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.
Electronic Preprint: arxiv:1606.01832
DOI https://doi.org/10.1007/s10468-017-9735-7 (resricted access)
updated 26 Sep 2017