Perfect complexes and completion

TL;DR

This paper establishes a criterion for when perfect complexes over the I-adic completion of a ring are equivalent to dualizable I-complete complexes, generalizing recent results in the noetherian local case.

Contribution

It provides a necessary and sufficient condition for the equivalence of perfect complexes and dualizable I-complete complexes, extending known results to broader classes of rings.

Findings

Criterion always holds for noetherian rings.

Recovers recent results for local noetherian rings with maximal ideal.

Characterizes perfect complexes in terms of dualizability in derived categories.

Abstract

Let $\hat{R}$ be the $I$-adic completion of a commutative ring $R$ with respect to a finitely generated ideal $I$. We give a necessary and sufficient criterion for the category of perfect complexes over $\hat{R}$ to be equivalent to the subcategory of dualizable objects in the derived category of $I$-complete complexes of $R$-modules. Our criterion is always satisfied when $R$ is noetherian. When specialized to $R$ local and noetherian and to $I$ the maximal ideal, our theorem recovers a recent result of Benson, Iyengar, Krause and Pevtsova.