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Difference between revisions of "Specialization of a point"

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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> Yu.I. Manin,   "Lectures on algebraic geometry" , '''1''' , Moscow (1970) (In Russian)</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)</TD></TR></table>
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<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">[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>
  
  
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<table><TR><TD valign="top">[a1]</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"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>

Revision as of 21:56, 30 March 2012

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
How to Cite This Entry:
Specialization of a point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Specialization_of_a_point&oldid=23981
This article was adapted from an original article by V.V. Shokurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article