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 |
Specialization of a point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Specialization_of_a_point&oldid=23981