## You are here

Homevariety

## Primary tabs

# 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*.

Some authors also require $k$ to be algebraically closed, and some authors require curves to be nonsingular.

Calling $X$ a variety would appear to conflict with the preexisting notion of an affine 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

$\left<f_{1},\ldots,f_{m}\right>$ |

is identified with

$\Spec k[X_{1},\ldots,X_{n}]/\left<f_{1},\ldots,f_{m}\right>.$ |

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 $\left<f_{1},\ldots,f_{m}\right>$. 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

$\left<f_{1}(T_{0},\ldots,T_{n}),\ldots,f_{m}(T_{0},\ldots,T_{n})\right>.$ |

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.

For more on this, see Hartshorne’s book *Algebraic Geometry*; see the bibliography for algebraic geometry for more resources.

## Mathematics Subject Classification

14-00*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections