Derived Complete Complexes at Weakly Proregular Ideals

TL;DR

This paper explores the properties and structures of derived complete complexes in rings with weakly proregular ideals, extending classical results and examining their behavior under ring completion.

Contribution

It introduces new characterizations and structural insights into derived complete complexes associated with weakly proregular ideals, including a derived Nakayama theorem.

Findings

Weak proregularity often occurs in non-noetherian rings used in cohomology theories.

The paper characterizes derived complete complexes in terms of adic flatness.

Weak proregularity is preserved under ring completion.

Abstract

Weak proregularity of an ideal in a commutative ring is a subtle generalization of the noetherian property of the ring. Weak proregularity is of special importance for the study of derived completion, and it occurs quite often in non-noetherian rings arising in Hochschild and prismatic cohomologies. This paper is about several related topics: adically flat modules, recognizing derived complete complexes, the structure of the category of derived complete complexes, and a derived complete Nakayama theorem - all with respect to a weakly proregular ideal; and the preservation of weak proregularity under completion of the ring.