Unramified morphism

In algebraic geometry, an unramified morphism is a morphism $$f: X \to Y$$ of schemes such that (a) it is locally of finite presentation and (b) for each $$x \in X$$ and $$y = f(x)$$, we have that
 * 1) The residue field $$k(x)$$ is a separable algebraic extension of $$k(y)$$.
 * 2) $$f^{\#}(\mathfrak{m}_y) \mathcal{O}_{x, X} = \mathfrak{m}_x, $$ where $$f^{\#}: \mathcal{O}_{y, Y} \to \mathcal{O}_{x, X}$$ and $$\mathfrak{m}_y, \mathfrak{m}_x$$ are maximal ideals of the local rings.

A flat unramified morphism is called an étale morphism. Less strongly, if $$f$$ satisfies the conditions when restricted to sufficiently small neighborhoods of $$x$$ and $$y$$, then $$f$$ is said to be unramified near $$x$$.

Some authors prefer to use weaker conditions, in which case they call a morphism satisfying the above a G-unramified morphism.

Simple example
Let $$A$$ be a ring and B the ring obtained by adjoining an integral element to A; i.e., $$B = A[t]/(F)$$ for some monic polynomial F. Then $$\operatorname{Spec}(B) \to \operatorname{Spec}(A)$$ is unramified if and only if the polynomial F is separable (i.e., it and its derivative generate the unit ideal of $$A[t]$$).

Curve case
Let $$f: X \to Y$$ be a finite morphism between smooth connected curves over an algebraically closed field, P a closed point of X and $$Q = f(P)$$. We then have the local ring homomorphism $$f^{\#} : \mathcal{O}_Q \to \mathcal{O}_P$$ where $$(\mathcal{O}_Q, \mathfrak{m}_Q)$$ and $$(\mathcal{O}_P, \mathfrak{m}_P)$$ are the local rings at Q and P of Y and X. Since $$\mathcal{O}_P$$ is a discrete valuation ring, there is a unique integer $$e_P > 0$$ such that $$f^{\#} (\mathfrak{m}_Q) \mathcal{O}_P = {\mathfrak{m}_P}^{e_P}$$. The integer $$e_P$$ is called the ramification index of $$P$$ over $$Q$$. Since $$k(P) = k(Q)$$ as the base field is algebraically closed, $$f$$ is unramified at $$P$$ (in fact, étale) if and only if $$e_P = 1$$. Otherwise, $$f$$ is said to be ramified at P and Q is called a branch point.

Characterization
Given a morphism $$f: X \to Y$$ that is locally of finite presentation, the following are equivalent:
 * 1) f is unramified.
 * 2) The diagonal map $$\delta_f: X \to X \times_Y X$$ is an open immersion.
 * 3) The relative cotangent sheaf $$\Omega_{X/Y}$$ is zero.