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.
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 ).
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 . 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.
Given a morphism that is locally of finite presentation, the following are equivalent:
- 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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