Flatness and Completion Revisited

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