

We study singularity categories through Gorenstein objects in triangulated categories and silting theory. Let $ω$ be a semi-selforthogonal (or presilting) subcategory of a triangulated category $\mathcal{T}$. We introduce the notion of $ω$-Gorenstein objects, which is far extended version of Gorenstein projective modules and Gorenstein injective modules in triangulated categories. We prove that the stable category $\underline{\mathcal{G}_ω}$, where $\mathcal{G}_ω$ is the subcategory of all $ω$-Gorenstein objects, is a triangulated category and it is, under some conditions, triangle equivalent to the relative singularity category of $\mathcal{T}$ with respect to $ω$.

math.RT↗
arXiv2013

Wakamatsu-silting complexes

We introduce Wakamatsu-silting complexes (resp., Wakamatsu-tilting complexes) as a common generalization of both silting complexes (resp., tilting complexes) and Wakamatsu-tilting modules. Characterizations of Wakamatsu-silting complexes are given. In particular, we show that a complex $T$ is Wakamatsu-silting if and only if its dual $DT$ is Wakamatsu-silting. It is conjectured that all compact Wakamatsu-silting complexes are just silting complexes. We prove that the conjecture lies under the finitistic dimension conjecture.

math.RT↗
arXiv2012

Gorenstein homological theory for differential modules

We show that a differential module is Gorenstein projective if and only if its underlying module is Gorenstein projective. Dually, a differential module is Gorenstein injective if and only if its underlying module is Gorenstein injective.

math.RT↗
arXiv2011

Auslander Bounds and Homological Conjectures

Inspired by recent works on rings satisfying Auslander's conjecture, we study invariants, which we call Auslander bounds, and prove that they have strong relations to some homological conjectures.

math.RA↗
arXiv2011

Finitistic Dimension Conjecture and Conditions on Ideals

The notion of Igusa-Todorov classes is introduced in connection with the finitistic dimension conjecture. As application we consider conditions on special ideals which imply the Igusa-Todorov and other finiteness conditions on modules proving the finitistic dimension conjecture and related conjectures in those cases.

math.RA↗
arXiv2011

Derived categories and syzygies

We introduce syzygies for derived categories and study their properties. Using these, we prove the derived invariance of the following classes of artin algebras: (1) syzygy-finite algebras, (2) Igusa-Todorov algebras, (3) AC algebras, (4) algebras satisfying the finitistic Auslander conjecture, and (5) algebras satisfying the generalized Auslander-Reiten conjecture. In particular, Gorenstein CM-finite algebras are derived invariants.

math.RT↗
arXiv2008

Finitistic and Representation Dimensions

We prove that the finitistic dimension conjecture, the Gorenstein Symmetry Conjecture, the Wakamatsu-tilting conjecture and the generalized Nakayama conjecture hold for artin algebras which can be realized as endomorphism algebras of modules over representation-finite algebras. Note it is a question whether or not all artin algebras have such realizations. It was also shown that if every quasi-hereditary algebras has a left idealized extension which is a monomial algebra or an algebra whose representation dimension is not more than 3, then the finitistic dimension conjecture holds.

math.RT↗