Tag Archives: exact functors

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 31

Posted on April 19, 2020 by limsup

Flat Modules Recall from proposition 3 here: for an A-module M, is a right-exact functor. Definition. We say M is flat over A (or A-flat) if is an exact functor, equivalently, if A flat A-algebra is an A-algebra which is flat as … Continue reading →

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Commutative Algebra 27

Posted on April 14, 2020 by limsup

Free Modules All modules are over a fixed ring A. We already mentioned finite free modules earlier. Here we will consider general free modules. Definition. Let be any set. The free A-module on I is a direct sum of copies of … Continue reading →

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Commutative Algebra 24

Posted on April 11, 2020 by limsup

Quotient vs Localization Taking the quotient and localization are two sides of the same coin when we look at . Quotient removes the “small” prime ideals in – it only keeps the prime ideals containing . Localization removes the “large” … Continue reading →

Posted in Advanced Algebra | Tagged , , , , | 2 Comments