variety

Definition 1

Let $X$ be a scheme over a field $k$. Then $X$ is said to be an abstract variety over $k$ if it is integral, separated, and of finite type over $k$. Usually we simply say $X$ is a variety. If $X$ is proper over $k$, it is said to be complete  . If the dimension  of $X$ is one, then $X$ is said to be a curve.

Calling $X$ a variety would appear to conflict with the preexisting notion of an affine (http://planetmath.org/AffineVariety) or projective variety. However, it can be shown that if $k$ is algebraically closed, then there is an equivalence of categories between affine abstract varieties over $k$ and affine varieties  over $k$, and another between projective abstract varieties over $k$ and projective varieties over $k$.

This equivalence of categories identifies an abstract variety with the set of its $k$-points; this can be thought of as simply ignoring all the generic points. In the other direction, it identifies an affine variety with the prime spectrum of its coordinate ring: the variety in $\mathbb{A}^{n}$ defined by the ideal

 $\left$

is identified with

 $\operatorname{Spec}k[X_{1},\ldots,X_{n}]/\left.$

A projective variety is identified as the gluing together of the affine varieties obtained by taking the complements of hyperplanes   . To see this, suppose we have a projective variety in $\mathbb{P}^{n}$ given by the homogeneous ideal $\left$. If we delete the hyperplane $X_{i}=0$, then we obtain an affine variety: let $T_{j}=X_{j}/X_{i}$; then the affine variety is the set of common zeros of

 $\left.$

In this way, we can get $n+1$ overlapping affine varieties that cover our original projective variety. Using the theory of schemes, we can glue these affine varieties together to get a scheme; the result will be projective.

 Title variety Canonical name Variety Date of creation 2013-03-22 14:16:43 Last modified on 2013-03-22 14:16:43 Owner mps (409) Last modified by mps (409) Numerical id 9 Author mps (409) Entry type Definition Classification msc 14-00 Synonym abstract variety Related topic Scheme Related topic AffineVariety Related topic ProjectiveVariety Defines complete Defines curve