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

gollark: But it's more practical (from a real life perspective) to put it in the middle, away from micrometeoroid strikes or whatever.

gollark: You don't really need to look outside to pilot the ship.

gollark: Come to think of it, why put the bridge at the front at all?

gollark: Yes.

gollark: You don't actually need the fluxducts under all of them. They share power with adjacent ones.

References

Hartshorne, Ch II, Exercise 3.15. (a)
Hartshorne, Ch II, Exercise 3.15. (b)

Sources

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

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[1] Hartshorne, Ch II, Exercise 3.15. (a)

[2] Hartshorne, Ch II, Exercise 3.15. (b)