title: Flatness and Completion Revisited
author: Amnon Yekutieli
publication status: to appear in Algebras and Representation Theory

Abstract: We continue investigating the interaction between flatness and 𝔞-adic completion for infinitely generated modules over a commutative ring A. 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 \hat{A} is not flat. We also study 𝔞-adic systems, and prove that if the ideal 𝔞 is finitely generated, then the limit of any 𝔞-adic system is a complete module.

updated 26 Sep 2017