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Tag Archives: topology
Commutative Algebra 13
Zariski Topology for Rings In this article, we generalize earlier results in algebraic geometry to apply to general rings. Recall that points on an affine variety V correspond to maximal ideals . For general rings, we have to switch to … Continue reading
Posted in Advanced Algebra
Tagged homomorphism, maximal ideals, prime ideals, rings, spectrum, topology, zariski topology
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Commutative Algebra 5
Morphisms in Algebraic Geometry Next we study the “nice” functions between closed subspaces of . Definition. Suppose and are closed subsets. A morphism is a function which can be expressed as: for some polynomials . We also say f is a regular … Continue reading
Posted in Advanced Algebra
Tagged algebraic geometry, continuity, ideals, morphisms, rings, topology
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Commutative Algebra 3
Algebraic Geometry Concepts We have decided to introduce, at this early point, some basics of algebraic geometry in order to motivate the later concepts. In summary, algebraic geometry studies solutions to polynomial equations over a field. First we consider a … Continue reading
Posted in Advanced Algebra
Tagged algebraic geometry, commutative rings, ideals, nullstellensatz, radical ideals, rings, topology, zariski topology
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From Euler Characteristics to Cohomology (II)
Boundary Maps Here’s a brief recap of the previous article: we learnt that in refining a cell decomposition of an object M, we can, at each step, pick an i-dimensional cell and divide it in two. In this way, we … Continue reading
Posted in Notes
Tagged advanced, betti numbers, cell complexes, cellular homology, euler characteristics, homology, simplicial complexes, topology
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From Euler Characteristics to Cohomology (I)
[ Warning: this is primarily an expository article, so the proofs are not airtight, but they should be sufficiently convincing. ] The five platonic solids were well-known among the ancient Greeks (V, E, F denote the number of vertices, edges and faces respectively): … Continue reading
Topology: More on Algebra and Topology
We’ve arrived at the domain where topology meets algebra. Thus we have to proceed carefully to ensure that the topology of our algebraic constructions are well-behaved. Let’s look at topological groups again. Our first task is to show that the … Continue reading
Posted in Notes
Tagged algebra, connected spaces, groups, isomorphism theorems, open maps, orthogonal groups, quotient topology, rings, topology
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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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