Relations between semidualizing complexes

TL;DR

This paper explores the relationship between semidualizing complexes over rings, focusing on when vanishing Ext conditions imply reflexivity, and introduces an equivalence relation to classify these complexes.

Contribution

It introduces an equivalence relation on semidualizing complexes based on the derived Picard group and provides conditions linking Ext vanishing to reflexivity.

Findings

Conditions equivalent to C-reflexivity are identified.

An equivalence relation on semidualizing complexes is defined and characterized.

The original question is answered in specific cases.

Abstract

We study the following question: Given two semidualizing complexes B and C over a commutative noetherian ring R, does the vanishing of Ext^n_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 Sega. We begin by providing conditions equivalent to B being C-reflexive, each of which is slightly stronger than the condition Ext^n_R(B,C)=0 for all n>>0. We introduce and investigate an equivalence relation \approx 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^n_R(B,C)=0 for all n>>0. Finally, we answer our original question in some special cases.