# Generalized Auslander-Reiten Conjecture and Derived Equivalences

@article{Pan2011GeneralizedAC, title={Generalized Auslander-Reiten Conjecture and Derived Equivalences}, author={Shengyong Pan}, journal={Communications in Algebra}, year={2011}, volume={41}, pages={3695 - 3704} }

In this note, we prove that the generalized Auslander-Reiten conjecture is preserved under derived equivalences between Artin algebras.

#### 5 Citations

Singular equivalences and Auslander-Reiten conjecture.

- Mathematics
- 2020

Auslander-Reiten conjecture, which says that an Artin algebra does not have any non-projective generator with vanishing self-extensions in all positive degrees, is shown to be invariant under certain… Expand

Derived equivalences and Cohen-Macaulay Auslander algebras

- Mathematics
- 2015

Let A and B be Artin R-algebras of finite Cohen-Macaulay type. Then we prove that, if A and B are standard derived equivalent, then their Cohen-Macaulay Auslander algebras are also derived… Expand

Derived invariance by syzygy complexes

- 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 Equivalences for Φ-Cohen–Macaulay Auslander–Yoneda Algebras

- Mathematics
- 2014

In this paper, we introduce a class of algebras called Φ-Cohen–Macaulay Auslander–Yoneda algebras with Φ an admissible set of ℕ, and construct derived equivalences between these Φ-Cohen–Macaulay… Expand

Stable functors of derived equivalences and Gorenstein projective modules

- Mathematics
- 2015

From certain triangle functors, called non-negative functors, between the bounded derived categories of abelian categories with enough projective objects, we introduce their stable functors which are… Expand

#### References

SHOWING 1-10 OF 35 REFERENCES

Generalized Auslander-Reiten conjecture and tilting equivalences

- Mathematics
- 2009

We show that the generalized Auslander-Reiten conjecture is stable under tilting equivalences.

A Note on Auslander Bounds

- Mathematics
- 2008

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.

Finiteness of finitistic dimension is invariant under derived equivalences

- Mathematics
- 2009

Abstract In this note, we show that the finiteness of the finitistic dimension of a left coherent ring is invariant under derived equivalences.

The Generalized Auslander–Reiten Condition for the bounded derived category

- Mathematics
- 2012

We consider the bounded derived category Db(R mod) of a left Noetherian ring R. We give a version of the Generalized Auslander–Reiten Condition for Db(R mod) that is equivalent to the classical… Expand

Derived equivalences for Cohen-Macaulay Auslander algebras

- Mathematics
- 2010

Let $A$ and $B$ be Gorenstein Artin algebras of Cohen-Macaulay finite type. We prove that, if $A$ and $B$ are derived equivalent, then their Cohen-Macaulay Auslander algebras are also derived… Expand

ON A CONJECTURE OF AUSLANDER AND REITEN

- Mathematics
- 2003

Abstract In studying Nakayama's 1958 conjecture on rings of infinite dominant dimension, Auslander and Reiten proposed the following generalization: Let Λ be an Artin algebra and M a Λ -generator… Expand

Tilting invariance of the Auslander-Reiten conjecture

- Mathematics
- 2010

Let $R$ be an artin algebra and $T$ be a tilting or cotilting $R$-module with $S=\mr{End}_RT$. We show that $R$ satisfies the Auslander-Reiten conjecture if and only if so does $S$.

Derived categories and syzygies

- Mathematics
- 2011

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

Algebras that satisfy Auslander’s condition on vanishing of cohomology

- Mathematics
- 2008

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 to… Expand

Triangulated categories in the representation theory of finite dimensional algebras

- Mathematics
- 1988

Preface 1. Triangulated categories 2. Repetitive algebras 3. Tilting theory 4. Piecewise hereditary algebras 5. Trivial extension algebras References Index.