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Author Archives: limsup
Commutative Algebra 57
Continuing from the previous article, A denotes a noetherian ring and all A-modules are finitely generated. As before all completions are taken to be -stable for a fixed ideal . Noetherianness We wish to prove that the -adic completion of … Continue reading
Posted in Advanced Algebra
Tagged a-adic filtrations, algebraic geometry, analysis, completion, filtrations, hensels lemma, local rings, p-adic
1 Comment
Commutative Algebra 56
Throughout this article, A denotes a noetherian ring and is a fixed ideal. All A-modules are finitely generated. Consequences of Artin-Rees Lemma Suppose we have an exact sequence of finitely generated A-modules Let M be given the 𝔞-adic filtration; the induced filtration on … Continue reading
Commutative Algebra 55
Exactness of Completion Throughout this article, A denotes a filtered ring. Proposition 1. Let be a short exact sequence of A-modules. Suppose M is filtered, inducing filtrations on N and P. Then is also exact as -modules. Proof Without loss of … Continue reading
Posted in Advanced Algebra
Tagged a-adic filtrations, artin-rees lemma, blowup algebras, completions, filtrations, limits, noetherian
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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, p-adic, 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
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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
Posted in Advanced Algebra
Tagged adjoint functors, coinduced modules, colimits, duals, left-exact, limits, right-exact, tensor products
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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
Posted in Advanced Algebra
Tagged colimits, directed limits, directed sets, left-exact, limits, right-exact
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, left-exact, limits, right-exact, 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
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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
Posted in Advanced Algebra
Tagged category theory, colimits, coproducts, fibres, functors, pullbacks, pushouts
4 Comments
Mathematics and Such 