Tag Archives: colimits

Commutative Algebra 52

Posted on May 10, 2020 by limsup

Direct Limits of Rings Let be a directed system of rings. Regard them as a directed system of abelian groups (i.e. ℤ-modules) and take the direct limit A. Proposition 1. The abelian group A has a natural structure of a … Continue reading →

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Commutative Algebra 51

Posted on May 9, 2020 by limsup

Limits Are Left-Exact By example 6 and proposition 2 in the previous article, one is inclined to conclude that taking the colimit in is a right-exact functor, but there is a rather huge issue here: the functors are between and … Continue reading →

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Commutative Algebra 50

Posted on May 8, 2020 by limsup

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 →

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Commutative Algebra 49

Posted on May 7, 2020 by limsup

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 →

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Commutative Algebra 48

Posted on May 6, 2020 by limsup

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 , , , , , , | 4 Comments