Unramified morphism
In algebraic geometry, an unramified morphism is a morphism [math]\displaystyle{ f: X \to Y }[/math] of schemes such that (a) it is locally of finite presentation and (b) for each [math]\displaystyle{ x \in X }[/math] and [math]\displaystyle{ y = f(x) }[/math], we have that
- The residue field [math]\displaystyle{ k(x) }[/math] is a separable algebraic extension of [math]\displaystyle{ k(y) }[/math].
- [math]\displaystyle{ f^{\#}(\mathfrak{m}_y) \mathcal{O}_{x, X} = \mathfrak{m}_x, }[/math] where [math]\displaystyle{ f^{\#}: \mathcal{O}_{y, Y} \to \mathcal{O}_{x, X} }[/math] and [math]\displaystyle{ \mathfrak{m}_y, \mathfrak{m}_x }[/math] are maximal ideals of the local rings.
A flat unramified morphism is called an étale morphism. Less strongly, if [math]\displaystyle{ f }[/math] satisfies the conditions when restricted to sufficiently small neighborhoods of [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math], then [math]\displaystyle{ f }[/math] is said to be unramified near [math]\displaystyle{ x }[/math].
Some authors prefer to use weaker conditions, in which case they call a morphism satisfying the above a G-unramified morphism.
Simple example
Let [math]\displaystyle{ A }[/math] be a ring and B the ring obtained by adjoining an integral element to A; i.e., [math]\displaystyle{ B = A[t]/(F) }[/math] for some monic polynomial F. Then [math]\displaystyle{ \operatorname{Spec}(B) \to \operatorname{Spec}(A) }[/math] is unramified if and only if the polynomial F is separable (i.e., it and its derivative generate the unit ideal of [math]\displaystyle{ A[t] }[/math]).
Curve case
Let [math]\displaystyle{ f: X \to Y }[/math] be a finite morphism between smooth connected curves over an algebraically closed field, P a closed point of X and [math]\displaystyle{ Q = f(P) }[/math]. We then have the local ring homomorphism [math]\displaystyle{ f^{\#} : \mathcal{O}_Q \to \mathcal{O}_P }[/math] where [math]\displaystyle{ (\mathcal{O}_Q, \mathfrak{m}_Q) }[/math] and [math]\displaystyle{ (\mathcal{O}_P, \mathfrak{m}_P) }[/math] are the local rings at Q and P of Y and X. Since [math]\displaystyle{ \mathcal{O}_P }[/math] is a discrete valuation ring, there is a unique integer [math]\displaystyle{ e_P \gt 0 }[/math] such that [math]\displaystyle{ f^{\#} (\mathfrak{m}_Q) \mathcal{O}_P = {\mathfrak{m}_P}^{e_P} }[/math]. The integer [math]\displaystyle{ e_P }[/math] is called the ramification index of [math]\displaystyle{ P }[/math] over [math]\displaystyle{ Q }[/math].[1] Since [math]\displaystyle{ k(P) = k(Q) }[/math] as the base field is algebraically closed, [math]\displaystyle{ f }[/math] is unramified at [math]\displaystyle{ P }[/math] (in fact, étale) if and only if [math]\displaystyle{ e_P = 1 }[/math]. Otherwise, [math]\displaystyle{ f }[/math] is said to be ramified at P and Q is called a branch point.
Characterization
Given a morphism [math]\displaystyle{ f: X \to Y }[/math] that is locally of finite presentation, the following are equivalent:[2]
- f is unramified.
- The diagonal map [math]\displaystyle{ \delta_f: X \to X \times_Y X }[/math] is an open immersion.
- The relative cotangent sheaf [math]\displaystyle{ \Omega_{X/Y} }[/math] is zero.
See also
References
- ↑ Hartshorne 1977, Ch. IV, § 2.
- ↑ Grothendieck & Dieudonné 1967, Corollary 17.4.2.
- Grothendieck, Alexandre; Dieudonné, Jean (1967). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie". Publications Mathématiques de l'IHÉS 32. doi:10.1007/bf02732123. http://www.numdam.org/articles/PMIHES_1967__32__5_0.
- Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9
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