Projective algebraic set
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:
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:
In the case where is a homogeneous prime ideal, the projective algebraic set is called a projective variety.
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|[a2]||R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 MR0463157 Zbl 0367.14001|
Projective algebraic set. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Projective_algebraic_set&oldid=23932