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Tag Archives: universal properties
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 28
Tensor Products In this article (and the next few), we will discuss tensor products of modules over a ring. Here is a motivating example of tensor products. Example If and are real vector spaces, then is the vector space with … Continue reading
Posted in Advanced Algebra
Tagged bilinear maps, distributive property, modules, tensor product, universal properties
2 Comments
Commutative Algebra 24
Quotient vs Localization Taking the quotient and localization are two sides of the same coin when we look at . Quotient removes the “small” prime ideals in – it only keeps the prime ideals containing . Localization removes the “large” … Continue reading
Posted in Advanced Algebra
Tagged algebras, exact functors, induced modules, localization, universal properties
2 Comments
Commutative Algebra 22
Localization Recall that given an integral domain, there is a canonical way to construct the “smallest field containing it”, its field of fractions. Here, we will generalize this construction to arbitrary rings. We let A be a fixed ring throughout. Definition. … Continue reading
Posted in Advanced Algebra
Tagged field of fractions, ideals, localization, universal properties
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Commutative Algebra 9
Direct Sums and Direct Products Recall that for a ring A, a sequence of A-modules gives the A-module where the operations are defined component-wise. In this article, we will generalize the construction to an infinite collection of modules. Throughout this article, let denote … Continue reading
Posted in Advanced Algebra
Tagged direct products, direct sums, modules, rings, universal properties
5 Comments
Polynomials and Representations XXXVII
Notations and Recollections For a partition , one takes its Young diagram comprising of boxes. A filling is given by a function for some positive integer m. When m=d, we will require the filling to be bijective, i.e. T contains {1,…,d} and each element occurs exactly … Continue reading
Tensor Product and Linear Algebra
Tensor products can be rather intimidating for first-timers, so we’ll start with the simplest case: that of vector spaces over a field K. Suppose V and W are finite-dimensional vector spaces over K, with bases and respectively. Then the tensor product is the vector … Continue reading
Posted in Notes
Tagged bilinear maps, duals, linear algebra, tensor algebra, tensor products, universal properties
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Topology: Quotients of Topological Groups
Topology for Coset Space This is really a continuation from the previous article. Let G be a topological group and H a subgroup of G. The collection of left cosets G/H is then given the quotient topology. This quotient space, however, satisfies an additional … Continue reading
Posted in Notes
Tagged advanced, group quotients, open maps, quotient topology, topological groups, topology, universal properties
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Topology: Quotient Topology and Gluing
In topology, there’s the concept of gluing points or subspaces together. For example, take the closed interval X = [0, 1] and glue the endpoints 0 and 1 together. Pictorially, we get: That looks like a circle, but to prove it’s … Continue reading
Posted in Notes
Tagged advanced, gluing, klein bottle, mobius strip, quotient topology, topological groups, topology, universal properties
2 Comments
Mathematics and Such 