  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.
