اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدFlatness and Completion Revisited
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تأليف
Amnon Yekutieli
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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.
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