What is the étale fundamental group of projective space over an algebraically closed field?
In char = 0 it is trivial (Lefschetz principle), as well as in dimension 1 (Riemann-Hurwitz).
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1$\begingroup$ That's actually an interesting point even about P^1, given that A^1 is extremely not simply-connected. $\endgroup$– Ben Webster ♦Commented Jul 4, 2010 at 10:56
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1 Answer
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It is a birational invariant (for smooth proper connected schemes over a field, ultimately due to Zariski-Nagata purity of the branch locus), and its formation is compatible with products (for proper connected schemes over an algebraically closed field), so we can replace projective $n$-space with the $n$-fold product of copies of the projective line to conclude. Likewise, due to limit arguments and invariance of the etale site with respect to finite radiciel surjections (such as a finite purely inseparable extension of a ground field), it suffices to take the ground field to be separably closed rather than algebraically closed. This is all in SGA1.
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$\begingroup$ Can we also argue inductively using the diagram Unknown control sequence $\mathbf{P}^n \leftarrow \mathrm{Bl}_x{\mathbf{P}^n} \to \mathbf{P}^{n-1}$? $\endgroup$– user6960Commented Jul 4, 2010 at 12:07