Difference between revisions of "Algebraic geometry"
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== Affine Algebraic Varieties == | == Affine Algebraic Varieties == | ||
| − | One of the first basic objects studied in algebraic geometry is a [[variety]]. Let <math>\mathbb{A}^k<math> denote [[affine]] <math>k</math>-space, i.e. a [[vector space]] of [[dimension]] <math>k</math> over an algebraically closed field, such as the field <math>\mathbb{C}</math> of [[complex number]]s. (We can think of this as <math>k</math>-dimensional ``complex Euclidean'' space.) Let <math>R=\mathbb{C}[X_1,\ldots,X_k]</math> be the [[polynomial ring]] in <math>k</math> variables, and let <math>I</math> be a [[maximal ideal]] of <math>R</math>. Then <math>V(I)=\{p\in\mathbb{A}^k\mid f(p)=0\mathrm{\ for\ all\ } f\in I\}</math> is called an '''affine algebraic variety'''. | + | One of the first basic objects studied in algebraic geometry is a [[variety]]. Let <math>\mathbb{A}^k</math> denote [[affine]] <math>k</math>-space, i.e. a [[vector space]] of [[dimension]] <math>k</math> over an algebraically closed field, such as the field <math>\mathbb{C}</math> of [[complex number]]s. (We can think of this as <math>k</math>-dimensional ``complex Euclidean'' space.) Let <math>R=\mathbb{C}[X_1,\ldots,X_k]</math> be the [[polynomial ring]] in <math>k</math> variables, and let <math>I</math> be a [[maximal ideal]] of <math>R</math>. Then <math>V(I)=\{p\in\mathbb{A}^k\mid f(p)=0\mathrm{\ for\ all\ } f\in I\}</math> is called an '''affine algebraic variety'''. |
== Projective Varieties == | == Projective Varieties == | ||
Revision as of 19:43, 28 June 2006
Algebraic geometry is the study of solutions of polynomial equations by means of abstract algebra, and in particular ring theory. Algebraic geometry is most easily done over algebraically closed fields, but it can also be done more generally over any field or even over rings.
Affine Algebraic Varieties
One of the first basic objects studied in algebraic geometry is a variety. Let 
denote affine 
-space, i.e. a vector space of dimension over an algebraically closed field, such as the field 
of complex numbers. (We can think of this as -dimensional ``complex Euclidean space.) Let ![$R=\mathbb{C}[X_1,\ldots,X_k]$](http://latex.artofproblemsolving.com/4/9/7/4972be20db6d6721a720fa76fd05d0889388bbbe.png)
be the polynomial ring in variables, and let 
be a maximal ideal of 
. Then 
is called an affine algebraic variety.
Projective Varieties
(Someone here knows more algebraic geometry than I do.)
Schemes
(See above remark.)
This article is a stub. Help us out by expanding it.
