Difference between revisions of "Projective algebraic set"
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− | + | A subset of points of a projective space $ P ^ {n} $ | |
+ | defined over a field $ k $ | ||
+ | that has (in homogeneous coordinates) the form | ||
+ | |||
+ | $$ | ||
+ | V ( I) = \{ {( a _ {0} \dots a _ {n} ) \in P ^ {n} } : {f | ||
+ | ( a _ {0} \dots a _ {n} ) = 0 \textrm{ for any } f \in I } \} | ||
+ | . | ||
+ | $$ | ||
+ | |||
+ | Here $ I $ | ||
+ | is a homogeneous ideal in the polynomial ring $ k [ X _ {0} \dots X _ {n} ] $. | ||
+ | (An ideal $ I $ | ||
+ | is homogeneous if $ f \in I $ | ||
+ | and $ f = \sum f _ {i} $, | ||
+ | where the $ f _ {i} $ | ||
+ | are homogeneous polynomials of degree $ i $, | ||
+ | imply that $ f _ {i} \in I $.) | ||
Projective algebraic sets possess the following properties: | Projective algebraic sets possess the following properties: | ||
− | 1) | + | 1) $ V ( \sum _ {i \in S } I _ {i} ) = \cap _ {i \in S } V ( I _ {i} ) $; |
− | 2) | + | 2) $ V ( I _ {1} \cap I _ {2} ) = V ( I _ {1} ) \cup V ( I _ {2} ) $; |
− | 3) if | + | 3) if $ I _ {1} \subset I _ {2} $, |
+ | then $ V ( I _ {2} ) \subset V ( I _ {1} ) $; | ||
− | 4) | + | 4) $ V ( I) = V ( \sqrt I ) $, |
+ | where $ \sqrt I $ | ||
+ | is the radical of the ideal $ I $( | ||
+ | cf. [[Radical of an ideal|Radical of an ideal]]). | ||
− | It follows from properties 1)–3) that on | + | It follows from properties 1)–3) that on $ V ( I) $ |
+ | the [[Zariski topology|Zariski topology]] can be introduced. If $ I = \sqrt I $, | ||
+ | then $ I $ | ||
+ | can be uniquely represented as the intersection of homogeneous prime ideals: | ||
− | + | $$ | |
+ | I = \mathfrak B _ {1} \cap \dots \cap \mathfrak B _ {s} $$ | ||
and | and | ||
− | + | $$ | |
+ | V ( I) = V ( \mathfrak B _ {1} ) \cup \dots \cup V ( \mathfrak B _ {s} ) . | ||
+ | $$ | ||
− | In the case where | + | In the case where $ I $ |
+ | is a homogeneous prime ideal, the projective algebraic set $ V ( I) $ | ||
+ | is called a projective variety. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> O. Zariski, P. Samuel, "Commutative algebra" , '''1''' , Springer (1975) {{MR|0389876}} {{MR|0384768}} {{ZBL|0313.13001}} </TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> O. Zariski, P. Samuel, "Commutative algebra" , '''1''' , Springer (1975) {{MR|0389876}} {{MR|0384768}} {{ZBL|0313.13001}} </TD></TR></table> | ||
− | |||
− | |||
====Comments==== | ====Comments==== | ||
− | |||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D. Mumford, "Algebraic geometry" , '''1. Complex projective varieties''' , Springer (1976) {{MR|0453732}} {{ZBL|0356.14002}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D. Mumford, "Algebraic geometry" , '''1. Complex projective varieties''' , Springer (1976) {{MR|0453732}} {{ZBL|0356.14002}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table> |
Revision as of 08:08, 6 June 2020
A subset of points of a projective space $ P ^ {n} $
defined over a field $ k $
that has (in homogeneous coordinates) the form
$$ V ( I) = \{ {( a _ {0} \dots a _ {n} ) \in P ^ {n} } : {f ( a _ {0} \dots a _ {n} ) = 0 \textrm{ for any } f \in I } \} . $$
Here $ I $ is a homogeneous ideal in the polynomial ring $ k [ X _ {0} \dots X _ {n} ] $. (An ideal $ I $ is homogeneous if $ f \in I $ and $ f = \sum f _ {i} $, where the $ f _ {i} $ are homogeneous polynomials of degree $ i $, imply that $ f _ {i} \in I $.)
Projective algebraic sets possess the following properties:
1) $ V ( \sum _ {i \in S } I _ {i} ) = \cap _ {i \in S } V ( I _ {i} ) $;
2) $ V ( I _ {1} \cap I _ {2} ) = V ( I _ {1} ) \cup V ( I _ {2} ) $;
3) if $ I _ {1} \subset I _ {2} $, then $ V ( I _ {2} ) \subset V ( I _ {1} ) $;
4) $ V ( I) = V ( \sqrt I ) $, where $ \sqrt I $ is the radical of the ideal $ I $( cf. Radical of an ideal).
It follows from properties 1)–3) that on $ V ( I) $ the Zariski topology can be introduced. If $ I = \sqrt I $, then $ I $ can be uniquely represented as the intersection of homogeneous prime ideals:
$$ I = \mathfrak B _ {1} \cap \dots \cap \mathfrak B _ {s} $$
and
$$ V ( I) = V ( \mathfrak B _ {1} ) \cup \dots \cup V ( \mathfrak B _ {s} ) . $$
In the case where $ I $ is a homogeneous prime ideal, the projective algebraic set $ V ( I) $ is called a projective variety.
References
[1] | I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001 |
[2] | O. Zariski, P. Samuel, "Commutative algebra" , 1 , Springer (1975) MR0389876 MR0384768 Zbl 0313.13001 |
Comments
References
[a1] | D. Mumford, "Algebraic geometry" , 1. Complex projective varieties , Springer (1976) MR0453732 Zbl 0356.14002 |
[a2] | R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 MR0463157 Zbl 0367.14001 |
Projective algebraic set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Projective_algebraic_set&oldid=23932