On a generalization of two results of Happel to commutative rings

Abstract

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