# Affine variety

*affine algebraic variety*

A generalization of the concept of an affine algebraic set. An affine variety is a reduced affine scheme $ X $
of finite type over a field $ k $,
*i.e.* $ X = { \mathop{\rm Spec} } A $,
where $ A $
is a commutative $ k $-
algebra of finite type without nilpotent elements. The affine variety $ X = { \mathop{\rm Spec} } k[ T _ {1} \dots T _ {n} ] $,
where $ k[T _ {1} \dots T _ {n} ] $
is the ring of polynomials over $ k $,
is called affine space over $ k $
and is denoted by $ \mathbf A _ {k} ^ {n} $.
An affine scheme is an affine variety if and only if it is isomorphic to a reduced closed subscheme of an affine space. Each system of generators $ x _ {1} \dots x _ {n} $
of a $ k $-
algebra $ A $
defines a surjective homomorphism $ \phi : k[ T _ {1} \dots T _ {n} ] \rightarrow A $,
defined by the formula $ \phi ( T _ {i} ) = x _ {i} $.
Let $ \overline{k}\; $
be the algebraic closure of $ k $.
The subset of the set $ \overline{k}\; ^ {n} $
consisting of the common zeros of all the polynomials of the ideal $ { \mathop{\rm ker} } \phi $
is an affine algebraic set over $ k $.
The coordinate ring of such an affine algebraic set is isomorphic to the ring $ A $.
Each affine algebraic set over $ k $
in turn defines an algebraic variety $ { \mathop{\rm Spec} } k[X] $,
where $ k[X] $
is the coordinate ring of $ X $.
The set of points of an affine variety is in a one-to-one correspondence with the irreducible subvarieties of the corresponding affine algebraic set.

To each affine variety $ X = { \mathop{\rm Spec} } A $ a functor on the category of $ k $- algebras is assigned. It is defined by the correspondence:

$$ B \rightarrow X (B) = { \mathop{\rm Hom} } _ {k- \mathop{\rm alg} } ( A , B ). $$

If $ B = \overline{k}\; $( respectively, if $ B = k $), the elements of the set $X(\overline{k})$ (respectively, $ X(k) $) are called geometric (respectively, rational) points of $ X $. The set $ X( \overline{k}) $ is in a bijective correspondence with the set of maximal ideals $ { \mathop{\rm Specm} } (A) $ of the ring $ A $, and with the set of points of an algebraic set $ V $ whose coordinate ring is isomorphic to $ A $. The spectral topology in the space $ X $ induces on the everywhere-dense subset $ { \mathop{\rm Specm} } (A) $ a topology which corresponds to the Zariski topology on $ V $.

#### Comments

Frequently the name "variety" means a reduced and irreducible scheme of finite type over an algebraically closed field.

#### References

- [a1] I.R. Shafarevich, "Basic algebraic geometry", Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001

**How to Cite This Entry:**

Affine variety.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Affine_variety&oldid=52610