Geometrically (algebraic geometry)

In algebraic geometry, especially in scheme theory, a property is said to hold geometrically over a field if it also holds over the algebraic closure of the field. In other words, a property holds geometrically if it holds after a base change to a geometric point. For example, a smooth variety is a variety that is geometrically regular.

Geometrically irreducible and geometrically reduced

Given a scheme X that is of finite type over a field k, the following are equivalent:^[1]

• X is geometrically irreducible; i.e., [math]\displaystyle{ X \times_k \overline{k} := X \times_{\operatorname{Spec} k} {\operatorname{Spec} \overline{k}} }[/math] is irreducible, where [math]\displaystyle{ \overline{k} }[/math] denotes an algebraic closure of k.
• [math]\displaystyle{ X \times_k k_s }[/math] is irreducible for a separable closure [math]\displaystyle{ k_s }[/math] of k.
• [math]\displaystyle{ X \times_k F }[/math] is irreducible for each field extension F of k.

The same statement also holds if "irreducible" is replaced with "reduced" and the separable closure is replaced by the perfect closure.^[2]

References

Sources

• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9 

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