Fltness nd Completion Revisited Academic Article uri icon

abstract

  • Abstract 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, 2017