Generic point

The concept of the generic point belongs to the mathematical branch of set- theoretical topology , but is mainly used in algebraic geometry .

definition

A point of a topological space is called generic if it is the closure of the subset . Equivalent to this is the condition that every open subset contains unequal to the empty set. ${\displaystyle \eta }$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle \{\eta \}}$ ${\displaystyle \eta }$

properties

• Spaces that have a generic point are always irreducible .
• If a space fulfills the axiom of separation T ₀ , it has at most one generic point.
• In Hausdorff spaces that contain more than one point, there are no generic points.
• Is a point of any topological space , then the conclusion of in an irreducible subset of , and is a generic point of . ${\displaystyle x}$ ${\displaystyle X}$ ${\displaystyle \{x\}}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle x}$ ${\displaystyle Y}$

Example from algebraic geometry

If an integrity ring is , then the null ideal is the (only) generic point of the spectrum ; the remainder class field of the generic point is the quotient field of . ${\displaystyle A}$ ${\displaystyle \{0\}}$ ${\displaystyle \operatorname {Spec} A}$ ${\displaystyle A}$

Significance for algebraic geometry

If an irreducible schema and its generic point, then statements about open subsets of are often equivalent to the corresponding statements for . For example, if a coherent sheaf is on , it is equivalent to that for all in a suitable open subset of . ${\displaystyle X}$ ${\displaystyle \eta }$ ${\displaystyle X}$ ${\displaystyle \eta }$ ${\displaystyle M}$ ${\displaystyle X}$ ${\displaystyle M_{\eta }=0}$ ${\displaystyle M_{x}=0}$ ${\displaystyle x}$ ${\displaystyle X}$

Related terms

If every irreducible subset in a topological space has a generic point, the space is called sober .

literature

• Ernst Kunz : Introduction to algebraic geometry (= Vieweg study. Vol. 87). Vieweg, Braunschweig et al. 1997, ISBN 3-528-07287-3 , pp. 69-70.