Difference between revisions of "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 $V$.