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Difference between revisions of "Affine variety"

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A generalization of the concept of an [[Affine algebraic set|affine algebraic set]]. An affine variety is a reduced [[Affine scheme|affine scheme]]  $  X $
 
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 $,  
 
of finite type over a field  $  k $,  
i.e.  $  X = { \mathop{\rm Spec} }  A $,  
+
''i.e.'' $  X = { \mathop{\rm Spec} }  A $,  
 
where  $  A $
 
where  $  A $
 
is a commutative  $  k $-
 
is a commutative  $  k $-
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If  $  B = \overline{k}\; $(
 
If  $  B = \overline{k}\; $(
 
respectively, if  $  B = k $),  
 
respectively, if  $  B = k $),  
the elements of the set $ X ( \overline{k}\; ) $(
+
the elements of the set $X(\overline{k})$ (respectively,  $  X(k) $)  
respectively,  $  X(k) $)  
 
 
are called geometric (respectively, rational) points of  $  X $.  
 
are called geometric (respectively, rational) points of  $  X $.  
The set  $  X( \overline{k}\; ) $
+
The set  $  X( \overline{k}) $
 
is in a bijective correspondence with the set of maximal ideals  $  { \mathop{\rm Specm} }  (A) $
 
is in a bijective correspondence with the set of maximal ideals  $  { \mathop{\rm Specm} }  (A) $
 
of the ring  $  A $,  
 
of the ring  $  A $,  
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The spectral topology in the space  $  X $
 
The spectral topology in the space  $  X $
 
induces on the everywhere-dense subset  $  { \mathop{\rm Specm} }  (A) $
 
induces on the everywhere-dense subset  $  { \mathop{\rm Specm} }  (A) $
a topology which corresponds to the Zariski topology on  $  V $.
+
a topology which corresponds to the [[Zariski topology]] on  $  V $.
  
 
====Comments====
 
====Comments====
Frequently the name "variety" means a reduced and irreducible scheme of finite type over an algebraically closed field.
+
Frequently the name "variety" 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"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR></table>
+
 
 +
* {{Ref|a1}} I.R. Shafarevich, "Basic algebraic geometry", Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}}

Latest revision as of 20:48, 15 March 2023


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=52531
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article