
Recent Posts
Archives
 March 2023
 January 2023
 May 2020
 April 2020
 March 2020
 June 2018
 July 2016
 June 2016
 May 2016
 March 2015
 February 2015
 January 2015
 December 2014
 December 2013
 November 2013
 July 2013
 June 2013
 May 2013
 March 2013
 February 2013
 January 2013
 December 2012
 November 2012
 October 2012
 September 2012
 August 2012
 April 2012
 March 2012
 February 2012
 January 2012
 December 2011
 November 2011
 October 2011
Categories
Meta
Pages
Tag Archives: localization
Commutative Algebra 58
We have already seen two forms of unique factorization. In a UFD, every nonzero element is a unique product of irreducible (also prime) elements. In a Dedekind domain, every nonzero 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
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
Commutative Algebra 44
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 nonzero … Continue reading
Posted in Advanced Algebra
Tagged fractional ideals, ideal division, invertible ideals, localization, picard group, principal ideals
5 Comments
Commutative Algebra 40
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
Flat Modules Recall from proposition 3 here: for an Amodule M, is a rightexact functor. Definition. We say M is flat over A (or Aflat) if is an exact functor, equivalently, if A flat Aalgebra is an Aalgebra which is flat as … Continue reading
Commutative Algebra 29
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 welldefined: since only finitely many are nonzero, only finitely … Continue reading
Posted in Advanced Algebra
Tagged hom functor, induced modules, localization, rightexact, tensor products, yoneda lemma
2 Comments
Commutative Algebra 27
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 Amodule on I is a direct sum of copies of … Continue reading
Posted in Advanced Algebra
Tagged exact functors, free modules, leftexact, localization, projective modules, splitting lemma
Leave a comment
Commutative Algebra 25
Arbitrary Collection of Modules Finally, we consider the case where we have potentially infinitely many modules. Proposition 1. For a collection of Amodules , 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
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