Geometrically (algebraic geometry)

Given a scheme X that is of finite type over a field k, the following are equivalent: • 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. == References ==