On a generalization of two results of Happel to commutative rings

TL;DR

This paper generalizes two of Happel's results to commutative rings, characterizing when certain derived and homotopy categories have Auslander-Reiten triangles based on properties like regularity and Gorenstein conditions.

Contribution

It extends Happel's theorems to commutative rings, linking AR-triangles in derived and homotopy categories to regularity and Gorenstein properties.

Findings

D^b_f(mod A) has AR-triangles iff A is regular.

K^b_f(proj A) has AR-triangles if A is complete and Gorenstein.

If K^b_f(proj A) has AR-triangles and A is Cohen-Macaulay or dim A=1, then A is Gorenstein.

Abstract

In this paper we extend two results of Happel to commutative rings. Let $(A, \mathfrak{m})$ be a commutative Noetherian local ring. Let $D^b_f(mod \ A)$ be the bounded derived category of complexes of finitely generated modules over $A$ with finite length cohomology. We show $D^b_f( mod \ A)$ has Auslander-Reiten(AR)-triangles if and only if $A$ is regular. Let $K^b_f(proj \ A)$ be the homotopy category of finite complexes of finitely generated free $A$-modules with finite length cohomology. We show that if $A$ is complete and if $A$ is Gorenstein then $K^b_f( proj \ A)$ has AR triangles. Conversely we show that if $K^b_f(proj \ A)$ has AR triangles and if $A$ is Cohen-Macaulay or if $\dim A = 1$ then $A$ is Gorenstein.