Unramified morphism

In algebraic geometry, an unramified morphism is a morphism of schemes such that (a) it is locally of finite presentation and (b) for each and , we have that

  1. The residue field is a separable algebraic extension of .
  2. where and are maximal ideals of the local rings.

A flat unramified morphism is called an étale morphism. Less strongly, if satisfies the conditions when restricted to sufficiently small neighborhoods of and , then is said to be unramified near .

Some authors prefer to use weaker conditions, in which case they call a morphism satisfying the above a G-unramified morphism.

Simple example

Let be a ring and B the ring obtained by adjoining an integral element to A; i.e., for some monic polynomial F. Then is unramified if and only if the polynomial F is separable (i.e., it and its derivative generate the unit ideal of ).

Characterization

Given a morphism that is locally of finite presentation, the following are equivalent:[1]

  1. f is unramified.
  2. The diagonal map is an open immersion.
  3. The relative cotangent sheaf is zero.

See also

References

  1. EGA IV, Corollary 17.4.2.
  • Grothendieck, Alexandre; Dieudonné, Jean (1967). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie". Publications Mathématiques de l'IHÉS. 32. doi:10.1007/bf02732123. MR 0238860.
  • 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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