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
×
k
k
¯
=
X
×
Spec
k
Spec
k
¯
{\displaystyle X\times _{k}{\overline {k}}=X\times _{\operatorname {Spec} k}{\operatorname {Spec} {\overline {k}}}}
is irreducible, where k ¯ {\displaystyle {\overline {k}}}
denotes an algebraic closure of k.
-
X
×
k
k
s
{\displaystyle X\times _{k}k_{s}}
is irreducible for a separable closure k s {\displaystyle k_{s}}
of k.
-
X
×
k
F
{\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
- Hartshorne 1977, Ch II, Exercise 3.15. (a)
- 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