Specialization of a point

of a topological space

A point for which the inclusion holds (this is equivalent to the inclusion ). A point is called generic if any point of is a specialization of it, that is, if . The other extreme case is that of a closed point: A point which has a unique specialization, namely the point itself.

For the affine scheme of a ring , a point is a specialization of a point if for the corresponding prime ideals of the inclusion holds. When is a ring without zero divisors, the point is the generic one. The relation of specialization distributes into levels: the highest are the closed points, on the next level are the points whose specializations are closed, and on the -th level are the points whose specializations belong to the levels with labels . For example, for there are levels: closed points, generic points of curves, generic points of surfaces the generic point of the -dimensional affine space.

References

[1]	Yu.I. Manin, "Lectures on algebraic geometry" , 1 , Moscow (1970) (In Russian) MR0284434 Zbl 0204.21302
[2]	A. Grothendieck, J. Dieudonné, "Eléments de géometrie algébrique" , I. Le langage des schémes , Springer (1971) MR0217085 {ZBL|0203.23301}}

Comments

Of course, denotes the closure of the set . The closure of a point is an irreducible subset of , and conversely, every irreducible subset of has a generic point.

References

[a1]	R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 MR0463157 Zbl 0367.14001