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    Étale covers, trivial covers, and simple connectedness of $\PP^1$

    D-IV2ETCOV

    (Étale cover)

    A morphism \(f : X \to Y\) is an étale cover if it is finite and étale. It is a trivial cover if \(X \cong \coprod_{i \in I} Y\) for a finite index set \(I\), with \(f\) the identity on each copy. \(Y\) is simply connected if every étale cover of \(Y\) is trivial, equivalently \(\pi_1^{\Et}(Y) = 0\).

    \(\PP^1_k\) over \(k = \kbar\) is simply connected, in every characteristic.

    Let \(f : X \to \PP^1\) be an étale cover; it is enough to treat \(X\) connected, and then to show \(f\) is an isomorphism. \(X\) is regular because \(f\) is étale and \(\PP^1\) is smooth over \(k\), and a connected regular scheme is irreducible, so \(X\) is a curve; \(f\) is finite, so \(X\) is projective; \(f\) is unramified, so \(k(X)/k(\PP^1)\) is separable. Riemann–Hurwitz therefore applies, and \(R = 0\) since \(f\) is unramified, so with \(n = \deg f\), \[ 2 g_X - 2 = n(2 \cdot 0 - 2) = -2n . \] Then \(g_X \geq 0\) gives \(-2n \geq -2\), so \(n = 1\) and \(g_X = 0\), and a finite morphism of degree \(1\) between curves is an isomorphism.

    (Topological analogue)

    For a connected covering space \(S\to S^2\) of degree \(n\), \(\chi(S)=n\chi(S^2)=2n\), and \(\chi(S)\le2\) for a compact connected surface, so \(n=1\); the Riemann–Hurwitz proof for \(\PP^1\) replaces \(\chi\) by \(2-2g\) and the covering space by a finite étale morphism.

    (The affine line in characteristic $p$)

    Let \(\operatorname{char}k=p>0\). The Artin–Schreier cover \(\Spec k[x,y]/(y^p - y - x)\to\AA^1\), \((x,y)\mapsto x\), is finite étale of degree \(p\) and connected, because \(d(y^p - y) = -dy\) never vanishes; so \(\AA^1_k\) is not simply connected. The cover extends to a morphism \(\PP^1\to\PP^1\) that is totally and wildly ramified at \(\infty\) (D-IV2RAM). Over \(\CC\), \(\AA^1\) has no nontrivial finite étale covers.

    (Finite étale covers)

    Étale morphisms are defined in D-MORETALE. For a proper nonempty open subset \(U\subseteq Y\), the morphism \(U\sqcup Y\to Y\) is étale and surjective but not finite, and it is not of the form \(\coprod_{i\in I}Y\to Y\).

    What is an étale cover, and what is a trivial one?
    What does it mean for a curve to be simply connected?
    Prove that $\PP^1$ has no nontrivial étale covers.
    Is $\AA^1$ simply connected?

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