Adic Foxby Classes

TL;DR

This paper explores the properties and stability of Foxby classes associated with adic semidualizing complexes over noetherian rings, extending foundational results like Foxby Equivalence and examining their behavior under various algebraic operations.

Contribution

It advances the understanding of Foxby classes by establishing new stability results, change of rings, and local-global properties for these classes in the context of adic semidualizing complexes.

Findings

Established Foxby Equivalence for adic semidualizing complexes.

Proved stability results under tensor operations with finite flat dimension complexes.

Analyzed change of rings and local-global properties of Foxby classes.

Abstract

We continue our work on adic semidualizing complexes over a commutative noetherian ring $R$ by investigating the associated Auslander and Bass classes (collectively known as Foxby classes), following Foxby and Christensen. Fundamental properties of these classes include Foxby Equivalence, which provides an equivalence between the Auslander and Bass classes associated to a given adic semidualizing complex. We prove a variety of stability results for these classes, for instance, with respect to $F\otimes^{\mathbf{L}}_R-$ where $F$ is an $R$-complex finite flat dimension, including special converses of these results. We also investigate change of rings and local-global properties of these classes.