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. $f^{\mathrm{\#}}({\mathfrak{𝔪}}_y){{𝒪}}_{x,X}={\mathfrak{𝔪}}_x,$ where $f^{\mathrm{\#}}:{{𝒪}}_{y,Y}\to {{𝒪}}_{x,X}$ 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 $\mathrm{Spec}(B)\to \mathrm{Spec}(A)$ 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 $f^{\mathrm{\#}}:{{𝒪}}_Q\to {{𝒪}}_P$ where $({{𝒪}}_Q,{\mathfrak{𝔪}}_Q)$ and $({{𝒪}}_P,{\mathfrak{𝔪}}_P)$ are the local rings at Q and P of Y and X. Since ${{𝒪}}_P$ is a discrete valuation ring, there is a unique integer $e_P>0$ such that $f^{\mathrm{\#}}({\mathfrak{𝔪}}_Q){{𝒪}}_P={{\mathfrak{𝔪}}_P}^{e_P}$. 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 ${\mathrm{\Omega }}_{X/Y}$ is zero.

See also

• Finite extensions of local fields
• Ramification (mathematics)

References

• 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. MR 0238860.
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157