Difference between revisions of "Affine variety"
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− | + | ''affine algebraic variety'' | |
− | + | A generalization of the concept of an [[Affine algebraic set|affine algebraic set]]. An affine variety is a reduced [[Affine scheme|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==== | ====Comments==== | ||
− | Frequently the name | + | Frequently the name "varietyvariety" means a reduced and irreducible scheme of finite type over an algebraically closed field. |
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR></table> |
Revision as of 16:09, 1 April 2020
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 "varietyvariety" 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 |
Affine variety. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Affine_variety&oldid=15053