# 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., 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.[2]

## References

- Hartshorne, Ch II, Exercise 3.15. (a)
- Hartshorne, 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, MR 0463157

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