Weak proregularity, derived completion, adic flatness, and prisms

We begin by recalling the role that weak proregularity of an ideal in a commutative ring has in derived completion and adic flatness. We also introduce the new concepts of idealistic and sequential derived completion, and prove a few results about them, including the fact that these two concepts agree iff the ideal is weakly proregular. Next we study the local nature of weak proregularity, and its behavior w.r.t. ring quotients. These results allow us to prove our main theorem, which states that weak proregularity occurs in the context of bounded prisms. Prisms belong to the new groundbreaking theory of perfectoid rings, developed by Scholze and his collaborators. Since perfectoid ring theory makes heavy use of derived completion and adic flatness, we anticipate that our results shall help simplify and improve some of the more technical aspects of this theory.