Difference between revisions of "Algebraic geometry"
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== Projective Varieties == | == Projective Varieties == | ||
| − | + | Let k be a field. A projective variety over k is a projective scheme over k. Projective varieties are algebraic varieties. | |
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== Schemes == | == Schemes == | ||
Revision as of 10:49, 11 February 2008
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
Let k be a field. A projective variety over k is a projective scheme over k. Projective varieties are algebraic varieties.
Schemes
(See above remark.)
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