Tag Archives: flat modules

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 →

Posted in Advanced Algebra | Tagged , , , , , , , | 3 Comments

Commutative Algebra 35

Posted on April 23, 2020 by limsup

Noetherian Modules Through this article, A is a fixed ring. For the first two sections, all modules are over A. Recall that a submodule of a finitely generated module is not finitely generated in general. This will not happen if we constrain … Continue reading →

Posted in Advanced Algebra | Tagged , , , , , | Leave a comment

Commutative Algebra 34

Posted on April 22, 2020 by limsup

Nakayama’s Lemma The following is a short statement which has far-reaching applications. Since its main applications are for local rings, we will state the result in this context. Throughout this section, is a fixed local ring. Theorem (Nakayama’s Lemma). Let … Continue reading →

Posted in Advanced Algebra | Tagged , , , , , , | Leave a comment

Commutative Algebra 32

Posted on April 20, 2020 by limsup

Torsion and Flatness Definition. Let A be a ring and M an A-module; let . If satisfies , we call it an –torsion element. If is an -torsion for some non-zero-divisor we call it a torsion element. M is said … Continue reading →

Posted in Advanced Algebra | Tagged , , , , | Leave a comment

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 →

Posted in Advanced Algebra | Tagged , , , , , , | Leave a comment