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., ${\displaystyle X\times _{k}{\overline {k}}:=X\times _{\operatorname {Spec} k}{\operatorname {Spec} {\overline {k}}}}$ is irreducible, where ${\displaystyle {\overline {k}}}$ denotes an algebraic closure of k.
• ${\displaystyle X\times _{k}k_{s}}$ is irreducible for a separable closure ${\displaystyle k_{s}}$ of k.
• ${\displaystyle X\times _{k}F}$ 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

1. Hartshorne 1977, Ch II, Exercise 3.15. (a)
2. Hartshorne 1977, Ch II, Exercise 3.15. (b)

Sources

• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157