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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.{{harvnb|Hartshorne|1977|loc=Ch. IV, § 2.}} 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:{{harvnb|Grothendieck|Dieudonné|1967|loc=Corollary 17.4.2.}}

  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.

See also

References

{{reflist}}

  • {{EGA|book=4-4}}
  • {{Hartshorne AG}}

Category:Algebraic geometry

Category:Morphisms

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