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