
Recent Posts
Archives
 March 2023
 January 2023
 May 2020
 April 2020
 March 2020
 June 2018
 July 2016
 June 2016
 May 2016
 March 2015
 February 2015
 January 2015
 December 2014
 December 2013
 November 2013
 July 2013
 June 2013
 May 2013
 March 2013
 February 2013
 January 2013
 December 2012
 November 2012
 October 2012
 September 2012
 August 2012
 April 2012
 March 2012
 February 2012
 January 2012
 December 2011
 November 2011
 October 2011
Categories
Meta
Pages
Tag Archives: zariski topology
Commutative Algebra 36
In this article, we will study the topology of Spec A when A is noetherian. For starters, let us consider irreducible topological spaces in greater detail. Irreducible Spaces Recall that an irreducible topological space is a nonempty space X satisfying any of the … Continue reading
Commutative Algebra 23
Localization and Spectrum Recall that the ideals of correspond to a subset of the ideals of A. If we restrict ourselves to prime ideals, we get the following nice bijection. Theorem 1. The above gives a bijection between Useful trick If … Continue reading
Posted in Advanced Algebra
Tagged algebraic geometry, local rings, localization, prime ideals, rational functions, spectrum, zariski topology
12 Comments
Commutative Algebra 14
Basic Open Sets For , let , an open subset of Spec A. Note that . Proposition 1. The collection of over all forms a basis for the topology of . Proof Let be an open subset of Spec A. Suppose … Continue reading
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
Leave a comment
Commutative Algebra 4
More Concepts in Algebraic Geometry As before, k denotes an algebraically closed field. Recall that we have a bijection between radical ideals of and closed subsets of . The bijection reverses the inclusion so if and only if . Not too … Continue reading
Posted in Advanced Algebra
Tagged algebraic geometry, closed subsets, ideals, irreducible spaces, rings, zariski topology
7 Comments
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
Leave a comment