Unramified morphism

In algebraic geometry, an unramified morphism is a morphism ${\displaystyle f:X\to Y}$ of schemes such that (a) it is locally of finite presentation and (b) for each ${\displaystyle x\in X}$ and ${\displaystyle y=f(x)}$, we have that

1. The residue field ${\displaystyle k(x)}$ is a separable algebraic extension of ${\displaystyle k(y)}$.
2. ${\displaystyle f^{\#}({\mathfrak {m}}_{y}){\mathcal {O}}_{x,X}={\mathfrak {m}}_{x},}$ where ${\displaystyle f^{\#}:{\mathcal {O}}_{y,Y}\to {\mathcal {O}}_{x,X}}$ and ${\displaystyle {\mathfrak {m}}_{y},{\mathfrak {m}}_{x}}$ are maximal ideals of the local rings.

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

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

Simple example

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

Curve case

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

Characterization

Given a morphism ${\displaystyle f:X\to Y}$ that is locally of finite presentation, the following are equivalent:[2]

1. f is unramified.
2. The diagonal map ${\displaystyle \delta _{f}:X\to X\times _{Y}X}$ is an open immersion.
3. The relative cotangent sheaf ${\displaystyle \Omega _{X/Y}}$ is zero.

See also

• Finite extensions of local fields
• Ramification (mathematics)

References

1. Hartshorne, Ch. IV, § 2. harvnb error: no target: CITEREFHartshorne (help)
2. EGA IV, Corollary 17.4.2. harvnb error: no target: CITEREFEGA_IV (help)

• 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, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157