ar X iv : 0 71 2 . 32 75 v 1 [ m at h . A C ] 1 9 D ec 2 00 7 RELATIONS BETWEEN SEMIDUALIZING COMPLEXES

We study the following question: Given two semidualizing complexes B and C over a commutative noetherian ring R, does the vanishing of Ext R (B, C) for n ≫ 0 imply that B is C-reflexive? This question is a natural generalization of one studied by Avramov, Buchweitz, and Şega. We begin by providing conditions equivalent to B being C-reflexive, each of which is slightly stronger than the condition Ext R (B, C) = 0 for all n ≫ 0. We introduce and investigate an equivalence relation ≈ on the set of isomorphism classes of semidualizing complexes. This relation is defined in terms of a natural action of the derived Picard group and is well-suited for the study of semidualizing complexes over nonlocal rings. We identify numerous alternate characterizations of this relation, each of which includes the condition Ext R (B, C) = 0 for all n ≫ 0. Finally, we answer our original question in some special cases.