
Recent Posts
Archives
 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
Author Archives: limsup
Commutative Algebra 54
Filtered Rings Definition. Let A be a ring. A filtration on A is a sequence of additive subgroups such that for any . A filtered ring is a ring with a designated filtration. Note Since , in fact each is … Continue reading
Posted in Advanced Algebra
Tagged completions, filtrations, formal power series, limits, metric spaces, padic, ultrametric
5 Comments
Commutative Algebra 53
Graded Rings Definition. A grading on a ring A is a collection of additive subgroups such that as abelian groups, and for any , i.e.. A graded ring is a ring A with a specified grading. Note The notation means every … Continue reading
Posted in Advanced Algebra
Tagged graded modules, graded rings, graded submodules, UFDs
Leave a comment
Commutative Algebra 52
Direct Limits of Rings Let be a directed system of rings. Regard them as a directed system of abelian groups (i.e. ℤmodules) and take the direct limit A. Proposition 1. The abelian group A has a natural structure of a … Continue reading
Posted in Advanced Algebra
Tagged adjoint functors, coinduced modules, colimits, duals, leftexact, limits, rightexact, tensor products
Leave a comment
Commutative Algebra 51
Limits Are LeftExact By example 6 and proposition 2 in the previous article, one is inclined to conclude that taking the colimit in is a rightexact functor, but there is a rather huge issue here: the functors are between and … Continue reading
Posted in Advanced Algebra
Tagged colimits, directed limits, directed sets, leftexact, limits, rightexact
2 Comments
Commutative Algebra 50
Adjoint Functors Adjoint functors are a general construct often used for describing universal properties (among other things). Take two categories and . Definition. Covariant functors and are said to be adjoint if we have isomorphisms which are natural in A and … Continue reading
Posted in Advanced Algebra
Tagged adjoint functors, category theory, colimits, hom functor, leftexact, limits, rightexact, tensor products, universal properties
2 Comments
Commutative Algebra 49
Morphism of Diagrams Throughout this article denotes a category and J is an index category. Definition Given diagrams , a morphism is a natural transformation . Thus we have the category of all diagrams in of type J, which we … Continue reading
Posted in Advanced Algebra
Tagged category theory, colimits, coproducts, limits, products, pullbacks, pushouts, universal properties
Leave a comment
Commutative Algebra 48
Introduction For the next few articles we are back to discussing category theory to develop even more concepts. First we will look at limits and colimits, which greatly generalize the concept of products and coproducts and cover loads of interesting … Continue reading
Posted in Advanced Algebra
Tagged category theory, colimits, coproducts, fibres, functors, pullbacks, pushouts
4 Comments
Commutative Algebra 47
Minkowski Theory: Introduction Suppose is a finite extension and is the integral closure of in K. In algebraic number theory, there is a classical method by Minkowski to compute the Picard group of (note: in texts on algebraic number theory, this … Continue reading
Commutative Algebra 46
Properties of Dedekind Domains Throughout this article, A denotes a Dedekind domain. Proposition 1. Every fractional ideal of A can be written as where each is a maximal ideal. The expression is unique up to permutation of terms. Note In … Continue reading
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