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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
5 Comments
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
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
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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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Topology: Limits and Convergence
Following what we did for real analysis, we have the following definition of limits. Definition of Limits. Let X, Y be topological spaces and . If f : X{a} → Y is a function, then we write if the function: is … Continue reading
Posted in Notes
Tagged advanced, continuity, convergence, extended reals, Hausdorff, limits, sequences, topology
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Basic Analysis: Limits and Continuity (3)
Let’s consider multivariate functions where . To that end, we need the Euclidean distance function on Rn. If x = (x1, x2, …, xn) is in Rn, we define: Note that x = 0 if and only if x is the zero vector 0. Now we are ready … Continue reading
Posted in Notes
Tagged advanced, analysis, continuity, limits, multivariate, open balls, open subsets, topology
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Basic Analysis: Limits and Continuity (2)
Previously, we defined continuous limits and proved some basic properties. Here, we’ll try to port over more results from the case of limits of sequences. Monotone Convergence Theorem. If f(x) is increasing on the open interval (c, a) and has … Continue reading
Posted in Notes
Tagged advanced, analysis, continuity, limits, monotone convergence theorem, points of accumulation, squeeze theorem
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