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
- X is geometrically irreducible; i.e., is irreducible, where denotes an algebraic closure of k.
- is irreducible for a separable closure of k.
- 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.