Flatness and completion revisited Academic Article uri icon


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

publication date

  • January 1, 2018