Namespaces
Variants
Actions

Difference between revisions of "Specialization of a point"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (links)
(better)
Line 1: Line 1:
 
''$x$ in a topological space $X$''
 
''$x$ in a topological space $X$''
  
A point $y \in X$ for which the inclusion $y \in \overline{\{x\}}$ holds (this is equivalent to the inclusion $\overline{\{y\}} \subseteq \overline{\{x\}}$). A point $x$ is called ''generic'' if any point of $X$ is a specialization of it, that is, if $\overline{\{x\}} = X$. The other extreme case is that of a closed point: a point which has a unique specialization, namely the point itself.
+
A point $y \in X$ for which the inclusion $y \in \overline{\{x\}}$ holds; equivalently the inclusion $\overline{\{y\}} \subseteq \overline{\{x\}}$; every [[neighbourhood]] of $x$ is a [[neighbourhood]] of $y$.
 +
 
 +
A point $x$ is called ''generic'' if any point of $X$ is a specialization of it, that is, if $\overline{\{x\}} = X$. 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]] $\mathrm{Spec}(A)$ of a ring $A$, a point $y$ is a specialization of a point $x$ if for the corresponding prime ideals of $A$ the inclusion $\mathfrak{p}_x \subseteq \mathfrak{p}_y$ holds. When $A$ is a ring without zero divisors, the point $\{0\}$ 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 $i$-th level are the points whose specializations belong to the levels with labels $\le i-1$. For example, for $\mathrm{Spec}(\mathbf{C}[T_1,\ldots,T_n]$ there are $n+1$ levels: closed points, generic points of curves, generic points of surfaces,$\ldots$, the generic point of the $n$-dimensional [[affine space]].
 
For the [[affine scheme]] $\mathrm{Spec}(A)$ of a ring $A$, a point $y$ is a specialization of a point $x$ if for the corresponding prime ideals of $A$ the inclusion $\mathfrak{p}_x \subseteq \mathfrak{p}_y$ holds. When $A$ is a ring without zero divisors, the point $\{0\}$ 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 $i$-th level are the points whose specializations belong to the levels with labels $\le i-1$. For example, for $\mathrm{Spec}(\mathbf{C}[T_1,\ldots,T_n]$ there are $n+1$ levels: closed points, generic points of curves, generic points of surfaces,$\ldots$, the generic point of the $n$-dimensional [[affine space]].
Line 8: Line 10:
 
<table>
 
<table>
 
<TR><TD valign="top">[1]</TD> <TD valign="top"> Yu.I. Manin, "Lectures on algebraic geometry" , '''1''' , Moscow (1970) (In Russian) {{MR|0284434}} {{ZBL|0204.21302}} </TD></TR>
 
<TR><TD valign="top">[1]</TD> <TD valign="top"> Yu.I. Manin, "Lectures on algebraic geometry" , '''1''' , Moscow (1970) (In Russian) {{MR|0284434}} {{ZBL|0204.21302}} </TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top"> A. Grothendieck, J. Dieudonné, "Eléments de géometrie algébrique" , '''I. Le langage des schémes''' , Springer (1971) {{MR|0217085}} {ZBL|0203.23301}} </TD></TR>
+
<TR><TD valign="top">[2]</TD> <TD valign="top"> A. Grothendieck, J. Dieudonné, "Eléments de géometrie algébrique" , '''I. Le langage des schémes''' , Springer (1971) {{MR|0217085}} {{ZBL|0203.23301}} </TD></TR>
 
</table>
 
</table>
  

Revision as of 17:12, 1 January 2016

$x$ in a topological space $X$

A point $y \in X$ for which the inclusion $y \in \overline{\{x\}}$ holds; equivalently the inclusion $\overline{\{y\}} \subseteq \overline{\{x\}}$; every neighbourhood of $x$ is a neighbourhood of $y$.

A point $x$ is called generic if any point of $X$ is a specialization of it, that is, if $\overline{\{x\}} = X$. 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 $\mathrm{Spec}(A)$ of a ring $A$, a point $y$ is a specialization of a point $x$ if for the corresponding prime ideals of $A$ the inclusion $\mathfrak{p}_x \subseteq \mathfrak{p}_y$ holds. When $A$ is a ring without zero divisors, the point $\{0\}$ 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 $i$-th level are the points whose specializations belong to the levels with labels $\le i-1$. For example, for $\mathrm{Spec}(\mathbf{C}[T_1,\ldots,T_n]$ there are $n+1$ levels: closed points, generic points of curves, generic points of surfaces,$\ldots$, the generic point of the $n$-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, $\overline{\{x\}}$ denotes the closure of the singleton set $\{x\}$. The closure of a point is an irreducible subset of $X$, and conversely, every irreducible subset of $X$ has a generic point.

References

[a1] R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 MR0463157 Zbl 0367.14001
How to Cite This Entry:
Specialization of a point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Specialization_of_a_point&oldid=37222
This article was adapted from an original article by V.V. Shokurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article