Category Archives: Advanced Algebra

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

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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

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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. 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

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

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

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

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

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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

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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

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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

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