Flatness and Completion Revisited

We continue investigating the interaction between flatness and a-adic completion for infinitely generated A-modules. Here A is a commutative ring and a is a finitely generated ideal in it. We introduce the concept of a-adic flatness, which is weaker than flatness. We prove that a-adic flatness is preserved under completion when the ideal a is weakly proregular. We also prove that when A is noetherian, a-adic flatness coincides with flatness (for complete modules). An example is worked out of a non-noetherian ring A, with a weakly proregular ideal a, for which the completion  is not flat. We also study a-adic systems, and prove that if the ideal a is finitely generated, then the limit of every a-adic system is a complete module.