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 . 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: f is unramified. The diagonal map is an open immersion. The relative cotangent sheaf is zero. See also Finite extensions of local fields Ramification (mathematics) References Algebraic geometry Morphisms