The set of semidualizing complexes is a nontrivial metric space

Abstract

We show that the set $\s(R)$ of shift-isomorphism classes of semidualizing complexes over a local ring $R$ admits a nontrivial metric. We investigate the interplay between the metric and several algebraic operations. Motivated by the dagger duality isometry, we prove the following: If $K,L$ are homologically bounded below and degreewise finite $R$-complexes such that $K\lotimes_R K\lotimes_R L$ is semidualizing, then $K$ is shift-isomorphic to $R$. In investigating the existence of nontrivial open balls in $\s(R)$, we prove that $\s(R)$ contains elements that are not comparable in the reflexivity ordering if and only if it contains at least three distinct elements.