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

- The residue field is a separable algebraic extension of .
- 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 ).

## Curve case

Let be a finite morphism between smooth connected curves over an algebraically closed field, *P* a closed point of *X* and . We then have the local ring homomorphism where and are the local rings at *Q* and *P* of *Y* and *X*. Since is a discrete valuation ring, there is a unique integer such that . The integer is called the **ramification index** of over .^{[1]} Since as the base field is algebraically closed, is unramified at (in fact, étale) if and only if . Otherwise, is said to be ramified at *P* and *Q* is called a branch point.

## Characterization

Given a morphism that is locally of finite presentation, the following are equivalent:^{[2]}

*f*is unramified.- The diagonal map is an open immersion.
- The relative cotangent sheaf is zero.

## See also

## References

**^**Hartshorne, Ch. IV, § 2.**^**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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