Tag Archives: a-adic filtrations

Commutative Algebra 57

Posted on May 15, 2020 by limsup

Continuing from the previous article, A denotes a noetherian ring and all A-modules are finitely generated. As before all completions are taken to be -stable for a fixed ideal . Noetherianness We wish to prove that the -adic completion of … Continue reading →

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Commutative Algebra 56

Posted on May 14, 2020 by limsup

Throughout this article, A denotes a noetherian ring and is a fixed ideal. All A-modules are finitely generated. Consequences of Artin-Rees Lemma Suppose we have an exact sequence of finitely generated A-modules Let M be given the 𝔞-adic filtration; the induced filtration on … Continue reading →

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Commutative Algebra 55

Posted on May 13, 2020 by limsup

Exactness of Completion Throughout this article, A denotes a filtered ring. Proposition 1. Let be a short exact sequence of A-modules. Suppose M is filtered, inducing filtrations on N and P. Then is also exact as -modules. Proof Without loss of … Continue reading →

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