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Tag Archives: limits
Commutative Algebra 55
Exactness of Completion Throughout this article, A denotes a filtered ring. Proposition 1. Let be a short exact sequence of Amodules. 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 aadic filtrations, artinrees 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, padic, ultrametric
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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. 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
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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
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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
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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Topology: Nets and Points of Accumulation
Recall that a sequence in a topological space X converges to a in X if the function f : N* → X which takes is continuous at . Unrolling the definition, it means that for any open subset U of X containing a, the set contains (N, ∞] for some N. In … Continue reading
Posted in Notes
Tagged advanced, closed subsets, continuity, convergence, limits, metric spaces, nets, points of accumulation, sequences, topology
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