Unramified morphism


In algebraic geometry, an unramified morphism is a morphism of schemes such that it is locally of finite presentation and 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.

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