Tag Archives: limits

Commutative Algebra 55

Posted on May 13, 2020 by limsup

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 | Leave a comment

Commutative Algebra 54

Posted on May 12, 2020 by limsup

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 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 →

Posted in Advanced Algebra | Tagged adjoint functors, coinduced modules, colimits, duals, left-exact, limits, right-exact, tensor products | Leave a comment

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 →

Posted in Advanced Algebra | Tagged colimits, directed limits, directed sets, left-exact, limits, right-exact | 2 Comments

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 →

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

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 →

Posted in Advanced Algebra | Tagged category theory, colimits, coproducts, limits, products, pullbacks, pushouts, universal properties | Leave a comment

Topology: Nets and Points of Accumulation

Posted on February 5, 2013 by limsup

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 | Leave a comment

Topology: Limits and Convergence

Posted on February 2, 2013 by limsup

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 | Leave a comment

Basic Analysis: Limits and Continuity (3)

Posted on November 25, 2012 by limsup

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 | Leave a comment

Basic Analysis: Limits and Continuity (2)

Posted on November 23, 2012 by limsup

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 | Leave a comment