Flatness and Completion Revisited

TL;DR

This paper explores the relationship between flatness and $rak{a}$-adic completion in modules over rings, introducing $rak{a}$-adic flatness, and establishing conditions under which it aligns with flatness, especially in noetherian contexts.

Contribution

It introduces the concept of $rak{a}$-adic flatness, analyzes its preservation under completion, and compares it with flatness in various ring settings, including non-noetherian cases.

Findings

$rak{a}$-adic flatness is preserved under completion for weakly proregular ideals.

In noetherian rings, $rak{a}$-adic flatness coincides with flatness for complete modules.

An example shows that $rak{a}$-adic flatness does not imply flatness in non-noetherian rings.

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.