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

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