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
# The residue field $k(x)$ is a separable algebraic extension of $k(y)$.
# $f^(\mathfrak_y) \mathcal_ = \mathfrak_x,$ where $f^: \mathcal_ \to \mathcal_$ and $\mathfrak_y, \mathfrak_x$ 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 (F) for some monic polynomial ''F''. Then $\operatorname(B) \to \operatorname(A)$ is unramified if and only if the polynomial ''F'' is separable (i.e., it and its derivative generate the unit ideal of A /math>).

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 $f^ : \mathcal_Q \to \mathcal_P$ where $(\mathcal_Q, \mathfrak_Q)$ and $(\mathcal_P, \mathfrak_P)$ are the local rings at ''Q'' and ''P'' of ''Y'' and ''X''. Since $\mathcal_P$ is a discrete valuation ring, there is a unique integer $e_P > 0$ such that $f^ (\mathfrak_Q) \mathcal_P = ^$. 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:
# ''f'' is unramified.
# The diagonal map $\delta_f: X \to X \times_Y X$ is an open immersion.
# The relative cotangent sheaf $\Omega_$ is zero.

See also

* Finite extensions of local fields
*Ramification (mathematics)

References

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Algebraic geometry
Morphisms

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