Derived complete complexes at weakly proregular ideals

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.