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Difference between revisions of "Projective algebraic set"

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====References====
 
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.R. Shafarevich,   "Basic algebraic geometry" , Springer (1977) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> O. Zariski,   P. Samuel,   "Commutative algebra" , '''1''' , Springer (1975)</TD></TR></table>
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<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>
  
  
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D. Mumford,   "Algebraic geometry" , '''1. Complex projective varieties''' , Springer (1976)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Hartshorne,   "Algebraic geometry" , Springer (1977) pp. Sect. IV.2</TD></TR></table>
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<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 21:55, 30 March 2012

A subset of points of a projective space defined over a field that has (in homogeneous coordinates) the form

Here is a homogeneous ideal in the polynomial ring . (An ideal is homogeneous if and , where the are homogeneous polynomials of degree , imply that .)

Projective algebraic sets possess the following properties:

1) ;

2) ;

3) if , then ;

4) , where is the radical of the ideal (cf. Radical of an ideal).

It follows from properties 1)–3) that on the Zariski topology can be introduced. If , then can be uniquely represented as the intersection of homogeneous prime ideals:

and

In the case where is a homogeneous prime ideal, the projective algebraic set 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
How to Cite This Entry:
Projective algebraic set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Projective_algebraic_set&oldid=23932
This article was adapted from an original article by Vik.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article