Auslander bounds and homological conjectures.

@article{Wei2008AuslanderBA,
  title={Auslander bounds and homological conjectures.},
  author={Jiaqun Wei},
  journal={Revista Matematica Iberoamericana},
  year={2008},
  volume={27},
  pages={871-884}
}
  • Jiaqun Wei
  • Published 2008
  • Mathematics
  • Revista Matematica Iberoamericana
Inspired by recent works on rings satisfying Auslander’s conjecture, we study invariants, called Auslander bounds, and prove that they have strong relations to some homological conjectures. 
Tilting complexes and Auslander–Reiten conjecture
We studied the properties of tilting complexes and proved that derived equivalences preserve the validity of the Auslander–Reiten conjecture.
Auslander-Reiten conjecture and Auslander-Reiten duality
Abstract Motivated by a result of Araya, we extend the Auslander–Reiten duality theorem to Cohen–Macaulay local rings. We also study the Auslander–Reiten conjecture, which is rooted in NakayamaʼsExpand
On the Generalized Auslander–Reiten Conjecture under Certain Ring Extensions
Abstract We show that under some conditions a Gorenstein ring $R$ satisfies the Generalized Auslander–Reiten conjecture if and only if $R\left[ x \right]$ does. When $R$ is a local ring we prove theExpand
Derived invariance by syzygy complexes
  • Jiaqun Wei
  • Mathematics
  • Mathematical Proceedings of the Cambridge Philosophical Society
  • 2017
Abstract We study derived invariance through syzygy complexes. In particular, we prove that syzygy-finite algebras and Igusa--Todorov algebras are invariant under derived equivalences. Consequently,Expand
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)Expand
On the Auslander–Reiten conjecture for Cohen–Macaulay rings and path algebras
ABSTRACT In this note, it is shown that the validity of the Auslander–Reiten conjecture for a given d-dimensional Cohen–Macaulay local ring R depends on its validity for all direct summands of d-thExpand
The Auslander-Reiten conjecture for certain non-Gorenstein Cohen-Macaulay rings.
The Auslander-Reiten conjecture is a notorious open problem about the vanishing of Ext modules. In a Cohen-Macaulay local ring $R$ with a parameter ideal $Q$, the Auslander-Reiten conjecture holdsExpand
Auslander-Reiten conjecture for non-Gorenstein Cohen-Macaulay rings
Let $R$ be a Cohen-Macaulay local ring and $Q$ be a parameter ideal of $R$. Due to M. Auslander, S. Ding, and \O. Solberg, the Auslander-Reiten conjecture holds for $R$ if and only if it holds forExpand
When the kernel of a complete hereditary cotorsion pair is the additive closure of a tilting module
Abstract In this paper, we study when the kernel of a complete hereditary cotorsion pair is the additive closure of a tilting module. Applications go in three directions. The first is to characterizeExpand
Complete cohomology for extriangulated categories
Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a proper class $\xi$ of $\mathbb{E}$-triangles. In this paper, we study complete cohomology of objects inExpand

References

SHOWING 1-10 OF 22 REFERENCES
Algebras that satisfy Auslander’s condition on vanishing of cohomology
Auslander conjectured that every Artin algebra satisfies a certain condition on vanishing of cohomology of finitely generated modules. The failure of this conjecture—by a 2003 counterexample due toExpand
Nonvanishing cohomology and classes of Gorenstein rings
Abstract We give counterexamples to the following conjecture of Auslander: given a finitely generated module M over an Artin algebra Λ , there exists a positive integer n M such that for all finitelyExpand
Wakamatsu tilting modules
Abstract We study a generalization of tilting modules to modules of possibly infinite projective dimension, introduced by Wakamatsu in [J. Algebra 114 (1988) 106–114]. In particular, we characterizeExpand
Finite Flat and Projective Dimension
ABSTRACT It is proved that over a right-noetherian algebra with a dualizing complex, a left-module with finite flat dimension has a finite projective dimension.
Tilting modules of finite projective dimension
Tilting modules are well known, and are useful in the representation theory of artinian algebras ([2-4, 6, 11]). The purpose of this paper is to define tilting modules of finite projective dimensionExpand
Symmetry in the vanishing of Ext over Gorenstein rings
We investigate symmetry in the vanishing of Ext for finitely generated modules over local Gorenstein rings. In particular, we define a class of local Gorenstein rings, which we call AB rings, andExpand
On modules with trivial self-extensions
Throughout this note, all rings will be self-basic connected artin algebras over a fixed commutative local artin ring k and all modules will be finitely generated. Homomorphisms will be written onExpand
Symmetry in the vanishing of Ext over stably symmetric algebras
Abstract A Frobenius algebra over a field k is called symmetric if the Nakayama automorphism is an inner automorphism. A stably symmetric algebra is defined to be a generalization of a symmetricExpand
Local Limitations of the Ext Functor Do Not Exist
In this note it is shown that for $k$ a field, and for the four-dimensional algebra $\Lambda=k\langle x,y\rangle /\langle x^2,y^2,xy+qyx\rangle$ when $q^n\neq 1,0$ for all $n$ , there exist aExpand
Handbook of Tilting Theory
1. Introduction 2. Basic results of classic tilting theory L. Angeleri Hugel, D. Happel and H. Krause 3. Classification of representation-finite algebras and their modules T. Brustle 4. A spectralExpand
...
1
2
3
...