Unramified morphism

In algebraic geometry, an unramified morphism is a morphism formula_1 of schemes such that (a) it is locally of finite presentation and (b) for each formula_2 and formula_3, we have that
A flat unramified morphism is called an étale morphism. Less strongly, if formula_9 satisfies the conditions when restricted to sufficiently small neighborhoods of formula_10 and formula_11, then formula_9 is said to be unramified near formula_10.
Some authors prefer to use weaker conditions, in which case they call a morphism satisfying the above a G-unramified morphism.
Simple example.
Let formula_14 be a ring and "B" the ring obtained by adjoining an integral element to "A"; i.e., formula_15 for some monic polynomial "F". Then formula_16 is unramified if and only if the polynomial "F" is separable (i.e., it and its derivative generate the unit ideal of formula_17).
Curve case.
Let formula_1 be a finite morphism between smooth connected curves over an algebraically closed field, "P" a closed point of "X" and formula_19. We then have the local ring homomorphism formula_20 where formula_21 and formula_22 are the local rings at "Q" and "P" of "Y" and "X". Since formula_23 is a discrete valuation ring, there is a unique integer formula_24 such that formula_25. The integer formula_26 is called the ramification index of formula_27 over formula_28. Since formula_29 as the base field is algebraically closed, formula_9 is unramified at formula_27 (in fact, étale) if and only if formula_32. Otherwise, formula_9 is said to be ramified at "P" and "Q" is called a branch point.
Characterization.
Given a morphism formula_1 that is locally of finite presentation, the following are equivalent: