Derived equivalences for Cohen–Macaulay Auslander algebras

Shengyong Pan

doi:10.1016/j.jpaa.2011.06.017

Abstract

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

Highlights

Triangulated categories and derived categories were introduced by Grothendieck and Verdier [27]
In the representation theory of algebras, we will restrict our attention to the equivalences of derived categories, that is, derived equivalences
Hochschild homology and cohomology [26], finiteness of finitistic dimension [23] have been shown to be invariant under derived equivalences

Summary

Introduction

Triangulated categories and derived categories were introduced by Grothendieck and Verdier [27] Today, they have widely been used in many branches: algebraic geometry, stable homotopy theory, representation theory, etc. It is of interest to construct a new derived equivalence from given one by finding a suitable tilting complex. In [17] they constructed new derived equivalences between Φ-Auslander-Yoneda algebras from a given almost ν-stable equivalence. In [17, Corollary 3.13] Hu and Xi proved that, if two representation finite self-injective Artin algebras are derived equivalent, their Auslander algebras are derived equivalent. We generalize their result and prove that, if two Cohen-Macaulay finite Gorenstein Artin algebras are derived equivalent, their Cohen-Macaulay Auslander algebras are derived equivalent.

Preliminaries

Cohen-Macaulay Auslander algebras

The proof of the main result