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×kk‾:=X×Spec⁡kSpec⁡k‾ is irreducible, where k‾ denotes an algebraic closure of k.
* X×kks is irreducible for a separable closure ks of k.
* X×kF 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

1. ↑ Hartshorne 1977, Ch II, Exercise 3.15. (a)
2. ↑ Hartshorne 1977, 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

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