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''affine algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a0109402.png" />-set''
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''affine algebraic $k$-set''
  
The set of solutions of a given system of algebraic equations. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a0109403.png" /> be a field and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a0109404.png" /> be its algebraic closure. A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a0109405.png" /> of the Cartesian product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a0109406.png" /> is said to be an affine algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a0109408.png" />-set if its points are the common zeros of some family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a0109409.png" /> of the [[Ring of polynomials|ring of polynomials]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094010.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094011.png" /> of all polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094012.png" /> that vanish on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094013.png" /> forms an ideal, the so-called ideal of the affine algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094015.png" />-set. The ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094016.png" /> coincides with the radical of the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094017.png" /> generated by the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094018.png" />, i.e. with the set of polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094019.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094020.png" /> for some natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094021.png" /> (Hilbert's Nullstellensatz; cf. [[Hilbert theorem|Hilbert theorem]] 3)). Two affine algebraic sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094023.png" /> coincide if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094024.png" />. The affine algebraic set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094025.png" /> can be defined by a system of generators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094026.png" />. In particular, any affine algebraic set can be defined by a finite number of polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094027.png" />. The equalities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094028.png" /> are called the equations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094029.png" />. The affine algebraic sets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094030.png" /> form a lattice with respect to the operations of intersection and union. The ideal of the intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094031.png" /> is identical with the sum of their ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094032.png" />, while the ideal of the union <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094033.png" /> is identical with the intersection of their ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094034.png" />. Any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094035.png" /> is an affine algebraic set, called an affine space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094036.png" /> and denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094037.png" />; to it corresponds the zero ideal. The empty subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094038.png" /> is also an affine algebraic set with the unit ideal. The quotient ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094039.png" /> is called the coordinate ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094040.png" />. It is identical with the ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094041.png" />-regular functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094042.png" />, i.e. with the ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094043.png" />-valued functions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094044.png" />, for which there exists a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094045.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094046.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094047.png" />. An affine algebraic set is said to be irreducible if it is not the union of two affine algebraic proper subsets. An equivalent definition is that the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094048.png" /> is prime. Irreducible affine algebraic sets together with projective algebraic sets were the subjects of classical algebraic geometry. They were called, respectively, affine algebraic varieties and projective algebraic varieties over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094049.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094050.png" />-varieties). Affine algebraic sets have the structure of a topological space. The affine algebraic subsets are the closed sets of this topology (the [[Zariski topology|Zariski topology]]). An affine algebraic set is irreducible if and only if it is irreducible as a topological space. Further development of the concept of an affine algebraic set leads to the concepts of an [[Affine variety|affine variety]] and an [[Affine scheme|affine scheme]].
+
The set of solutions of a given system of algebraic equations. Let $k$
 +
be a field and let $\bar k$ be its algebraic closure. A subset $X$ of the
 +
Cartesian product ${\bar k}^n$ is said to be an affine algebraic $k$-set if its
 +
points are the common zeros of some family $S$ of the
 +
[[Ring of polynomials|ring of polynomials]] $k[T]=k[T_1,\dots,T_n]$. The set ${\mathfrak A}_X$ of all
 +
polynomials in $k[T_1,\dots,T_n]$ that vanish on $X$ forms an ideal, the so-called
 +
ideal of the affine algebraic $k$-set. The ideal ${\mathfrak A}_X$ coincides with
 +
the radical of the ideal $I(S)$ generated by the family $S$, i.e. with
 +
the set of polynomials $f\in k[T_1,\dots,T_n]$ such that $f^m \in I(S)$ for some natural number $m$
 +
(Hilbert's Nullstellensatz; cf.
 +
[[Hilbert theorem|Hilbert theorem]] 3)). Two affine algebraic sets $X$
 +
and $Y$ coincide if and only if ${\mathfrak A}_X = {\mathfrak A}_Y$. The affine algebraic set $X$ can
 +
be defined by a system of generators of ${\mathfrak A}_X$. In particular, any affine
 +
algebraic set can be defined by a finite number of polynomials
 +
$f_1,\dots,f_k\in k[T]$. The equalities $f_1 = \dots = f_k = 0$ are called the equations of $X$. The affine
 +
algebraic sets of ${\bar k}^n$ form a lattice with respect to the operations of
 +
intersection and union. The ideal of the intersection $X\cap Y$ is identical
 +
with the sum of their ideals ${\mathfrak A}_X + {\mathfrak A}_Y$, while the ideal of the union $X\cup Y$ is
 +
identical with the intersection of their ideals ${\mathfrak A}_X \cap {\mathfrak A}_Y$. Any set ${\bar k}^n$ is an
 +
affine algebraic set, called an affine space over $k$ and denoted by
 +
$A_k^n$; to it corresponds the zero ideal. The empty subset of ${\bar k}^n$ is also
 +
an affine algebraic set with the unit ideal. The quotient ring $k[X]=k[T]/{\mathfrak A}_X$ is
 +
called the coordinate ring of $X$. It is identical with the ring of
 +
$k$-regular functions on $X$, i.e. with the ring of $k$-valued
 +
functions, $f:X \to {\bar k}$, for which there exists a polynomial $F\in k[T]$ such that $f(x)=F(x)$
 +
for all $x\in X$. An affine algebraic set is said to be irreducible if it
 +
is not the union of two affine algebraic proper subsets. An equivalent
 +
definition is that the ideal ${\mathfrak A}_X$ is prime. Irreducible affine
 +
algebraic sets together with projective algebraic sets were the
 +
subjects of classical algebraic geometry. They were called,
 +
respectively, affine algebraic varieties and projective algebraic
 +
varieties over the field $k$ (or $k$-varieties). Affine algebraic sets
 +
have the structure of a topological space. The affine algebraic
 +
subsets are the closed sets of this topology (the
 +
[[Zariski topology|Zariski topology]]). An affine algebraic set is
 +
irreducible if and only if it is irreducible as a topological
 +
space. Further development of the concept of an affine algebraic set
 +
leads to the concepts of an
 +
[[Affine variety|affine variety]] and an
 +
[[Affine scheme|affine scheme]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> O. Zariski,   P. Samuel,   "Commutative algebra" , '''2''' , Springer (1975)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.R. Shafarevich,   "Basic algebraic geometry" , Springer (1977) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> R. Hartshorne,   "Algebraic geometry" , Springer (1977)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD
 +
valign="top"> O. Zariski, P. Samuel, "Commutative algebra" , '''2''' ,
 +
Springer (1975)</TD></TR><TR><TD valign="top">[2]</TD> <TD
 +
valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer
 +
(1977) (Translated from Russian)</TD></TR><TR><TD
 +
valign="top">[3]</TD> <TD valign="top"> R. Hartshorne, "Algebraic
 +
geometry" , Springer (1977)</TD></TR></table>
  
