Tag Archives: localization

Commutative Algebra 60

Posted on May 18, 2020 by limsup

Primary Decomposition of Ideals Definition. Let be a proper ideal. A primary decomposition of is its primary decomposition as an A-submodule of A: where each is -primary for some prime , i.e. . Here is a quick way to determine if … Continue reading →

Posted in Advanced Algebra | Tagged associated primes, embedded primes, localization, minimal primes, primary decompositions, primary ideals | Leave a comment

Commutative Algebra 58

Posted on May 16, 2020 by limsup

We have already seen two forms of unique factorization. In a UFD, every non-zero element is a unique product of irreducible (also prime) elements. In a Dedekind domain, every non-zero ideal is a unique product of maximal ideals. Here, we … Continue reading →

Posted in Advanced Algebra | Tagged annihilators, associated primes, localization, module division, modules, supports | 4 Comments

Commutative Algebra 45

Posted on May 3, 2020 by limsup

Invertibility is Local In this article, we again let A be an integral domain and K its field of fractions. We continue our discussion of invertible fractional ideals of A. Proposition 1. A fractional ideal M of A is invertible if and … Continue reading →

Posted in Advanced Algebra | Tagged dedekind domains, discrete valuation rings, invertible ideals, local properties, localization, nakayamas lemma, normal domains | 7 Comments

Commutative Algebra 44

Posted on May 2, 2020 by limsup

Fractional Ideals Throughout this article, let A be an integral domain and K its field of fractions. We do not assume the ring to be noetherian. The objective here is to develop the theory of multiplying and dividing certain classes of non-zero … Continue reading →

Posted in Advanced Algebra | Tagged fractional ideals, ideal division, invertible ideals, localization, picard group, principal ideals | 5 Comments

Commutative Algebra 40

Posted on April 28, 2020 by limsup

More on Integrality Lemma 1. Let be an integral extension. If is an ideal and , the resulting injection is an integral extension. Proof Any element of can be written as , . Then x satisfies a monic polynomial relation: . … Continue reading →

Posted in Advanced Algebra | Tagged closed maps, fibres, finite extensions, going up, integral extensions, krull dimension, localization | 4 Comments

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 direct sums, exact functors, flat modules, local properties, localization, projective modules, tensor products | Leave a comment

Commutative Algebra 29

Posted on April 16, 2020 by limsup

Distributivity Finally, tensor product is distributive over arbitrary direct sums. Proposition 1. Given any family of modules , we have: Proof Take the map which takes . Note that this is well-defined: since only finitely many are non-zero, only finitely … Continue reading →

Posted in Advanced Algebra | Tagged hom functor, induced modules, localization, right-exact, tensor products, yoneda lemma | 2 Comments

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 →

Posted in Advanced Algebra | Tagged exact functors, free modules, left-exact, localization, projective modules, splitting lemma | Leave a comment

Commutative Algebra 25

Posted on April 12, 2020 by limsup

Arbitrary Collection of Modules Finally, we consider the case where we have potentially infinitely many modules. Proposition 1. For a collection of A-modules , we have Proof First claim: we will show that the LHS satisfies the universal property for … Continue reading →

Posted in Advanced Algebra | Tagged direct products, direct sums, exact sequences, local properties, local rings, localization | Leave a comment

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 algebras, exact functors, induced modules, localization, universal properties | 2 Comments