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Algebras and Representation Theory

, Volume 21, Issue 4, pp 717–736 | Cite as

Flatness and Completion Revisited

Article

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.

Keywords

Adic completion Adic system Flat module Noetherian ring Weakly proregular ideal 

Mathematics Subject Classification (2010)

Primary 13J10 Secondary 13B35 13E05 13C10 13C11 13D07 

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Notes

Acknowledgments

Thanks to Liran Shaul, Sean Sather-Wagstaff, Asaf Yekutieli, Steven Kleiman, Brian Conrad, Ofer Gabber, Lorenzo Ramero, Pierre Deligne, Johan de Jong, Ilya Tyomkin and Leonid Positselski for helpful discussions. We also wish to thank the anonymous referee, for reading the paper carefully and suggesting several improvements.

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Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Department of MathematicsBen Gurion UniversityBe’er ShevaIsrael

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