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.
Given a scheme X that is of finite type over a field k, the following are equivalent: [1]
The same statement also holds if "irreducible" is replaced with " reduced" and the separable closure is replaced by the perfect closure. [2]
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.
Given a scheme X that is of finite type over a field k, the following are equivalent: [1]
The same statement also holds if "irreducible" is replaced with " reduced" and the separable closure is replaced by the perfect closure. [2]