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''affine algebraic 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]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a0111601.png" /> of finite type over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a0111602.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a0111603.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a0111604.png" /> is a commutative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a0111605.png" />-algebra of finite type without nilpotent elements. The affine variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a0111606.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a0111607.png" /> is the ring of polynomials over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a0111608.png" />, is called affine space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a0111609.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a01116010.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a01116011.png" /> of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a01116012.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a01116013.png" /> defines a surjective homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a01116014.png" />, defined by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a01116015.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a01116016.png" /> be the algebraic closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a01116017.png" />. The subset of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a01116018.png" /> consisting of the common zeros of all the polynomials of the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a01116019.png" /> is an affine algebraic set over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a01116020.png" />. The coordinate ring of such an affine algebraic set is isomorphic to the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a01116021.png" />. Each affine algebraic set over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a01116022.png" /> in turn defines an algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a01116023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a01116024.png" /> is the coordinate ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a01116025.png" />. 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.
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To each affine variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a01116026.png" /> a functor on the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a01116027.png" />-algebras is assigned. It is defined by the correspondence:
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''affine algebraic variety''
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a01116028.png" /></td> </tr></table>
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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 $
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of finite type over a field  $  k $,
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i.e.  $  X = { \mathop{\rm Spec} }  A $,
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where  $  A $
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is a commutative  $  k $-
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algebra of finite type without nilpotent elements. The affine variety  $  X = { \mathop{\rm Spec} }  k[ T _ {1} \dots T _ {n} ] $,
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where  $  k[T _ {1} \dots T _ {n} ] $
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is the ring of polynomials over  $  k $,
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is called affine space over  $  k $
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and is denoted by  $  \mathbf A _ {k}  ^ {n} $.
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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} $
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of a  $  k $-
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algebra  $  A $
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defines a surjective homomorphism  $  \phi : k[ T _ {1} \dots T _ {n} ] \rightarrow A $,
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defined by the formula  $  \phi ( T _ {i} ) = x _ {i} $.
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Let  $  \overline{k}\; $
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be the algebraic closure of  $  k $.
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The subset of the set  $  \overline{k}\; ^ {n} $
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consisting of the common zeros of all the polynomials of the ideal  $  { \mathop{\rm ker} }  \phi $
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is an affine algebraic set over  $  k $.
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The coordinate ring of such an affine algebraic set is isomorphic to the ring  $  A $.  
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Each affine algebraic set over  $  k $
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in turn defines an algebraic variety  $  { \mathop{\rm Spec} }  k[X] $,
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where  $  k[X] $
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is the coordinate ring of  $  X $.  
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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.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a01116029.png" /> (respectively, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a01116030.png" />), the elements of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a01116031.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a01116032.png" />) are called geometric (respectively, rational) points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a01116033.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a01116034.png" /> is in a bijective correspondence with the set of maximal ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a01116035.png" /> of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a01116036.png" />, and with the set of points of an algebraic set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a01116037.png" /> whose coordinate ring is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a01116038.png" />. The spectral topology in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a01116039.png" /> induces on the everywhere-dense subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a01116040.png" /> a topology which corresponds to the Zariski topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011160/a01116041.png" />.
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To each affine variety  $  X = { \mathop{\rm Spec} }  A $
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a functor on the category of $  k $-
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algebras is assigned. It is defined by the correspondence:
  
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$$
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B  \rightarrow  X (B)  =  { \mathop{\rm Hom} } _ {k-  \mathop{\rm alg}  } ( A , B ).
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$$
  
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If  $  B = \overline{k}\; $(
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respectively, if  $  B = k $),
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the elements of the set  $  X ( \overline{k}\; ) $(
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respectively,  $  X(k) $)
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are called geometric (respectively, rational) points of  $  X $.
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The set  $  X( \overline{k}\; ) $
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is in a bijective correspondence with the set of maximal ideals  $  { \mathop{\rm Specm} }  (A) $
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of the ring  $  A $,
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and with the set of points of an algebraic set  $  V $
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whose coordinate ring is isomorphic to  $  A $.
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The spectral topology in the space  $  X $
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induces on the everywhere-dense subset  $  { \mathop{\rm Specm} }  (A) $
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a topology which corresponds to the Zariski topology on  $  V $.
  
 
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Latest 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
How to Cite This Entry:
Affine variety. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Affine_variety&oldid=45051
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article