Adic semidualizing complexes

TL;DR

This paper introduces and characterizes a new class of $rak{a}$-adic semidualizing modules and complexes, extending existing concepts and providing a novel proof of dualizing complexes' existence over complete local rings.

Contribution

It defines $rak{a}$-adic semidualizing modules and complexes, offers their characterizations, and presents a new proof for the existence of dualizing complexes without Cohen's theorem.

Findings

Introduction of $rak{a}$-adic semidualizing modules and complexes

Equivalent characterizations including semidualizing property

Proof of dualizing complexes existence over complete local rings

Abstract

We introduce and study a class of objects that encompasses Christensen and Foxby's semidualizing modules and complexes and Kubik's quasi-dualizing modules: the class of $\mathfrak{a}$-adic semidualizing modules and complexes. We give examples and equivalent characterizations of these objects, including a characterization in terms of the more familiar semidualizing property. As an application, we give a proof of the existence of dualizing complexes over complete local rings that does not use the Cohen Structure Theorem.