  
  
 
====Comments====
 
====Comments====
A topological space is irreducible if it is not the union of two closed proper subspaces.
+
A topological space is irreducible if it is not the
 +
union of two closed proper subspaces.

Revision as of 08:35, 12 September 2011

affine algebraic $k$-set

The set of solutions of a given system of algebraic equations. Let $k$ be a field and let $\bar k$ be its algebraic closure. A subset $X$ of the Cartesian product ${\bar k}^n$ is said to be an affine algebraic $k$-set if its points are the common zeros of some family $S$ of the ring of polynomials $k[T]=k[T_1,\dots,T_n]$. The set ${\mathfrak A}_X$ of all polynomials in $k[T_1,\dots,T_n]$ that vanish on $X$ forms an ideal, the so-called ideal of the affine algebraic $k$-set. The ideal ${\mathfrak A}_X$ coincides with the radical of the ideal $I(S)$ generated by the family $S$, i.e. with the set of polynomials $f\in k[T_1,\dots,T_n]$ such that $f^m \in I(S)$ for some natural number $m$ (Hilbert's Nullstellensatz; cf. Hilbert theorem 3)). Two affine algebraic sets $X$ and $Y$ coincide if and only if ${\mathfrak A}_X = {\mathfrak A}_Y$. The affine algebraic set $X$ can be defined by a system of generators of ${\mathfrak A}_X$. In particular, any affine algebraic set can be defined by a finite number of polynomials $f_1,\dots,f_k\in k[T]$. The equalities $f_1 = \dots = f_k = 0$ are called the equations of $X$. The affine algebraic sets of ${\bar k}^n$ form a lattice with respect to the operations of intersection and union. The ideal of the intersection $X\cap Y$ is identical with the sum of their ideals ${\mathfrak A}_X + {\mathfrak A}_Y$, while the ideal of the union $X\cup Y$ is identical with the intersection of their ideals ${\mathfrak A}_X \cap {\mathfrak A}_Y$. Any set ${\bar k}^n$ is an affine algebraic set, called an affine space over $k$ and denoted by $A_k^n$; to it corresponds the zero ideal. The empty subset of ${\bar k}^n$ is also an affine algebraic set with the unit ideal. The quotient ring $k[X]=k[T]/{\mathfrak A}_X$ is called the coordinate ring of $X$. It is identical with the ring of $k$-regular functions on $X$, i.e. with the ring of $k$-valued functions, $f:X \to {\bar k}$, for which there exists a polynomial $F\in k[T]$ such that $f(x)=F(x)$ for all $x\in X$. An affine algebraic set is said to be irreducible if it is not the union of two affine algebraic proper subsets. An equivalent definition is that the ideal ${\mathfrak A}_X$ is prime. Irreducible affine algebraic sets together with projective algebraic sets were the subjects of classical algebraic geometry. They were called, respectively, affine algebraic varieties and projective algebraic varieties over the field $k$ (or $k$-varieties). Affine algebraic sets have the structure of a topological space. The affine algebraic subsets are the closed sets of this topology (the Zariski topology). An affine algebraic set is irreducible if and only if it is irreducible as a topological space. Further development of the concept of an affine algebraic set leads to the concepts of an affine variety and an affine scheme.

References

[1] O. Zariski, P. Samuel, "Commutative algebra" , 2 , Springer (1975)
[2] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian)
[3] R. Hartshorne, "Algebraic geometry" , Springer (1977)


Comments

A topological space is irreducible if it is not the union of two closed proper subspaces.

How to Cite This Entry:
Affine algebraic set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Affine_algebraic_set&oldid=13073
This article was adapted from an original article by I.V. DolgachevV.A. Iskovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article